Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 63, pp. 1–32. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.63 COMPLETE CLASSIFICATION OF SELF-SIMILAR SOLUTIONS FOR SINGULAR POLYTROPIC FILTRATION EQUATIONS YANZHI ZHENG, JINGXUE YIN, SHANMING JI Abstract. This article concerns the complete classification of self-similar solutions to the sin- gular polytropic filtration equation. We establish the existence and uniqueness of self-similar solutions of the form u(x, t) = (βt)−α/βw((βt) − 1 β |x|), and the regularity or singularity at x = 0, with α, β ∈ R and β = p−α(1−mp+m). The asymptotic behaviors of the solutions near 0 or ∞ are also described. Specifically, when β < 0, there always exist blow up solutions or oscillatory solutions. When β > 0, oscillatory solutions appear if α > N , 0 < m < 1 and 1 < p < 2. The main technical issue for the proof is to overcome the difficulty arising from the doubly nonlinear non-Newtonian polytropic filtration diffusion div(|∇um|p−2∇um). 1. Introduction The aim of this article is to study the singular polytropic filtration equation ut − div ( |∇(|u|m−1u)|p−2∇(|u|m−1u) ) = 0, (1.1) in RN with N ≥ 1, m > 0, p > 1, and m(p − 1) < 1. We are concerned with the self-similar solutions for the equation (1.1). The equation (1.1) with m(p− 1) ≤ 1 corresponds to the infinite diffusion property while the case with m(p − 1) > 1 corresponds to the finite propagation speed property. The equation (1.1) has possible singularity or degeneracy for both cases depending on both m and p. Two typical cases of the equation (1.1) with m(p − 1) < 1 are the well-known singular porous medium equation (with m < 1 and p = 2) and the singular p-Laplace equation (with m = 1 and p < 2). There exist blow-up solutions and oscillatory solutions in the singular polytropic filtration equation, which is not presented in the case m(p − 1) > 1. For the positive solutions of the equation (1.1), it can be simplified into ut − div ( |∇um|p−2∇um ) = 0. (1.2) The equation (1.2) has a distinctive of practical applications in physics, chemistry, biology and other fields. Nonlinear diffusion equations, as an important area in the direction of partial differential equations, have received much attention in recent years. Many scholars study the properties of the equation (1.2), including the existence and uniqueness, the regularity and the asymptotic behavior of the solutions. The interior and boundary Hölder continuity for bounded weak solutions for quasi-linear equa- tion ut = div(|u|m−1|∇u|p−2∇u) has been proved by Porzio and Vespri [28]. The existence and asymptotic behavior of the traveling wave solutions for the doubly nonlinear diffusion equation with Fisher-KPP reaction in RN × (0,∞) have been addressed by Audrito and Vázquez [3]. The global stability of sharp traveling waves for the degenerate porous medium equation with Fisher- KPP reaction was proved in [35]. Qualitative theory of non-linear degenerate parabolic equations related to the polytropic filtration equation (1.1) was established in [24, 25, 29]. For the fast 2020 Mathematics Subject Classification. 35K67, 35C06, 35K92, 35B40. Key words and phrases. Polytropic filtration equation; self-similar solutions; singularity; phase plane analysis; asymptotic behavior. ©2025. This work is licensed under a CC BY 4.0 license. Submitted April 29, 2025. Published June 26, 2025. 1 2 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 diffusion case, asymptotic behaviors of the solutions were studied in [2, 11, 14, 18, 19]. Systematic analysis of qualitative theory for nonlinear parabolic equations can be found in [12, 31, 34]. Regarding the polytropic filtration equation (1.1), the extinction and nonnegative solutions for initial data has been shown by Yuan et al[37], which provided the existence of extinction solution in the degenerate case and nonnegative solution in the singular case. Jin and Yin discussed the critical exponents and non-extinction property for a nonlinear boundary value problem of the above equation (1.1) in singular case where p has a lower bound with p ≥ 1 + 1/(m+ 1) [22]. Then Li and Mu extended their results to p > 1 and gave the blow-up set and upper bound of the blow-up rate for the non-global solutions in [26]. From the above articles, it is clear that the smoothness of the solution of the polytropic filtration equation is not as good as that of the solution of the p-Laplace equation and the classification of the solution is more complicated. In addition to the study of the related properties of the diffusion equation, some people have also considered the diffusion equations from the perspectives of source term, convection, etc (see for example [21]). The reason is that after introducing the source term or the convection, their interactions with diffusive phenomena may lead to more diversified asymptotic behaviors of the solutions, such as instantaneous shrinking of the supports of solution, extinction and other phe- nomena. Moreover, a large amount of scholars are interested in diffusion equations with sources or nonlocal sources, and studied the existence of global solutions, critical extinction and blow-up exponents and etc, see for example [16, 32, 38, 39]. Jin et al. considered for the fast diffusive polytropic filtration equation with sources, proving its critical extinction and blow-up exponents in [23]. Propagation profile for a multi-dimensional non-Newtonian polytropic filtration equation with orientated convection was investigated by Ye and Yin in [36]. It was Bidaut-Véron who first studied the complete classification of self-similar solutions to the singular p-Laplace equation in 2006 [5]. Moreover, he also considered the degenerate case in 2009 [6], and the large time behavior of the self-similar solutions for the equation with a source in the same year [7]. Classification of self-similar solutions of the degenerate polytropic filtration equations (with m(p − 1) > 1) was investigated in [27]. Since the self-similar solutions make a great important in describing the regularity and stability of the solutions, the existence and further properties of the self-similar solutions has been well-studied in [1, 8, 9, 13, 20, 33]. Motivated by the previous works, we are looking for self-similar solutions of the singular poly- tropic filtration equation (1.1). We try to extend the classification results of the self-similar solu- tions for singular p-Laplace heat equation by Bidaut-Véron [5] to the singular polytropic filtration equation (1.1) with m(p−1) < 1. We note that the doubly nonlinear diffusion div(|∇um|p−2∇um) is more complicated than the single porous medium diffusion div(∇um) and the single p-Laplace diffusion div(|∇u|p−2∇u). The variables related to u and ∇u are not separable in the analysis of phase portrait, particularly there is a singular or degenerate factor |y| 1 m−1 in the corresponding differential system (see (2.14) in Section 2). Two main difficulties arise in the classification of self-similar solutions of the singular polytropic filtration equation (1.1): • The construction of energy functionals related only to Y and Y ′ is inapplicable, therefore we combine the energy functional and the Bendixson-Dulac criterion to show the non-existence result of periodic orbit or limit cycle. • The discussion of monotonicity of Y is more subtle. Different from the previous works, we cannot determine the monotonicity property according to the comparison principle based on the sign of Y ′′. The aim of this article is to present a complete description of the self-similar solutions to the singular polytropic filtration equation (1.1). We state the main results and related differential systems in Section 2. The local and global solutions together with their asymptotic behavior to the differential equations and the differential systems are formulated in Section 3. The existence, monotonicity and the asymptotic behavior of the self-similar solutions are classified in Section 4. 2. Main results and related systems In this article, we are concerned with the self-similar solutions for the singular polytropic filtration equation (1.1) in RN with N ≥ 1, m > 0, p > 1, such that m(p− 1) < 1. Note that for EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 3 the singular p-Laplace equation with p ∈ (1, 2) (i.e., the case m = 1 for the polytropic filtration equation (1.1)), the classification of self-similar solutions was investigated by Bidaut-Véron [5]. Here for the singular polytropic filtration equation (1.1) with m(p − 1) < 1, both m and p have influence on the asymptotic behavior and classification of self-similar solutions. We define δ = mp 1−mp+m , η = N − p p− 1 ; therefore δ −mN m(p− 1) = δ − η = mN − η 1−mp+m . (2.1) Two critical values P1, P2 are involved in the problem P1 = N +mN 1 +mN , P2 = N +mN 1 +m+mN ; they are connected to δ by the relations p > P1 ⇔ δ > mN, p > P2 ⇔ δ +mδ −mN > 0. If u(x, t) is a solution, then for any α, β ∈ R, uλ(x, t) = λαu(λx, λβt) is a solution of (1.1) if and only if β = p− α(1−mp+m) = (1−mp+m) ( δ m − α ) , noticing that β > 0 ⇔ α < δ m . Given α ∈ R such that α ̸= δ m , we search for self-similar solutions, radially symmetric in x, of the form u = u(x, t) = (βt)−α/βw(r), r = (βt)− 1 β |x|. (2.2) The time variable t in (2.2) can be replaced by t − T . Hence, for any real time T , we obtain solutions defined for any t > T when β > 0, or t < T when β < 0. The following second order non-autonomous differential equation arises: mp−1 [ ( ∣∣|w|m−1w′∣∣p−2|w|m−1w′)′ + N − 1 r ∣∣|w|m−1w′∣∣p−2|w|m−1w′ ] + αw + rw′ = 0, (Ew) with r ∈ (0,∞). We will show a complete description of all the possible solutions of (1.1) with constant or changing sign. 2.1. Explicit solutions. Solution U. The simplest positive solutions of equation (Ew), existing for any α such that (δ −mN)(δ −mα) > 0, are given by w(r) = ℓ1/mr−δ/m, where ℓ = ( δp−1 δ −mN δ −mα ) m 1−mp+m > 0. (2.3) They are associated to a unique solution u of (1.1) called U , singular at x = 0, for any |t| > 0: U(x, t) = ( Ct |x|p ) 1 1−mp+m , C = pδp−2(δ −mN). (2.4) Case α = N . Equation (Ew) has a first integral w +mp−1r−1||w|m−1w′|p−2|w|m−1w′ = Cr−N . (2.5) All the solutions for C = 0 are given by w = wK,1(r) = (δ−1rp ′ +K) − δ mp′ + , such that u = uK,1(x, t) = (βN t) −N/βN (δ−1(βt)−p′/βN |x|p ′ +K) − δ mp′ + , K ∈ R, (2.6) with β = βN = ( δ m − N)(1 −mp +m). Set p > N(1 −mp +m) and K > 0. The solutions are usually called Barenblatt solutions [4]. For every c > 0, the function uK,1, defined on RN × (0,∞), 4 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 admits a unique solution of equation (1.1) with initial data u(0) = cδ0, where δ0 is the Dirac mass at 0, and K is determined by ∫ RN uK(x, t)dx = c. Case α = η m . Equation (Ew) can be explicitly solved: w(r) = Cr−η/m, u(t, x) = C|x|−η/m, C ̸= 0. (2.7) The solutions u are independent of t. Case N = 1 and α = p−1 1−m(p−1) > 0. The equation (Ew) admits the explicit solutions in the form: w(r) = ±(Kr −mp−1αp−1|K|p)−α, where Kis a constant. They correspond to the unique solutions u of (1.1) is u(x, t) = ±(K|x| −mp−1αp−1|K|pt)−α. 2.2. Main results. Before introducing our main results for equation (1.1), we give some defini- tions of different kinds of solutions in terms of regular solutions, singular solutions and solutions with a reduced domain. Considering equation (Ew). The solution w is defined on a maximal interval (Rw,∞) with Rw ≥ 0, whose local existence and uniqueness near any point r1 > 0 are easy to be proved. Returning to solution u of equation (1.1) corresponding to w by (2.2), it is defined on a subset of (RN \ {0})× (0,±∞): Dw = { (x, t) | x ∈ RN , βt > 0, |x| > (βt) 1 βRw } . When w is defined on (0,∞), then u is defined on (RN \ {0})× (0,±∞). (i) Regular solutions. Among the solutions of (Ew) defined near 0, we also show the existence and uniqueness of solutions w = w(·; a) ∈ C2([Rw,∞)) with Rw = 0, such that for some a ∈ R, w(0) = a, w′(0) = 0, called regular solutions. If w is regular, then Dw = RN × (0,±∞), and u(·, t) ∈ C1(RN ) for t ̸= 0, u is called regular. This does not mean the regularity up to t = 0: indeed u presents a singularity at time t = 0 if and only if 0 < α < δ m . In the paper we would not consider the trivial solution w ≡ 0, associated to a = 0. (ii) Singular solutions. If Rw = 0, and w is not regular, then u presents a singularity at x = 0 for t ̸= 0, called a standing singularity. Following [10, 30], for such a solution, x = 0 is said to be of weak singularity if x 7→ w(|x|) ∈ L1 loc(RN ), or equivalently u(·, t) ∈ L1 loc(RN ) for t ̸= 0; and of strong singularity if not. If u has a strong (resp. weak) singularity, and limt→0 u(t, x) = 0 for any x ̸= 0, u is called a strong (resp. weak) razor blade. If u(·, t) ∈ L1(RN ) for t ̸= 0, u is called integrable. (iii) Solutions with a reduced domain. If Rw > 0, we will say that u and w have a reduced domain. Then Dw has a lateral boundary of the form Σw = {|x| = C(βt) 1 β }, of parabolic type if β > 0, of hyperbolic type if β < 0, and u has an explosion near Σw. We calculate the blow-up rate, which is of the order of d(x, t)− p−1 1−mp+m , where d(x, t) is the distance to Σw, by Proposition 3.13. Next, let us give the main results with respect to the function u. For simplicity, we avoid the particular case (for example N = 1, mα = δ, or p = P1) and do not mention the existence of solutions with a reduced domain. All of them and the detailed results in terms of the function w can be found in section 4. An important critical value of α is involved: α∗ = δ m + δ(mN − δ) m(p− 1)(mδ + δ −mN) , which appears when p > P2, such that α∗ > 0. Remark 2.1. To return from w to u, consider any solution w of (2.28) defined on (0,∞), such that for some λ ≥ 0 and µ ∈ R, limr→0 r λw = c ̸= 0 and limr→∞ rµw = c′ ̸= 0. Then EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 5 (i) For fixed t, u has a singularity at x = 0, and u(x) ∼ O(|x|−λ) as |x| → 0, u(x) ∼ O(|x|−µ) as |x| → ∞. Thus x = 0 is of weak singularity if and only if λ < N , and u is integrable if and only if λ < N < µ. (ii) For fixed x = 0, the behaviour of u near t = 0, depends on the sign of β: lim t→0 |x|µ|t|(α−µ)/βu(x, t) = C ̸= 0, if mα < δ, lim t→0 |x|λ|t|(α−λ)/βu(x, t) = C ̸= 0, if mα > δ. Our main results of the classification of self-similar solutions to the polytropic filtration equation (1.1) are listed for the cases β > 0 and β < 0 separately. We follow the same line as the self- similar solutions for p-Laplace equation in [5], which is the special case m = 1 for the polytropic filtration equation (1.1). Compared with the single p-Laplace equation in [5], we note that for non-Newtonian polytropic filtration equation (1.1), there is a singular or degenerate factor |y| 1 m−1 in the corresponding autonomous system via logarithmic change of the variable (see the system (2.14)), which is the main cause of the difficulties of this paper. (i) Solutions defined for β > 0. Here we consider the solutions u of (1.1) on (RN \{0})×(0,∞) of the form (2.2), which means t > 0, or equivalently mα < δ. Now we discuss with respect to the sign of p− P1. For the proofs, we refer the reader to Theorems 4.2, 4.4, and 4.6. Theorem 2.2. Assume −∞ < mα < δ, and p > P1(N ≤ 2). Then U is a solution on (RN \ {0}) × (0,∞), which is a strong razor blade. There exist also positive solutions with a strong singularity, and u ∼ O(|x|−δ/m) as |x| → 0, and limt→0 |x|αu = L > 0 (for x ̸= 0). For α ≥ N , there exists at most one zero in any function u(·, t) at time t. (1) For α < N , the regular solutions on RN × (0,∞) have a constant sign, are not integrable, and they are solutions of (1.1) with initial data L|x|−α ∈ L1 loc(RN ). There exist positive integrable razor blades, with u ∼ O(|x|−η/m) as |x| → 0. There exist also positive solutions with a weak regularity, and u ∼ O(|x|−η/m) as |x| → 0, and limt→0 |x|αu = L (in particular if α = η m , then u ≡ C|x|−η/m). There exist solutions with one zero and a weak or a strong singularity. (2) For α = N , the regular Barenblatt solutions are integrable with a constant sign. There exist solutions with one zero and of weak singularity. (3) For α > N and 0 < m < 1, 1 < p < 2, the regular solutions have at least one zero. If α < α∗, then any solution has a finite number of zeros. If N < α∗, there exist α̂ ∈ (α, δ) such that if α̂ < α, the regular solutions are oscillating around 0 for large |x|, and rδ|w|m−1w is asymptotically periodic in ln r; and there exists exactly a solution u such that rδ|w|m−1w is periodic in ln r. Theorem 2.3. Assume −∞ < mα < δ, and p < P1. Then the regular solutions on RN × (0,∞) have a constant sign, are not integrable, and are solutions of (1.1) with initial data L|x|−α ∈ L1 loc(RN ). The non-regular solutions are not defined on (RN \ {0})× (0,∞). Observe that if α > 0, all the solutions w tend to 0 at ∞, while if α < 0, some of the solutions are unbounded near ∞. (ii) Solutions defined for β < 0. Next we search for the solutions of (1.1) on (RN \{0})×(0,∞). They are associated to mα > δ, equivalently t < 0. One of our most important results is the existence of a critical value αcrit > 0 for P2 < p < P1, for which the regular solutions ucrit are positive, integrable, and vanish identically at time 0. Another new phenomena is the existence of positive solutions such that C1U ≤ u ≤ C2U for some C1, C2 > 0, with a periodic property, according to Theorem 2.5. From Theorems 4.7 when p > P1 and 4.10, 4.12 and 4.14 when p < P1, we deduce the following results. Theorem 2.4. Assume δ < mα, p > P1 (N ≥ 2). Then any solution u on (RN \ {0})× (0,−∞), in particular the regular ones, is oscillating around 0 for fixed t < 0 and large |x|, and rδ|w|m−1w is asymptotically periodic in ln r. There exist weak integrable razor blades, with u ∼ O(|x|−η/m) as |x| → 0. 6 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 Theorem 2.5. Assume δ < mα, p < P1. Then U is a solution on (RN \ {0})× (0,−∞), it is a weak razor blade. Moreover (1) If p < P2, the regular solutions on RN × (0,−∞) have a constant sign, are not integrable, and vanish identically at t = 0, with ∥u(·, t)∥L∞(RN ) ≤ C|t| α |β| . All the solutions have a finite number of zeros. (2) For mα < η, the regular solutions have a constant sign, with the same behaviour (given by (2.6) if α = N). There exists a positive solution u, which is not integrable, with u ∼ O(|x|−α) as |x| → 0 (a strong singularity if and only if α ≥ N), and limt→0 |x|αu = L. If α = η, then u(x, t) = C|x|−η/m is a solution with a strong singularity. (3) If p > P2 and 1 < p < 2, there exists a critical value αcrit such that η m < αcrit < α∗ and the regular solutions ucrit have a constant sign, are integrable, and vanish identically at t = 0, with ∥u(·, t)∥L∞(RN ) ≤ C|t| α |β| . (4) If α ∈ (αcrit, α ∗) and 1 < p < 2, there exist positive solutions u such that rδ|w|m−1w is periodic in ln r, thus C1U ≤ u ≤ C2U for some C1, C2 > 0. There exist positive solutions u, with the same bounds, such that rδ|w|m−1w is asymptotically peri- odic in ln r near 0. There exist positive integrable solutions u such that rδ|w|m−1w is asymptotically periodic near 0. (5) If αcrit < α and 1 < p < 2, the regular solutions are oscillating around 0 for fixed t < 0 and large |x|, and rδ|w|m−1w is asymptotically periodic in ln r. There exist solutions oscillating around 0, such that rδ|w|m−1w is periodic. If α∗ < α, there exist positive integrable razor blades, with u ∼ O(|x|−δ/m) as |x| → 0. 2.3. Different formulations of the problem. Defining JN (r) = rN (w +mp−1r−1||w|m−1w′|p−2|w|m−1w′), Jα(r) = rα−NJN (r), (2.8) equation (Ew) is equivalent to the form J ′ N (r) = rN−1(N − α)w, J ′ α(r) = −(N − α)mp−1rα−2||w|m−1w′|p−2|w|m−1w′. (2.9) If α = N , then JN is constant, which leads to (2.5). Let us use the following logarithmic substitution: given d ∈ R, and defined yd(τ) = rd|w|m−1w, Yd = −mp−1r(d+1)(p−1)||w|m−1w′|p−2|w|m−1w′, τ = ln r. (2.10) Then the equation (Ew) is rewritten in the form yd ′ = dyd − |Yd| 2−p p−1Yd, Yd ′ = (p− 1)(d− η)Yd + e(p+d(p−1)− d m )τ |yd| 1 m−1(αyd − 1 m |Yd| 2−p p−1Yd). (2.11) Thus yd satisfies y′′d + (η − 2d)y′d − d(η − d)yd + e(p+d(p−1)− d m )τ m(p− 1) |y′d − dyd|2−p|yd| 1 m−1(y′d + (mα− dyd)) = 0. (2.12) The most important case is the special value d = δ: setting y = yδ, y(τ) = rδ|w|m−1w, Y = −mp−1r(δ+1)(p−1)||w|m−1w′|p−2|w|m−1w′, (2.13) we are led to the autonomous system y′ = δy − |Y | 2−p p−1Y, Y ′ = ( δ m −N)Y + |y| 1 m−1(αy − 1 m |Y | 2−p p−1Y ). (2.14) EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 7 Hence equation (2.12) takes the form y′′ − (δ + δ −mN m(p− 1) )y′ + δ(δ −mN) m(p− 1) y + 1 m(p− 1) |δy − y′|2−p|y| 1 m−1(y′ + (mα− δ)y) = 0. (2.15) Remark 2.6. Since (2.14) is autonomous, for any solution w of the problem (Ew), all the functions wξ(r) = ξ−γw(ξr), ξ > 0, are also solutions. System (2.14) will be considered by applying the phase plane analysis method, and shows our main results. Notice that the set of trajectories of system (2.14) in the phase plane (y, Y ) is symmetric with respect to (0, 0). In the phase plane (y, Y ) we define M = { (y, Y ) ∈ R2 : |Y | 2−p p−1Y = δy } , (2.16) which is the set of the extremal points of y. We denote the four quadrants by Q1 = (0,∞)× (0,∞), Q2 = (−∞, 0)× (0,∞), Q3 = −Q1, Q4 = −Q2. Remark 2.7. The vector field at any point (0, ξ), ξ > 0 satisfies y′ = −ξ 1 p−1 < 0, thus points to Q2; moreover, Y ′ = ( δ m − N)ξ, if δ > mN , then Y ′ > 0, otherwise Y ′ < 0. The field at any point (φ, 0), φ > 0 satisfies Y ′ = αφ1/m, thus points to Q1 if α > 0 and to Q4 if α < 0; moreover y′ = δφ > 0. Remark 2.8. The couple (y, Y ) is related to JN by the identity JN (r) = rN− δ m (|y| 1 m−1y − Y ), τ = ln r, (2.17) and the formulae (2.8) can be rewritten again corresponding to the relations (|y| 1 m−1y − Y )′ = ( δ m − α)(|y| 1 m−1y − Y ) + (N − α)Y = ( δ m −N)(|y| 1 m−1y − Y ) + (N − α)|y| 1 m−1y. (2.18) 2.4. Stationary points and energy functionals of system (2.14). In this subsection, we first discuss stationary points of system (2.14) and research their local behaviour in order to facilitate the following study of the behavior of trajectories in phase plane. If δ = mN = mα, then system (2.14) admits an infinity many of stationary points, given by ±(k, (δk)p−1), k > 0. Apart from this case, if (δ − mN)(δ − mα) ≤ 0, there exists a unique stationary point (0, 0) in system (2.14). If (δ −mN)(δ −mα) > 0, there exist three stationary points: (0, 0), Mℓ = (ℓ, (δℓ)p−1) ∈ Q1, Mℓ ′ = −Mℓ ∈ Q3, where ℓ = (δp−1 δ−mN δ−mα ) m 1−mp+m . (i) Local behaviour at (0, 0). The linearized problem at (0, 0) is given by y′ = δy, Y ′ = ( δ m −N)Y, and has the eigenvalues λ1 = δ and λ2 = δ m −N . Thus (0, 0) is a saddle point when δ −mN < 0, and a source when δ −mN > 0. (ii) Local behaviour at Mℓ. Setting y = ȳ + ℓ, Y = Ȳ + (δℓ)p−1, (2.19) system (2.14) is equivalent in Q1 to ȳ′ = δȳ − (δℓ)2−p p− 1 Ȳ +Φ(ȳ, Ȳ ), Ȳ ′ = ( δ m −N)Ȳ + 1 m2 (mα+ (m− 1)δ)ℓ 1 m−1ȳ − 1 m(p− 1) δ2−pℓ1+ 1 m−pȲ +Ψ(ȳ, Ȳ ), (2.20) 8 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 where Φ(ȳ, Ȳ ) = −(Ȳ + (δℓ)p−1) 1 p−1 + δℓ+ (δℓ)2−pȲ p− 1 , (2.21) and Ψ(ȳ, Ȳ ) = 1 m (ȳ + ℓ) 1 m−1[mα(ȳ + ℓ)− (Ȳ + (δℓ)p−1) 1 p−1 ] + ( δ m −N)(δℓ)p−1 − 1 m2 (mα+ (m− 1)δ)ℓ 1 m−1ȳ + 1 m(p− 1) δ2−pℓ1+ 1 m−pȲ , Ȳ > −(δℓ)p−1. (2.22) The linearized problem is ȳ′ = δȳ − 1 p− 1 (δℓ)2−pȲ , Ȳ ′ = 1 m2 (mα+ (m− 1)δ)ℓ 1 m−1ȳ + ( δ m −N − 1 m(p− 1) δ2−pℓ1+ 1 m−p)Ȳ . Its eigenvalues λ1 ≤ λ2 are the solutions of equation det(λI −A) = λ2 − (δ + δ m −N − δ m(p− 1) · mN − δ mα− δ )λ− p m(p− 1) (δ −mN) = 0. (2.23) The discriminant ∆ of the equation (2.23) is ∆ = (δ + δ m −N − δ m(p− 1) mN − δ mα− δ )2 + 4p m(p− 1) (δ −mN) = [δ − δ m +N + δ(mN − δ) m(p− 1)(mα− δ) ]2 − 4δ(mN − δ) m(p− 1)(mα− δ) mα+ (m− 1)δ m . (2.24) The critical value α∗ of α from the discriminant ∆ is α∗ = δ m + δ(mN − δ) m(p− 1)(mδ + δ −mN) , (2.25) it appears when δ > m m+1N : α = α∗ ⇔ λ1 + λ2 = 0. When δ < mN , then δ < mα, Mℓ is sink when δ ≤ mN 1+m or δ > mN 1+m and α < α∗; Mℓ is source when δ > mN 1+m and α > α∗. When δ > mN , then Mℓ is always a saddle point, but as we will find the value α∗ also plays an important role in the sequel. Moreover, the sign of α∗ and its position with respect to N or η are important in the sequel. By calculations, α∗ = p′(δ2 + (mN −N −m− 2)mδ + 2m2N) m(1 +m)(mδ + δ −mN) = η m + (δ −mN)2 m2(p− 1)(mδ + δ −mN) = N + (δ −mN)(δ2 −m(m+N + 2)δ +m2N) m(δ −m)(mδ + δ −mN) . (2.26) Thus α∗ > η m > 0 if N > p and mδ + δ −mN > 0, if N = 1, α∗ > 0 if p > 2+2m 2m+1 . Also, when ∆ > 0 one can choose a basis of eigenvectors u1 = (− (δℓ)2−p p−1 , λ1 − δ) and u2 = ( (δℓ) 2−p p−1 , δ − λ2). We can obtain the stability at the stationary points by researching their local behaviour, and then we can know the motion of the trajectories near stationary points. In order to obtain a global distribution of the trajectories, we need to construct energy functionals to analyse the properties of the solutions in the whole space. A classical energy function is associated with equation (Ew) is E(r) = 1 p′ ||w|m−1w′|p + m1−pα m+ 1 |w|m−1w2, (2.27) EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 9 which is non-increasing, since E′(r) = −(N − 1)r−1||w|m−1w′|p − m1−pr|w|m−1w′2. It is not sufficient in the study, we construct a new energy function adapted to y and Y following [5]. Define W (τ) =W (y(τ), Y (τ)), where W (y, Y ) = (mδ + δ −mN)δp−1 p |y|p + m|Y |p′ p′ −mδyY + m(mα− δ) m+ 1 |y| 1 m−1y2. (2.28) It satisfies W ′(τ) = (δy − |Y | 2−p p−1Y )(|δy|p−2δy − Y )(mδ + δ −mN)− |y| 1 m−1(δy − |Y | 2−p p−1Y )2 = (δy − |Y | 2−p p−1Y )(|δy|p−2δy − Y ) ( mδ + δ −mN − |y| 1 m−1 δy − |Y | 2−p p−1Y |δy|p−2δy − Y ) . When mδ + δ −mN ≤ 0, W is non-increasing. When mδ + δ −mN > 0, we consider the curve L = { (y, Y ) ∈ R2 : H(y, Y ) = |y| 1 m−1 δy − |Y | 2−p p−1Y |δy|p−2δy − Y = mδ + δ −mN } , where by convention the quotient takes the value δ2−p p−1 y 1−mp+m m if |δy|p−2δy = Y . It is a closed curve surrounding (0, 0), symmetric in reference to (0, 0). Let SL be the domain with boundary L and containing (0, 0): SL = { (y, Y ) ∈ R2 : H(y, Y ) ≤ mδ + δ −mN } . (2.29) Then W ′(τ) ≥ 0 if (y(τ), Y (τ)) ∈ SL and W ′(τ) ≤ 0 if (y(τ), Y (τ)) /∈ SL. We notice that SL is bounded if 1 < p < 2 and 0 < m < 1: indeed for any (y, Y ) ∈ R2, H(y, Y ) ≥ 1 2 ((δy)2−p + |Y | 2−p p−1 )|y| 1 m−1. Also SL is connected, more precisely for any (y, Y ) ∈ SL and any θ ∈ [0, 1], (θy, θp−1Y ) ∈ SL. 2.5. Other systems for positive solutions. When w has a constant sign, we set two functions corresponding to (y, Y ): ζ(τ) = |Y | 2−p p−1Y y (τ) = −mrw ′ w , σ(τ) = Y |y| 1 m−1y = −m p−1||w|m−1w′|p−2|w|m−1w′ rw . (2.30) Thus ζ describes the behaviour of w′ w . They convert (2.14) into system ζ ′ = ζ(ζ − η) + mα− ζ m(p− 1) |ζ|2−p|y|1−p+ 1 m = ζ ( ζ − η + mα− ζ m(p− 1)σ ) , σ′ = (α−N) + ( 1 m |σ| 2−p p−1σ|y| 1−mp+m m(p−1) N ) (σ − 1) = (α−N) + ( ζ m −N ) (σ − 1). (2.31) In particular, System (2.31) provides a short proof of the local existence and uniqueness of the regular solutions, corresponding to its stationary point (0, α N ). Also, if w and w′ have a strict constant sign, we can define a new function in any quadrant Qi: setting ψ = 1 σ = |y| 1 m−1y Y . then we obtain a new system on (ζ, ψ): ζ ′ = ζ ( ζ − η + (mα− ζ)ψ m(p− 1) ) , ψ′ = ψ ( N − ζ m + ψ( ζ m − α) ) . (2.32) That means we can convert system (2.31) into a polynomial system (2.32) without singularity. We get two stationary points (η, 0) and (0,±∞) from system (2.32). 10 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 We also consider another system in any quadrant Qi: setting ζ = −1 g , σ = −s, dτ = gsdν = |y| 1−m m |Y | p−2 p−1 dν, then we are led to the new system dg dν = g ( s(1 + ηg) + 1 +mαg m(p− 1) ) , ds dν = −s ( 1 m + αg + s( 1 m +Ng) ) . (2.33) We obtain stationary points (mα, 0), (∞, 1) and (±∞, 0) from system (2.33). 3. Global solutions We study the differential equation (Ew) and the related differential systems in Section 2. Most of the proofs of this section are parallel and similar to that of self-similar solution for p-Laplace equations in [5], except that the non-Newtonian polytropic filtration diffusion div(|∇um|p−2∇um) in (1.1) changes the asymptotic behaviors, such that there is a singular or degenerate factor |y| 1 m−1 in the corresponding differential system (2.14), which causes the main difficulties of this paper. 3.1. Existence of solutions of equation (Ew). Proposition 3.1. For any r1 > 0 (r1 ≥ 0, if N = 1), and a, b ∈ R. There exists a unique solution w to the equation (Ew) in a neighborhood V of r1, such that w ∈ C2(V) and w(r1) = a, w′(r1) = b. And w can extend to a maximal interval of the form (Rw,∞), where Rw ≥ 0. Moreover if Rw > 0, then w is monotone near Rw with an infinite limit. Proof. We first prove the local existence and uniqueness. If 0 < m < 1, and 1 < p < 2, local existence and uniqueness near r1 > 0 follow directly from the existence and uniqueness theorem applied to equation (2.14), since the map ξ 7→ fp(ξ) = |ξ| 2−p p−1 ξ and η 7→ gm(η) = |η| 1 m−1 are of class C1. If N=1, setting r1 = 0, we can get a local solution in a neighborhood of 0 in R and extend it to [0,∞). Consider the case of 1 < p < 2 and 1 < m < 1 p−1 . For a = 0, the system is a priori singular on the line y = 0 since m > 1. Notice that it is only singular at (0, 0). In fact, near any point (0, ξ) with ξ ̸= 0, we can consider y as a variable, and dY dy = F (y, Y ), F (Y, y) = ( δ m −N)Y + |y| 1 m−1(αy − 1 m |Y | 2−p p−1Y ) δy − |Y | 2−p p−1Y , where F is continuous in y and C1 in Y , hence local existence and uniqueness hold. If 0 < m < 1, and 2 < p < 1 m + 1, and if b = 0, the system is only singular at (0, 0). In fact, near any point (ξ, 0) with ξ ̸= 0, we can consider Y as a variable, and dy dY = G(Y, y), G(Y, y) = δy − |Y | 2−p p−1Y ( δ m −N)Y + |y| 1 m−1(αy −m−1|Y | 2−p p−1Y ) , in the same way, where G is continuous in Y and C1 in y, thus local existence and uniqueness hold. Next, we show that any local solution around r1 can extend uniquely to a maximal interval of the form (Rw,∞) with Rw ≥ 0 and the solution is monotone near Rw with an infinite limit. These are proved in the same way as Theorem 2.2(i) in [5]. Here we omit the details for the sake of simplicity. □ Notation For any point P0 = (y0, Y0) ∈ R2 \ {(0, 0)}, T[P0] denotes the unique trajectory in the phase plane (y, Y ) of system (2.14) running through P0. By symmetry, T[−P0] = −T[P0]. EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 11 Theorem 3.2. (i) For any a ∈ R, a ̸= 0, the equation (Ew) admits a unique solution w = w(·, a) in an interval [0, r0), such that w ∈ C1([0, r0)) and ||w|m−1w′|p−2|w|m−1w′ ∈ C1([0, r0)) and w(0) = a, w′(0) = 0; and then lim r→0 ||w|m−1w′|p−2|w|m−1w′ rw = −m 1−pα N . (3.1) It implies that in the phase plane (y, Y ), a unique trajectory corresponding to the solution is denoted by Tr, satisfying limτ→−∞ y = 0 and limτ→−∞ Y |y| 1 m −1y = α N . (ii) If N ≥ 2, any solution defined near 0 and bounded is regular. If N = 1, it satisfies limr→0 w ′ = b ∈ R, and limr→0 w = a ∈ R. Proof. The idea of the proof is from [7], we show the process of the proof here for the sake of completeness. (i) Assume a > 0, w > 0. Let ρ > 0. From (2.8) and (2.9), any regular solution w ∈ C1([0, ρ]) of the problem, such that ∣∣|w|m−1w′ ∣∣p−2|w|m−1w′ ∈ C1([0, ρ]), satisfies w = T (w), where T (w(r)) = ( am +m ∫ r 0 |H(w)| 2−p p−1H(w) ds )1/m , and H(w(r)) = −m1−p(rw − r1−NJN (r)) = −m1−p ( rw − r1−N ∫ r 0 sN−1j(w(s)) ds ) , (3.2) with j(w) = (N − α)w. Conversely, the mapping T is well defined from C([0, ρ]) into itself. If w ∈ C([0, ρ]), and w = T (w), then w ∈ C1((0, ρ]), and ∣∣|w|m−1w′ ∣∣p−2|w|m−1w′ = H(w); then |wm−1w′|p−2wm−1w′ ∈ C1((0, ρ]) and w satisfies (Ew) in (0, ρ]. Furthermore, limr→0 j(w(r)) = (N − α)a. Then ||w|m−1w′|p−2|w|m−1w′ = −m1−pαa N r(1 + o(1)). In particular, limr→0 w ′(r) = 0, and ∣∣|w|m−1w′∣∣p−2|w|m−1w′ ∈ C1([0, ρ]). From (Ew), lim r→0 ||w|m−1w′|p−2|w|m−1w′ rw = −m 1−pα N . We consider the ball BR,M = { w ∈ C([0, ρ]) : ∥w − a∥C([0,R]) ≤M } , where M is a parameter such that 0 < M < a 2 . Notice that j is continuous, hence T is a strict contraction from Bρ,M into itself for ρ and M small enough. For any w ∈ Bρ,M , and any r ∈ [0, ρ] (ρ ≤ R), from (3.2), −m1−p ( a+M − N − α N (a−M) ) r ≤ H(w(r)) ≤ −m1−p ( a−M − N − α N (a+M) ) r. We define µ(a) = ( 1 + |N−α| N ) (a+M) > 0, then |H(w(r))| < m1−pµ(a)r. Thus we obtain ∥T (w)− a∥C([0,R]) ≤ µ(a) 1 m(p−1) ρ p m(p−1) , and T (w) ∈ Bρ,M for ρ = ρ(a) small enough. Now for any w1, w2 ∈ Bρ,M , and any r ∈ [0, ρ], then |Tm(w1(r))− Tm(w2(r))| ≤ m ∫ r 0 ||H(w1)| 2−p p−1H(w1)− |H(w2)| 2−p p−1H(w2)|(s) ds. 12 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 For any s ∈ [0, r], we have ||H(w1)| 2−p p−1H(w1)− |H(w2)| 2−p p−1H(w2)|(s) ≤ mp−2µ(a) 2−p p−1 s 2−p p−1 |H(w1)−H(w2)|(s) ≤ m−1µ(a) 2−p p−1 s 1 p−1 ( |w1 − w2|+ |N − α|s−N ∫ s 0 σN−1|w1 − w2|dσ ) ≤ m−1C(a)s 1 p−1 ∥w1 − w2∥C[0,ρ] (3.3) with C(a) = µ(a) 2−p p−1 (1 + |N−α| N ), so ∥Tm(w1(r))− Tm(w2(r))∥C([0,ρ]) ≤ C(a)ρp ′ ∥w1 − w2∥C([0,ρ]) ≤ 1 2 ∥w1 − w2∥C([0,ρ]) , if ρ(a) is small enough. Then Tm is strict contradiction from Bρ,M into itself, thus T is strict contradiction from Bρ,M into itself. Hence existence and uniqueness hold in [0, ρ]. (ii) If w is defined in (0, ρ) and bounded, then J ′ N = rN (N − α)w is integrable; suppose limr→0 JN (r) = l, then ||w|m−1w′|p−2|w|m−1w′ = m1−plr1−N (1+o(1)). If N ≥ 2, it implies l = 0. This shows w is regular. Indeed if l ̸= 0, either l is finite or l = ∞, then w′ converge to ∞ and w converge to ∞, which contradicts to that w is bounded. If N = 1, then it admits limr→0 w ′ = b, and limr→0 w = a ∈ R. □ Definition 3.3. In the plane (y, Y ), the trajectory Tr starting from (0, 0) at −∞ and its opposite −Tr are called regular trajectories. We shall say that y is regular. Notice that Tr starts in Q1 if α > 0, and in Q4 if α < 0. Remark 3.4. From Theorem 3.2 and Remark ??, all regular solutions are obtained from one of them: w(r, a) = aw(a 1 δ r, 1). Thus they have the same behaviour near ∞. 3.2. Sign properties. Next we will present more details on the zeros of w or w′, by applying the monotonicity properties of the functions yd and Yd, in particular y, Y , ζ and σ. At any extremal point τ , they satisfy the following differential equations respectively m(p− 1)y′′d (τ) = (md(p− 1)(η − d) + (d−mα)e(p+d(p−1)− d m )τ |dyd|2−p|yd| 1−m m )yd, (3.4) m(p− 1)y′′(τ) = (−δ(δ −mN) + (δ −mα)|δy|2−p|y| 1−m m )y = −m|Y | 2−p p−1Y ′, (3.5) m2Y ′′(τ) = mα|y| 1−m m y′ + (m− 1)|y| 1−m m y′ζ, (3.6) m2(p− 1)ζ ′′(τ) = (1−mp+m)(mα− ζ)|ζ|2−p|y|1−p+ 1 m y−1y′ = (1−mp+m)(mα− ζ)(δ − ζ)|ζ|2−p|y| 1−mp+m m , (3.7) m2(p− 1)σ′′(τ) = (1−mp+m)(σ − 1)|σ| 2−p p−1Y |y| 2−2mp+2m−p m(p−1) y′ = m(p− 1)(σ − 1)ζ ′. (3.8) Proposition 3.5. Let w ̸≡ 0 be any solution of (Ew). (i) If α ≤ max(N, η m ), then w has at most one zero, and no zero if w is regular. (ii) If N < max( δ m , α) and w is regular, it has at least one zero. Proof. (i) Let us consider two consecutive zeros ρ0 < ρ1 of w, with w > 0 on (ρ0, ρ1), thus w′(ρ1) < 0 < w′(ρ0). If α ≤ N , then JN (ρ1)− JN (ρ0) = −mp−1(ρN−1 1 ||w(ρ1)|m−1w′(ρ1)|p−1 + ρN−1 2 ||w(ρ2)|m−1w′(ρ2)|p−1) = (N − α) ∫ ρ1 ρ0 sN−1w ds, which is contradictory; hence w has at most one zero. If w is regular with w(0) > 0, and ρ1 is a first zero, then N (ρ1) = (N − α) ∫ ρ1 0 sN−1w ds ≥ 0, EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 13 we also get a contradiction. Next suppose 0 < mα < η and apply the substitution (2.10), with d = mα. Then there exists at most one zero in ymα. In fact, if ymα has a maximal point τ where is positive, and is not constant, then following (3.4) y′′mα(τ) = mα(η −mα)ymα(τ), (3.9) y′′mα > 0, which is a contradictory. Similarly, the regular solution satisfies limτ→−∞ ymα = 0 for α > 0, and ymα has no maximal point, hence ymα is positive and increasing. (ii) If w > 0 on [0,∞). Consider the case N < α, JN (r) = (N − α) ∫ r 0 sN−1w ds < 0, thus wm−1− 1 p−1w′ + 1 mr 1 p−1 < 0. The function r 7→ δ−1rp ′ − wm− 1 p−1 is non-increasing and w = O(r−δ/m) at ∞, then y is bounded at ∞. For any r ≥ 1, we have JN (r) ≤ JN (1) < 0, since J ′ N (r) < 0. Thus y 1 m (τ) + |JN (1)|e( δ m−N)τ ≤ Y (τ) for any τ ≥ 0. Then limτ→∞ Y = ∞, thus limτ→∞ y′ = −∞ from (2.14), which is contradictory. Hence, ρ1 is the first zero of w, which means w has at least one zero, and JN (ρ1) < 0, w′ < 0. □ Proposition 3.6. Let y be any non-constant solution of (2.15), the corresponding solution w is defined on the maximal interval (Rw,∞) with Rw ≥ 0, where (y, Y ) ̸= (0, 0), and s be an extremity of the maximal interval. (i) If y has a constant sign near s, then Y also has a constant sign. (ii) If y is strictly monotone near s, then Y, ζ, σ are also monotone near s. (iii) If y is not strictly monotone near s, then s = ±∞, δ < mN < mα, y oscillates around ℓ. Proof. (i) The function w has at most one extremal point on (Rw,∞). Indeed, if w > 0, r is extremal point, we find (|wm−1w′|p−2wm−1w′)′ = −m1−pαw,α ̸= 0, then if α < 0, we know w′ is monotone increasing. Since w′ is continuous, r is a unique point. From (2.13), Y has a constant sign near s. (ii) Assume y is strictly monotone near s. Firstly, consider the function ζ, which satisfies (2.31). If there are τ0 such that ζ ′(τ0) = mα(mα − η). If mα ̸= η, then τ0 is unique, since ζ ′ is continuous, thus mα− ζ has a constant sign near s. Then at any extremal point τ of ζ, ζ ′′(τ) also has a constant sign from (3.7). Thus ζ is strictly monotone near s. If mα = η, one gets ζ ≡ mα. Next consider Y . At the extremal point τ of Y , then m2Y ′′(τ) = mα|y| 1−m m y′ + (m− 1)|y| 1−m m y′ζ for (3.6). We know Y ′′ is of constant sign near τ , since y and ζ are monotone. Then τ is unique and Y is monotone near s. At last consider σ, which satisfies (2.31). If there are τ0 satisfying σ(τ0) = 1, then σ′(τ) = α −N . If α ̸= N , then τ0 is unique, and σ − 1 has a constant sign near s. Thus at the extremal point τ of σ, σ′′(τ) has a constant sign from (3.8). Then σ is strictly monotone near s. If α = N , then σ(τ) ≡ 1. (iii) Assume y is not strictly monotone near s. Then we can find a strictly monotone sequence {τn}∞n=1 with converging to s, such that y′(τn) = 0, y′′(τ2n+1) < 0 < y′′(τ2n). Since y(τn) = δ−1|Y | 2−p p−1Y , we deduce Y > 0 near s from (i). From (3.5), one know (δ −mα)δ2−py(τ2n+1) 1+ 1 m−p < δ(δ −mN) < (δ −mα)δ2−py(τ2n) 1+ 1 m−p, thus δ < min(mN,mα), and y(τ2n) < ℓ < y(τ2n+1). It cannot happen if s is finite, since y tends to ∞. It is also impossible when α ≤ N . In fact there exist at least two points θ1 < θ2 such that y(θ1) = y(θ2) = ℓ and y > ℓ on (θ1, θ2), y ′(θ1) > 0 > y′(θ2). According to the system (2.14), Y (θ1) < (δℓ)p−1 < Y (θ2). From (2.18), (e(N− δ m )τ (y1/m − Y ))′ = (N − α)e(N− δ m )τy1/m, and we know one solution of (2.14) is the constant (ℓ, (δℓ)p−1), hence (e(N− δ m )τ (y1/m − ℓ1/m − Y + (δℓ)p−1))′ = (N − α)e(N− δ m )τ (y1/m − ℓ1/m) > 0 on (θ1, θ2). It is contradictory to integrate on this interval. □ 3.3. Behaviour of w near 0 or ∞. Here we suppose y is monotone, such that w has constant sign near 0 or ∞, we can assume w > 0. Proposition 3.7. Let (y, Y ) be a solution of system(2.14), such that y is strictly monotone and y > 0 near s = ±∞. Then lim τ→s ζ = λ, λ = 0,mα, η, or δ. 14 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 Moreover, one of the following eventualities holds (i) (y, Y ) converges to a stationary point different from (0, 0), then λ = δ, and (δ−mN)(δ− mα) > 0 or δ = mα = mN . (ii) (y, Y ) converges to (0, 0), then • either λ = 0, s = −∞, and y is regular, or N=1. • or λ = η, then either (s = ∞, δ < mN) or (s = ∞, δ = mN,α − N < 0) or (s = −∞, δ > mN) or (s = −∞, δ = mN,α−N > 0). (iii) limτ→s y = ∞, and λ = mα, then either (s = ∞,mα < δ) or (s = ∞, δ = mα, δ−mN < 0) or (s = −∞,mα > δ) or (s = −∞, δ = mα, δ −mN > 0). Proof. In the case, the function y is monotone, then Y, σ, ζ are also monotone from Proposition 3.6, thus ζ has a limit λ ∈ [−∞,∞] and σ has a limit µ ∈ [−∞,∞], and (y, Y ) converges to a stationary point, or lim y = ∞; then lim |Y | = ∞, from system (2.14). To use the L’Hospital’s rule, we consider the two quotients Y ′ (y1/m)′ = (δ −mN)σ +mα− ζ δ − ζ (3.10) and (|Y | 2−p p−1Y )′ y′ = ζ(δ −mN + (mα− ζ)/σ) m(p− 1)(δ − ζ) = ζ(δ −mN) + (mα− ζ)|ζy|2−p|y| 1 m−1 m(p− 1)(δ − ζ) . (3.11) (i) If (δ − mN)(δ − mα) > 0 and (y, Y ) converges to (ℓ, (δℓ)p−1), then obviously λ = δ. If δ = mα = mN and limτ→s y = k > 0, then limτ→s Y = (δk)p−1, thus λ = δ. (ii) (y, Y ) converges to (0, 0). Then λ is finite. Indeed, if λ = ±∞, then µ = 0, the quotient (3.11) converges to mN−δ m(p−1) , since |ζy| = |Y | 1 p−1 = o(y1/m) and y tends to 0, thus ζ = |Y | 2−p p−1 Y y has the same limit, applying L’Hospital’s rule, which is impossible. • If mN < δ, then (0, 0) is a source, so s = −∞. If µ is finite, either µ = 0, then λ = mα, from the quotient (3.10). The quotient (3.11) converges to (δ−mN)α (p−1)(δ−mα) , as mentioned above, we know |ζy| = o(y1/m) and y tends to 0, following L’Hospital’s rule, which is contradictory; or µ = α N , and |ζ|p−1 = |µ|y 1−mp+m m (1 + o(1)), then λ = 0 and w is regular. If µ = ±∞, then λ = λ(δ−mN) m(p−1)(δ−λ) from quotient (3.11), thus λ = 0 or λ = η. If λ = 0, then ζ′ ζ → −η from system (2.31), and s = −∞, thus necessarily η < 0, which means N = 1. • If mN > δ (thus N ≥ 2), we know (0, 0) is a saddle point, thus either s = −∞ and µ = α N , λ = 0, then w is regular. Or s = ∞, µ = ±∞, then λ = 0 or λ = η. Now if λ = 0, the quotient (3.10) converges to ∓∞, we obtain a contradiction. Thus λ = η. • If mN = δ (thus N ≥ 2), either λ = 0, thus y′ = mNy(1 + o(1)) > 0, thus s = −∞ and µ = α N from quotient (3.10). Or λ > 0, then λ = δ = mN = η from (3.11). Furthermore if s = ∞, then α−N < 0; if s = −∞, then α−N > 0. In fact, (y1/m−Y )′ = (N −α)y1/m can be rewritten as (Y − y1/m)′ = (α−N)y1/m, and y = m−1N−1Y 1 p−1 (1 + o(1)), (Y − y1/m) = Y (1 + o(1)) since λ = η = mN and µ = ∞, then (Y − y1/m)′ = (α−N)m−1/mN−1/mY 1 m(p−1) = (α−N)m−1/mN−1/m(Y − y1/m) 1 m(p−1) . If s = −∞, and α−N ≤ 0 or s = ∞, and α−N ̸= 0, which is impossible. (iii) y tends to ∞. If s = ∞, then y′ > 0, for y′ = y(δ − ζ) > 0, thus ζ < δ; if s = −∞, then y′ < 0, thus ζ > δ. If λ = ±∞, we find the quotient (3.11) converges to ∞; thus λ = ∞ and s = −∞. In any case, ζ ′ < 0, thus |µ| ≤ 1 m(p−1) from system (2.31), then |µ| = 1 from (3.10). But Y ′ = − 1 m |Y | 1−mp+m m(p−1) Y (1 −mα · o(1) − (δ −mN)|Y |− 1−mp+m m(p−1) ) since system (2.14), a contradiction follows by integration. Thus λ is finite, and λ ̸= 0. Indeed, if λ = 0, then µ = 0, since σ = |ζ|p−2ζy− 1−mp+m m , but µ = mα δ , from (3.10), which is a contradiction. • If mα ̸= δ, then λ = mα or λ = δ, from (3.11). In turn σ = |λ|p−2λy− 1−mp+m m (1 + o(1)), thus µ = 0. Necessarily λ = mα. Indeed, if λ = δ, ζ converges to ∞ from (3.11), which is contradictory. And if s = ∞, then y′ > 0, thus ζ < δ, hence mα < δ. If s = −∞, then similarly mα > δ. EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 15 • If mα = δ, then λ = mα = δ ̸= mN , and ( δ m − N)(δ − ζ) > 0 from (3.11); thus if s = ∞, then (δ −mN) < 0; if s = −∞, then δ −mN > 0. □ Next in order to show a precise behaviour of w in all cases, we improve Proposition 3.7. Proposition 3.8. Under the assumptions of Proposition 3.7, let w be the solution of (Ew) asso- ciated to y by (2.13). (i) If λ = δ, and (δ −mN)(δ −mα) > 0 or δ = mN = mα, then (near 0, or ∞), lim rδ|w|m−1w = ℓ. (ii) If λ = mα ̸= δ, then (near 0, or ∞) lim rαw = L > 0. (iii) If λ = η > 0, η ̸= mN , then lim r η mw = c > 0. (iv) If λ = mα = δ ̸= mN , then lim r δ m (ln r)− 1 1−mp+mw = κ =: ( 1 m |mN − δ|δp−1(1−mp+m) ) 1 1−mp+m . (3.12) (v) If λ = η = mN = δ ̸= mα, then lim rN (ln r) 1+mN 1+m w = ρ̃ =: ( 1 mN )1/m( (mN) 1/m mN −m (1 +m)|α−N | ) 1+mN 1+m . (3.13) (vi) If N = 1, λ = η = −1 or λ = 0 (near 0) then lim r→0 w = a ∈ R, lim r→0 w′ = b; (3.14) and b ̸= 0, and a = 0 (thus b > 0) if λ = −m. Proof. (i) This follows directly from (2.13) and Proposition 3.7. (ii) Let λ = mα ̸= δ. From system (2.31), rw′ = −αw(1 + o(1)). Next we employ Proposition 3.7: • Either s = ∞ and mα < δ; thus for any γ > 0, w = O(r−α+γ), 1 w = O(rα+γ) near ∞ and w′ = O(r−α−1+γ), J ′ α = O(rα(1−mp+m)−p−1+γ), thus J ′ α is integrable, hence Jα has a limit L, and lim rαw = L, for µ = 0 and Jα(r) = rαw(1 + o(1)). If L = 0, then rαw = O(rα(1−mp+m)−p+γ), hence w ≤ rα(1−mp+m)−p−α+γ and 1 w ≥ r−α(1−mp+m)+p+α+γ , we reach a contradiction by the estimate of 1 w = O(rα+γ) for γ small enough. Thus L > 0. •Or s = −∞, and δ < mα, and limτ→s y = ∞, w = O(r−α−γ), 1 w = O(rα−γ), w′ = O(r−α−1−γ) near 0, and J ′ α = O(rα(1−mp+m)−p−1−γ), thus J ′ α is still integrable; thus lim rαw = L ≥ 0. If L = 0, then rαw = O(rα(1−mp+m)−p−γ), we reach a contradiction by the estimate of 1 w . Then again L > 0. (iii) Let λ = η > 0, η ̸= mN . From Proposition 3.7 either (s = ∞, δ < mN) or (s = −∞, δ > mN). As mentioned above we obtain w = O(r− η m±γ) and 1 w = O(r η m±γ) near ∞ or 0 for any γ > 0. Here we substitute d = η into (2.10). Thus yη = O(e±γτ ), 1 yη = O(e±γτ ), y′η = O(e±γτ ), hence Yη = O(e±γτ ), and Y ′ η = O(e±γτ ) from (2.11). From (2.11), we deduce Y ′ η = O(e 1 m (1−mp+m)(δ−η)±γ)τ ). When s = ∞, then δ < η, when s = −∞, then δ > η from (2.1). Thus Y ′ η is integrable, and Yη has a limit denoted by k, and Yη − k = O(e 1 m (1−mp+m)(δ−η)±γ)τ ). Now, (e−ητyη) ′ = −e−ητY 1 p−1 η , thus yη has a limit denoted by c = 1 ηk 1 p−1 , it implies lim rηwm = c, which is equivalent to lim r η mw = c̄, c̄ = c1/m. If c = 0, for Yη = O(e 1 m (1−mp+m)(δ−η)±γ)τ ), then yη = O(e 1 m (1−mp+m)(δ−η)±γ)τ ), we reach a contradiction by 1 yη = O(e±γτ ) for γ small enough. (iv) Let λ = mα = δ ̸= mN , thus (s = ∞, δ − mN < 0) or (s = −∞, δ − mN > 0), and limτ→s y = ∞. Then Y = (δy)p−1(1 + o(1)), and µ = 0, thus (y1/m − Y ) = y1/m(1 + o(1)), and from (2.18), (y1/m − Y )′ = (N − α)Y = (N − δ)δp−1(y1/m − Y )m(p−1)(1 + o(1)). Then y = ( 1 m |mN − δ|δp−1(1−mp+m)|τ |) m 1−mp+m (1 + o(1)), which is equivalent to (3.12). 16 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 (v) Let λ = η = mN = δ ̸= mα, thus (s = ∞, α − N < 0) or (s = −∞, α − N > 0), and limτ→s y = 0. Then Y = (mNy)p−1(1 + o(1)), and µ = ∞, thus Y − y1/m = Y (1 + o(1)), and from (3.12), (Y − y1/m)′ = (α−N)y1/m = (α−N)(mN)−1/mY 1 m(p−1) = (α−N)(mN)−1/m(Y − y1/m) 1 m(p−1) (1 + o(1)). Therefore y = c|τ |− m(1+mN) 1+m , c = 1 mN ((mN)1/m mN−m (1+m)|α−N | ) m(1+mN) 1+m (where p = (1+m)N 1+mN ), which is equivalent to (3.13). (vi) Let λ = 0, mrw′ = o(w), by integrating, we know w+ r|w′| = O(r−k) for any k > 0. Then J ′ 1(r) = (1−α)w is integrable, thus J1 has a limit at 0, and limr→0 rw = 0, thus limr→0 w m−1w′ = c ∈ R, then limr→0 w = a ≥ 0, and limr→0 w ′ = b. Then b ̸= 0, since the regular solutions satisfy (3.1), and a ̸= 0. If a = 0, implies w = br(1+o(1)), then ζ = −m. If λ = η = −1, then from (2.30), w is increasing, thus w has a limit a ≥ 0 at 0, w′ = −aλm−1r−1(1 + o(1)), and by integrating, we know a = 0. And ((wm−1w′)p−1)′ = m1−p(1 − α)w(1 + o(1)), thus wm−1w′ is integrable, so w′ has a limit b ̸= 0. □ Proposition 3.9. (i) Suppose mN ≤ δ < mα, or mN < δ ≤ mα. Then any solution y has a infinite number of zeros near ∞. (ii) Suppose that y has a infinite number of zeros near ±∞. Then either mN < mα < δ and |y| < ℓ, |Y | < (δℓ)p−1 near ±∞ or mN < δ = mα or max(δ,mN, η) < mα. Besides, if δ < mN < mα, then |y| > ℓ at this extremal points. Proof. (i) Suppose that is not the case. Then assume y > 0 for large τ ; we know y is monotone from Proposition 3.6 (iii). Applying Proposition 3.7 with s = ∞, it is contradictory. (ii) Suppose that y is oscillating around 0 near ±∞. Then from (3.5), at the extremal points, |y| 1−mp+m m (δ −mα) < δp−1(δ −mN). (3.15) We can prove that the inequality is strict: if one equality holds, then y is constant. Thus |y| < ℓ from (3.15) and |Y | < (δℓ)p−1 for y′ = 0. And max(mN, η) < mα, from Proposition 3.5, then y has a infinite number of zeros near ±∞. If δ < mN < mα, then |y| > ℓ at its extremal points from (3.5). □ Proposition 3.10. Suppose that mδ + δ −mN ≤ 0. Then any solution y has a finite number of zeros near lnRw. If it is defined near ±∞, and no monotone, then it converges to ±Mℓ. There is no cycle and no homoclinic orbit in R2. Proof. (i) Suppose that y has an infinite number of zeros. Then mα > mN > 0 from Proposition 3.9. If there exists two consecutive zeros ρ1 < ρ2 of w, and τ ∈ (ln ρ1, ln ρ2) is a maximal point of |yd|, from (3.4), then m(p− 1)d(η − d) + (d−mα)e(p+d(p−1)− d m )τ |dyd|2−p|yd| 1−m m ≤ 0. That means, with ρ = eτ ∈ (ρ1, ρ2), ρp|w|1−mp+m(mα− d) ≥ m(p− 1)d(η − d). (3.16) First, fix d < mα, d < η. Then we consider the energy function E(r) defined by (2.27). It is non- increasing. Then E(r) is bounded on (lnRw,∞) ∩ [ρ1,∞), so w is bounded since α > 0. Suppose that there exists a strictly monotone sequence {rn}∞n=1 of consecutive zeros of w, converging to r̄ ∈ [0,∞). If r̄ > 0, we can find sequence {rn}∞n=1, satisfying w > 0 on (r2n−1, r2n) and w < 0 on (r2n, r2n+1). Then there exists sn ∈ (rn, rn+1) such that w′(sn) = 0, since w ∈ C1[0,∞), it implies w(r̄) = w′(r̄) = 0. Since E′(r) is non-increasing, then E(r) ≡ 0, w(r) ≡ 0, we obtain a contradiction from (3.16), because the left-hand side tends to 0, the right-hand side tends to a constant greater than 0. If r̄ = 0, that means Rw = 0. As above, we also reach a contradiction. (ii) Suppose that y is positive near ±∞, and non monotone. Since δ < mN < mα, following Proposition 3.6, we know y oscillates around ℓ. There exists a sequence of extremal points {τn}∞n=1, where y(τn) < ℓ, and |Y (τn)| 2−p p−1Y (τn) = δy(τn), thus (y(τn), Y (τn)) is bounded. It means that EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 17 w(ρn), w ′(ρn) (where ρn = eτn) are bounded, thus E(ρn) is bounded, and E(r) is bounded. Hence, (y, Y ) is bounded. Denote the vector field of system (2.14) by (f1, f2) and defined h(x, y) = 1. Then div(hf1, hf2) = δ+ δ m−N− 1 m(p−1) |y| 1−m m |Y | 2−p p−1 , which is negative ifmδ+δ−mN ≤ 0. Bendixson- Dulac criterion shows that there is no periodic orbit. Thus the trajectory converges toMℓ. Finally, if there exists a homoclinic orbit, then Tr is homoclinic, limτ→−∞(y, Y ) = limτ→∞(y, Y ) = (0, 0), thus limr→0(w,w ′) = limr→∞(w,w′) = (0, 0) and limr→0E = limr→∞E = 0, hence E ≡ 0 and w ≡ 0, which is contradictory. □ Proposition 3.11. Suppose that y is not monotone near ∞ (positive or changing sign), then y and Y are bounded. Proof. If y is changing sign and mN < mα < δ, then |y| is bounded by ℓ from Proposition 3.9. Moreover, if mδ + δ − mN ≤ 0, y also is changing sign, from Proposition 3.10. Thus δ ≤ m m+1N < mN , and δ < mα. If y stays positive, we know δ < mN < mα, following Proposition 3.6. In any case δ < mα. Here we apply the energy function W defined by (2.28). Further W (y, Y ) can be rewritten under the form W (y, Y ) = F (y, Y ) +G(y, Y ) with F (y, Y ) = m|Y |p′ p′ −mδyY + m|δy|p p , G(y, Y ) = m(mα− δ) m+ 1 |y| 1 m−1y2 + (δ −mN)δp−1 p |y|p. (3.17) We find F (y, Y ) ≥ 0, from Young inequality, thus W (y, Y ) ≥ G(y, Y ) > 0 for large y. Then W ′(τ) < 0 whenever (y(τ), Y (τ)) /∈ SL, where SL is given at (2.29). Denote τ0 be arbitrary in the interval of definition of y. Then W (τ) ≤ W (τ0) for any τ such that τ − τ0 ≥ 0 and (y(τ), Y (τ)) /∈ SL. Since SL is bounded, there exists k > 0 large enough such that W (τ) ≤ k for any τ such that τ − τ0 ≥ 0, and (y(τ), Y (τ)) ∈ SL. And we can choose k > W (τ0). Then W (τ) ≤ k for τ − τ0 ≥ 0, Thus y and Y are bounded near ∞. □ Next we show a further sign property. By applying Proposition 3.6 and Proposition 3.7, we can improve Proposition 3.5. Proposition 3.12. Assume −∞ < mα ≤ δ and α < N . Then the regular solutions have a constant sign, y is strictly monotone and limτ→∞ ζ = mα. Moreover, any solution has at most one zero, and then limτ→∞ ζ = mα. The above proposition is proved in the same manner as [5, Proposition 2.13]. 3.4. Behaviour of w near Rw > 0. Proposition 3.13. Let w be any solution of (Ew) with a reduced domain (Rw,∞) (Rw > 0). Then lim r→Rw |r −Rw| p−1 1−mp+mR 1 1−mp+m w w = ( p− 1 1−mp+m ) p−1 1−mp+m , and lim τ→lnRw σ = 1. (3.18) Proof. Following Proposition 3.1, assume that w is decreasing near Rw and limr→Rw w = ∞, thus y > 0 and Y > 0 near lnRw, and limτ→lnRw y = ∞. And σ also is monotone near lnRw, from Proposition 3.6. Thus σ has a limit µ such that µ ∈ [0,∞]. Suppose µ = 0, then Y = o(y1/m) = o(y1/m − Y ), from (2.18) we have (y1/m − Y )′ = ( δ m − α)(y1/m − Y ) + (N − α)Y = ( δ m − α+ o(1) ) (y1/m − Y ), hence y cannot blow up in finite time. Similarly, if µ = ∞, then y1/m = o(Y ) = o(Y − y1/m), and (y1/m − Y )′ = ( δ m −N)(y1/m − Y ) + (N − α)y1/m = ( δ m − α+ o(1) ) (y1/m − Y ), 18 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 which is also contradictory; thus µ ∈ (0,∞). Moreover, if λ is finite, σ = |ζ|p−2ζy− 1−mp+m m converges to 0, which is not true. Thus limτ→lnRw ζ = ∞, hence µ = 1 from equation (3.10). Therefore σ(τ) = −mp−1 |wm−1w′|p−2wm−1w′ rw = 1+o(1), then w− 1 p−1wm−1w′+m−1r 1 p−1 (1+o(1)) = 0, and (3.18) holds. □ 3.5. More information on the stationary points. (i) Höpf bifurcation point. When δ+mδ−mN > 0, a Höpf bifurcation appears at the critical value α = α∗. Then there exist some cycles appearing near α∗, from the Poincaré-Andronov-Hopf theorem in [15]. We find more precise results by applying the Lyapounov test for a week source; it requires an expansion up to the order 3 near Mℓ, in a suitable basis of eigenvectors, where the linearized problem has a rotation matrix. Theorem 3.14. Let δ < mN,mδ + δ − mN > 0 and 1 < p < 2, α = α∗, then Mℓ is a week source; moreover if α < α∗ and α∗ − α is small enough, then there exists a unique limit cycle in Q1, attracting at −∞. Proof. When α = α∗, we can obtain the eigenvalues given by λ1 = −ib, λ2 = ib, with b =√ p m(p−1) (mN − δ). Let u(α) = (δℓ)2−p p− 1 , v(α) = δ m(p− 1) · mN − δ mα− δ , then v(α∗) = δ(mN − δ) m(p− 1)(mα∗ − δ) = mδ + δ −mN m = δ2−pℓ 1−mp+m m m(p− 1) . First we apply the substitution (2.19), which leads to (2.20). The functions Φ and Ψ, which is defined at (2.21) and (2.22), can be expanded near (0, 0) of the form Φ(ȳ, Ȳ ) = B2Ȳ 2 +B3Ȳ 3 + · · · , and Ψ(ȳ, Ȳ ) = C2,0ȳ 2 + C1,1ȳȲ + C0,2Ȳ 2 + C3,0ȳ 3 + C2,1ȳ 2Ȳ + C1,2ȳȲ 2 + C0,3Ȳ 3 + · · · , where B2 = − 2− p 2(p− 1)2 (δℓ)3−2p, B3 = − (2− p)(3− 2p) 6(p− 1)3 (δℓ)4−3p, C2,0 = 1−m 2m3 ℓ 1 m−2(mα− δ + 2mδ), C1,1 = − 1−m m2(p− 1) δ2−pℓ 1 m−p = − 2m mα− δ + 2mδ C2,0u(α), C0,2 = − 2− p 2m(p− 1)2 δ3−2pℓ2−2p+ 1 m , δC0,2 = − (2− p)m2 (1−m)(mα− δ + 2mδ) C2,0u 2(α), C3,0 = (1−m)(1− 2m) 6m4 ℓ 1 m−3(mα− δ + 3mδ) = 2m(1− 2m)(mα− δ + 3mδ) 3(1−m)(mα− δ + 2mδ)2 · C2 2,0u(α) v(α∗) , C2,1 = − (1−m)(1− 2m) 2m3(p− 1) δ2−pℓ−1+ 1 m−p = − 2m2(1− 2m) (1−m)(mα− δ + 2mδ)2 C2 2,0u 2(α) v(α) , C1,2 = − (2− p)(1−m) 2m2(p− 1)2 δ3−2pℓ1+ 1 m−2p, δC1,2 = − 2(2− p)m3 (1−m)(mα− δ + 2mδ)2 C2 2,0u 3(α) v(α) , C0,3 = − (2− p)(3− 2p) 6m(p− 1)3 δ4−3pℓ3+ 1 m−3p, δ2C0,3 = − 2m4(2− p)(3− 2p) 3(1−m)2(mα− δ + 2mδ)2 C2 2,0u 4(α) v(α) , δB2 = − (2− p)m2 (1−m)(mα− δ + 2mδ) C2,0u 3(α) v(α) . Next we use the substitution τ = −θ b , ȳ(τ) = u(α)x1(θ), Ȳ = δx1(θ) + bx2(θ), EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 19 to obtain x′1(θ) = x2 − 1 bu(α) Φ(u(α)x1, δx1 + bx2), x′2(θ) = −x1 − 1 b2 Ψ(u(α)x1, δx1 + bx2) + δ b2u(α) Φ(u(α)x1, δx1 + bx2). They can be written the expansion of order 3 in the form x′1 = x2 + a2,0x 2 1 + a1,1x1x2 + a0,2x 2 2 + a3,0x 3 1 + a2,1x 2 1x2 + a1,2x1x 2 2 + a0,3x 3 2 + · · · , and x′2 = −x1 + b2,0x 2 1 + b1,1x1x2 + b0,2x 2 2 + b3,0x 3 1 + b2,1x 2 1x2 + b1,2x1x 2 2 + b0,3x 3 2 + · · · . We compute the Lyapounov coefficient L = 3a3,0 + a1,2 + b2,1 + 3b0,3 − a2,0a1,1 + b1,1b0,2 − 2a0,2b0,2 − a0,2a1,1 + 2a2,0b2,0 + b1,1b2,0. After simplification, we obtain b(1−m)2(mα− δ + 2mδ)2δ2 4m5C2 2,0u 4(α) L = 1− p mδ + δ −mN [ δp+ (2− p)b2 ] < 0. The property of Mℓ follows from [17], let us notice the fact that θ has the opposite sign of τ . Moreover there exists an unstable limit cycle attracting at −∞ for all α near α∗ such that Mℓ is a sink, that means α < α∗. □ (ii) Node points or spiral points. When system (2.14) has three stationary points, and if Mℓ is a source or sink, then δ < mN . It is a interesting point that Mℓ is a node point. Moreover α∗ corresponds to a spiral point, from (2.23). We find Mℓ is a node point when ∆ ≥ 0 from (2.24), that means δ ≤ mN 1+m − 2 √ mp′(mN−δ) 1+m or δ > mN 1+m − 2 √ mp′(mN−δ) 1+m and α ≤ α1, or δ > mN 1+m + 2 √ mp′(mN−δ) 1+m and α ≥ α2, one verifies α1 < α2, where α1 = δ m + δ(mN − δ) m(p− 1)(mδ + δ −mN + 2 √ mp′(mN − δ)) , α2 = δ m + δ(mN − δ) m(p− 1)(mδ + δ −mN − 2 √ mp′(mN − δ)) . (3.19) Remark 3.15. If δ < mN , then N ≤ α1, and N = α1 ⇔ N = δ m + p′. Also α1 < η m ⇔ δ2 + (4m−N + 3)mδ +m2N > 0. (iii) Nonexistence of cycles. If there exists a cycle O in system (2.14) in R2, then O surrounds at least one stationary point. If it surrounds (0, 0), the associated solutions y have changing sign. If it only surrounds Mℓ, then it stays in Q1, thus y stays positive. In fact, O cannot intersect {(φ, 0), φ > 0} at two points, and similarly {(0, ξ), ξ > 0}, from Remark ??. For suitable values of mα, δ,mN , we can verify that cycles cannot exist by applying Bendixson- Dulac criterion. System (2.14) can be rewritten under the form y′ = f1(y, Y ), Y ′ = f2(y, Y ), (3.20) with f1(y, Y ) = δy − |Y | 2−p p−1Y, f2(y, Y ) = ( δ m −N)Y + |y| 1−m m (αy − 1 m |Y | 2−p p−1Y ). Thus ∂f1 ∂y (y, Y ) + ∂f2 ∂Y (y, Y ) = δ + δ m −N − 1 m(p− 1) |y| 1−m m |Y | 2−p p−1 . (3.21) For example, if mδ + δ −mN ≤ 0, we find that there exists no periodic orbit in R2 by a direct consequence of Bendixson-Dulac criterion, which was proved in Proposition 3.10. Next we consider cycles in Q1. 20 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 Theorem 3.16. Let δ < mN , then δ < mα. When Mℓ is a node point, there exists no cycle or no homoclinic orbit in Q1 if m > 1 and p < 2− (m−1)δ mα−δ+2mδ < 2. Proof. Let us use the linearization (2.19), (2.20). Consider the line L given by the equation Aȳ + Ȳ = 0, where A is a real parameter. The points of L are in Q1 when −(δℓ)p−1 < Ȳ and −ℓ < ȳ. We obtain: Aȳ′ + Ȳ ′ = [ (δℓ)2−p p− 1 A2 + (δ − δ m +N + 1 m(p− 1) δ2−pℓ 1−mp+m m )A + mα+ (m− 1)δ m2 ℓ 1−m m ] ȳ +AΦ(ȳ, Ȳ ) + Ψ(ȳ, Ȳ ). From (2.24), we find an A such that (δℓ)2−p p− 1 A2 + ( δ − δ m +N + 1 m(p− 1) δ2−pℓ 1−mp+m m ) A+ mα+ (m− 1)δ m2 ℓ 1−m m = 0, and A ̸= 0. Furthermore, Φ(ȳ, Ȳ ) ≤ 0 is on L∩Q1. In fact (p−1)Φ(t)′ = ((δℓ)p−1+t) 2−p p−1 −(δℓ)2−p, thus Φ has a maximum 0 on (−(δℓ)p−1,∞) at point 0, and it is non-positive on this interval. Ψ(ȳ, Ȳ ) ≤ 0 is also on L ∩Q1. In fact, Ψȳ = α m (ȳ + ℓ) 1 m−1 − 1−m m2 (ȳ + ℓ) 1 m (Ȳ + (δℓ)p−1) 1 p−1 − (mα+ (m− 1)δ) m2 ℓ 1 m−1, and ΨȲ = − 1 m(p− 1) (ȳ + ℓ) 1 m−1(Ȳ + (δℓ)p−1) 2−p p−1 + 1 m(p− 1) δ2−pℓ1−p+ 1 m , by a computations, we have Ψȳ(0, 0) = 0, ΨȲ (0, 0) = 0, Ψȳȳ(0, 0) = (1−m)(mα− δ + 2mδ) m3 ℓ 1 m−2, ΨȳȲ (0, 0) = − 1−m m2(p− 1) δ2−pℓ 1 m−p, ΨȲ Ȳ (0, 0) = − 2− p m(p− 1)2 δ3−2pℓ2−2p+ 1 m . Hence, applying the extremum principle of binary function, then we know Ψ has a maximum 0 on (−l,∞)∩ (−(δℓ)p−1,∞) at point 0 if m > 1 and p < 2− (m−1)δ mα−δ+2mδ < 2, thus it is non-positive on the interval. Then the orientation of the vector field does not change along L ∩ Q1, which means that there exists no cycle in Q1; and similarly no homoclinic orbit can exist. In the case α = N , then Y ≡ y1/m ∈ [0, ℓ) defines the trajectory Tr, associated to the solutions given by (2.6) with K > 0, and there exists no cycle in Q1, otherwise it would intersect Tr. □ Theorem 3.17. Assume δ < mN , δ − mα < 0 < mδ + δ − mN , 0 < m < 1, 1 < p < 2 and (p − 2)(mα − δ)(mα − δ + 2mδ) +m(1 −m)δ2 < 0. If α − α∗ ≥ 0, there exists no cycle or no homoclinic orbit in Q1. Proof. In this case, Mℓ is a source or a weak source. Suppose there is a cycle in Q1 in system (2.14). Then any trajectory starting from Mℓ has a limit cycle in Q1, with attracting at ∞. Such a cycle is stable; it means that the Floquet integral on the period [0,P] is non-positive. Thus from (3.21),∫ P 0 (∂f1 ∂y (y, Y ) + ∂f2 ∂Y (y, Y ) ) dτ = ∫ P 0 ( δ + δ m −N − 1 m(p− 1) y 1−m m Y 2−p p−1 ) dτ ≤ 0. (3.22) Now from (2.20), 0 = δ ∫ P 0 ȳ dτ − (δℓ)2−p p− 1 ∫ P 0 Ȳ dτ + ∫ P 0 Φ(ȳ, Ȳ ) dτ, and 0 = mα+mδ − δ m2 ℓ 1 m−1 ∫ P 0 ȳ dτ + ( δ m −N − δ2−pℓ1+ 1 m−p m(p− 1) )∫ P 0 Ȳ dτ + ∫ P 0 Ψ(ȳ, Ȳ ) dτ ; EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 21 then ∫ P 0 Ψ(ȳ, Ȳ ) dτ + [δ −mN m − 1 m(p− 1) δ2−pℓ1+ 1 m−p ] p− 1 (δℓ)2−p ∫ P 0 Φ(ȳ, Ȳ ) dτ = [mN − δ m (p− 1)δp−1ℓp−2 − mα− δ m2 ℓ 1 m−1 ] ∫ P 0 ȳ dτ,∫ P 0 Ψ(ȳ, Ȳ ) dτ − mα+mδ − δ m2δ ∫ P 0 Φ(ȳ, Ȳ ) dτ = − (mα− δ m2 δ1−pℓ1+ 1 m−p − δ −mN m )∫ P 0 Ȳ dτ, Let F (ȳ, Ȳ ) = Ψ(ȳ, Ȳ ) + [δ −mN m − 1 m(p− 1) δ2−pℓ1+ 1 m−p p− 1 (δℓ)2−p ] Φ(ȳ, Ȳ ), G(ȳ, Ȳ ) = Ψ(ȳ, Ȳ )− mα+mδ − δ m2δ Φ(ȳ, Ȳ ). We can show that F (ȳ, Ȳ ) has a minimum at point (0, 0), if 0 < m < 1, 1 < p < 2 and (p − 2)(mα − δ)(mα − δ + 2mδ) +m(1 −m)δ2 < 0. Simultaneously, G(ȳ, Ȳ ) also has a minimum at point (0, 0). Thus F (ȳ, Ȳ ) and G(ȳ, Ȳ ) are nonnegative in Q1. Hence,∫ P 0 ȳ dτ < 0 ⇒ y < ℓ, ∫ P 0 Ȳ dτ < 0. Since y′ = δy − Y 1 p−1 , we have ∫ P 0 Y 1 p−1 dτ = δ ∫ P 0 y dτ < δℓP. (3.23) From (3.22), (3.23) and the Hölder inequality, we find that (δ + δ m −N)P ≤ 1 m(p− 1) ∫ P 0 y 1−m m Y 2−p p−1 dτ ≤ 1 m(p− 1) δ2−pℓ1+ 1 m−pP = δ(mN − δ) m(p− 1)(mα− δ) P, thus α−α∗ < 0, which is contradictory. Next suppose there exists an homoclinic orbit. From [17, P303 Theorem 9.3.3],the saddle connection is repelling, since the sum of the eigenvalues λ1, λ2 of the linearized problem at (0, 0) is δ + δ m −N(> 0). Then the solutions just inside the cycle spiral toward the loop near −∞. We know Mℓ is a source, or a weak source, such solutions have a limit cycle attracting at ∞, which is contradictory. □ Theorem 3.18. Assume δ < mN and δ−mα < 0 < δ+ δ m −N . If mα ≤ η, there exists no cycle or no homoclinic orbit in Q1. Proof. Suppose system (2.14) admits a cycle in Q1. (i) Let mα ≤ η. Mℓ is a sink when α < α∗, thus any trajectory, which converges to Mℓ at ∞, has a limit cycle O in Q1, attracting at −∞. Denote (y, Y ) be any solution of orbit O, of period P. Then O is unstable, thus the Floquet integral is nonnegative. Following (3.21) we obtain tP0 ( δ + δ m −N − 1 m(p− 1) y 1−m m Y 2−p p−1 ) dτ ≥ 0. Moreover y is bounded from above and below; thus ymα, defined by (2.10) with d = mα, satisfies limτ→−∞ ymα = 0, limτ→∞ ymα = ∞. From (3.9), we know y′′mα > 0 for mα ≤ η, then it has only 22 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 minimal points. Thus y′mα > 0 in R. From (2.11) and (2.12) with d = mα, y′′mα y′mα + (η − 2mα) + 1 m(p− 1) y 1−m m mα Y 2−p p−1 mα = mα(η −mα)ymα y′mα = mα(η −mα)ymα mαymα − Y 1 p−1 mα > η −mα. Integrating on [0,P] it implies η − 2mα + δ + δ m − N > η − mα, which is contradictory to δ −mN + δ −mα < 0. (ii) Suppose system (2.14) admits an homoclinic orbit in Q1. Since δ < mN , (0, 0) is a saddle point, thus Tr is the only trajectory starting from (0, 0) in Q1, and there exists the only one trajectory Ts converging to (0, 0) in Q1 for large τ , with limτ→∞ Y y1/m = ∞, and limr→∞ rηwm = c > 0. We know Tr satisfies limτ→−∞ e−mατymα(τ) = am > 0, thus limτ→−∞ ymα = 0; and ymα has only minimal points. As above, Tr is monotone increasing and positive, and Ts satisfies limτ→∞ e(η−mα)τymα = c > 0. If mα < η, then limτ→∞ ymα = 0, thus Tr ̸= Ts. If mα = η, Ts is a explicit solution given by (2.7), which means ymα is constant, thus again Tr ̸= Ts. □ (iv) Boundedness of cycles. When there exist cycles, apart from a few cases, we cannot prove their uniqueness, hence we pay attention to the following properties. Theorem 3.19. When it is nonempty, the set C of all the cycles of system (2.14) is bounded in R2. Proof. Assume that there exists a cycle in R2. From Proposition 3.5, 3.6, 3.9 and 3.10, it can happen only in three cases: mN < mα < δ,mN < δ = mα, max(δ,mN, η) < mα and δ > mN 1+m . In the first case, the set C is bounded and contained in (−ℓ, ℓ)×(−(δℓ)p−1, (δℓ)p−1), from Proposition 3.9. In other cases we apply the energy functionW . The trajectoryO corresponding to the solution is denoted by (y, Y ). Then W is periodic, and its extremal points are exactly the points of the curve L. In fact, if there exists an extremal point τ1 such that W ′(τ1) = 0 and (y(τ1), Y (τ1)) /∈ L, and it is on the curve M defined at (2.16). Thus, y′(τ1) = 0, and y′′(τ1) ̸= 0, since O can not reduced to a stationary point. Consequently, (δy−|Y | 2−p p−1Y )(|δy|p−2δy−Y ) > 0 near τ1, then W ′ has a constant sign, and τ1 is not an extremal point. By this means, we obtain estimates for W independent of the trajectory: max τ∈R |W (τ)| =M = max (y,Y )∈L |W (y, Y )|. At the maximal points τ of y, we obtain |Y (τ)| 2−p p−1Y (τ) = δy(τ), thus W (τ) = (mδ + δ −mN)δp−1 p |y|p + m|Y |p′ p′ −mδyY + m(mα− δ) m+ 1 |y| 1 m−1y2 = (δ −mN)δp−1 p |y|p + m(mα− δ) m+ 1 |y| 1 m−1y2. In any case, y is bounded and independent of the trajectory, and m|Y |p′ p′ ≤ mδyY + (mδ + δ −mN)δp−1 p |y|p + m(mα− δ) m+ 1 y 1 m−1y2 +M, Hence Y is also uniformly bounded, and C is bounded. □ 4. Classification of self-similar solutions We classify the self-similar solutions to the polytropic filtration equation (1.1) and list them in a similar way as the self-similar solutions for p-Laplace equation in [5], which is the special case m = 1 for the polytropic filtration equation (1.1). Note that for non-Newtonian polytropic filtration equation (1.1), bothm and p have influence on the asymptotic behavior and classification of self-similar solutions for the singular casem(p−1) < 1, which is the main cause of the difficulties of this paper. EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 23 4.1. General properties. Lemma 4.1. Assume −∞ < max(mα,mN) < δ. Then in the phase plane (y, Y ), there exist (i) a trajectory T1 converging to Mℓ at ∞, such that y is increasing as long as it is positive; (ii) a trajectory T2 in Q1∪Q4 converging toMℓ at −∞, and unbounded at ∞, with limτ→∞ ζ = mα; (iii) a trajectory T3 converging to Mℓ at −∞, such that y has at least one zero; (iv) a trajectory T4 in Q1, converging to Mℓ at ∞, with limτ→lnRw Y y1/m = 1 (Rw > 0); (v) a trajectory T5 in Q1 ∪Q4 unbounded at ±∞, with limτ→∞ ζ = mα,limτ→lnRw Y y1/m = 1. The proof is similarly to the proof of [5, Lemma 3.1]. We omit it. Next we study the various global behaviours, according to the value of α. We describe the solutions of (Ew) by phase plane analysis. -20 -10 10 20 y -20 -10 10 20 Y -20 -10 10 20 y -20 -10 10 20 Y α = 0, N = 2 α = 1, N = 2 -20 -10 10 20 y -20 -10 10 20 Y -40 -30 -20 -10 10 20 30 40 y -40 -30 -20 -10 10 20 30 40 Y α = −1, N = 2 α = 2, N = 2 Figure 1. Phase planes with m = 0.8 and p = 1.8) 4.2. Subcase mα ≤ mN < δ. Theorem 4.2. Suppose −∞ < mα ≤ mN < δ. Then the regular solutions w have a con- stant sign, and limr→∞ rα|w| = L > 0 if α < N , especially limr→∞ |w| = L > 0 if α = 0, limr→∞ rδ|w|m−1w = ℓ if α = N . And w(r) = ℓ1/mr−δ/m is also a solution. There exist solutions satisfying one of the following properties: 24 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 (1) (Only if α < N) w is positive, limr→0 r η mw = c > 0 if N ≥ 2 (and (3.14) holds with a > 0 > b if N = 1), and limr→∞ rδ|w|m−1w = ℓ; (2) w is positive, limr→0 r δ|w|m−1w = ℓ, limr→∞ rαw = L > 0; (3) w has precisely one zero, limr→0 r δ|w|m−1w = ℓ, limr→∞ rαw = L < 0; (4) w is positive, Rw > 0, limr→∞ rδ|w|m−1w = ℓ; (5) w is positive, Rw > 0, limr→∞ rαw = L > 0; (6) w has one zero, Rw > 0, limr→∞ rαw = L ̸= 0; (7) (Only if α < N) w is positive, limr→0 r η mw = c > 0 if N ≥ 2 (and (3.14) holds with a > 0 > b if N = 1), and limr→∞ rαw = L > 0; (8) w has one zero, with limr→0 r η mw = c > 0 if N ≥ 2 (and (3.14) holds with a > 0 > b if N = 1), and limr→∞ rαw = −L < 0; (9) N = 1, w > 0 and (3.14) holds with a ≥ 0, b > 0 and limr→∞ rαw = L. Up to symmetry, all the solutions of (Ew) are described. The proof of the above theorem is similarly to that of [5, Theorem 3.2]. See the illustration Figure 1. 4.3. Subcase mN < mα < δ. In this case, we show that system (2.14) admits some periodic trajectories, according to the value of α with respect to α∗. Notice that N < α∗ whenever δ2 − (m + N + 2)mδ + m2N > 0 from (2.26). Our adopted method is the Poincaré-Bendixson theorem, by applying the level curves of the energy function W . Lemma 4.3. Suppose mN < mα < δ. Consider the level curves Ck = {(y, Y ) ∈ R2 |W (y, Y ) = k}(k ∈ R) of the function W defined at (2.28), which are symmetric with respect to (0, 0). Let kℓ =W (ℓ, (δℓ)p−1) = m m+ 1 (δ −mN)δp−2ℓp. (4.1) If k > kℓ, there are two unbounded connected components in Ck. If 0 < k < kℓ, there are three connected components in Ck, and one of them is bounded. If k = kℓ, Ckℓ is connected with a double point at Mℓ. If k = 0, and one of the three connected components of C0 is {(0, 0)}. If k < 0, there are two unbounded connected components in Ck. Proof. The above assertion for the special case m = 1 is proved in [5]. Here we can only consider the study of Ck to the set y > 0. Define a function φ(s) = m|s|p ′ p′ −ms+ m p , for any s ∈ R, which has the inverse function. Then we analysis the properties of the function φ(s) and its inverse function to obtain the results. The rest proof is similar as the proof of [5, Lemma 3.3]. □ Theorem 4.4. Suppose 0 < m < 1, 1 < p < 2 and mN < mα < δ. Then w(r) = ℓ1/mr−δ/m is a solution. Moreover, (i) If α ≤ α∗, then any solution of (Ew) has at most a finite number of zeros. (ii) There exist α̂ such that max(mN,mα∗) < mα̂ < δ, if α > α̂, there is a cycle around (0, 0) in the phase plane (y, Y ). (iii) Consider any α such that there exists no such cycle. Then the regular solutions have a finite number of zeros and limr→∞ rαw = Lr ̸= 0 or limr→∞ rδ|w|m−1w = ℓ. There exist solutions of type (2)-(6) of Theorem 5.2, and other solutions have one of the following properties: (1’) (only if Lr ̸= 0) limr→∞ rαw = Lr ̸= 0 and limr→0 r η mw = c ̸= 0 (or (3.14) holds if N = 1); (7’) limr→0 r η mw = c ̸= 0 (or (3.14) holds if N = 1) and limr→∞ rαw = L ̸= 0. (iv) Consider any α such that there exists such a cycle, thus there exists solutions w with oscillating near 0 and ∞, satisfying rδ|w|m−1w is periodic in ln r. The regular solutions w oscillate near ∞, satisfying rδ|w|m−1w is asymptotically periodic in ln r. There exist solutions of type (2), (4), (5) of Theorem 4.2, and other solutions have one of the following properties: EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 25 -40 -20 20 40 y -40 -20 20 40 Y -100 -50 50 100 y -100 -50 50 100 Y N = 2 < α = 2.32 < δ m N = 2 < α = 3.8 < δ m Figure 2. Phase planes with m = 0.8 and p = 1.8 (1”) with precisely one zero, Rw > 0, and limr→∞ rδ|w|m−1w = ℓ; (3”) with limr→0 r δ|w|m−1w = ℓ, and oscillating near ∞; (9) with limr→0 r η mw = c ̸= 0 (or (3.14) holds if N = 1) and oscillating near ∞. (10) with precisely one zero, Rw > 0, and limr→∞ rαw = L ̸= 0; (11) with Rw > 0 and oscillating near ∞. Proof. First observe (Ew) always admits solutions of type (2), (4), (5) of Theorem 4.2, from Lemma 4.1. (i) Suppose α ≤ α∗ (Fig. 2). Consider any trajectory T . If y has an infinity of zeros near ±∞, then following Proposition 3.9, T is contained in the set D = { (y, Y ) ∈ R2 | |y| < ℓ, |Y | < (δℓ)p−1 } near ±∞. It means that T is bounded near ±∞, hence the limit set at ±∞ is contained D. However Mℓ /∈ D; and (0, 0), which is a source, a node point, can not belong to the limit set Γ at ∞. Indeed, according to Poincaré-Bendixson theorem in [17], Γ is a closed orbit, so system (2.14) admits a cycle. Moreover, from (3.20) and (3.21), ∂f1 ∂y (y, Y ) + ∂f2 ∂Y (y, Y ) = 1 m(p− 1) (D 1−m m (δD)2−p − |y| 1−m m |Y | 2−p p−1 ), thus applying Bendixson-Dulac criterion, there exists no cycle in the set {|y| < D, |Y | < (δD)p−1}, where D = (δp−2(p− 1)(mδ + δ −mN)) m 1−mp+m and 0 < m < 1, 1 < p < 2. Now we find that α ≤ α∗ ⇔ ℓ ≤ D, then (δℓ)p−1 ≤ (δD)p−1. (4.2) Hence there exists no cycle in D, which is contradictory. (ii) Assume α > max(N,α∗). The curve L intersects M at point (D, (δD)p−1). Then SL ∩M = { (θD, δp−1(θD)p−1) : θ ∈ [0, 1] } ; and D < ℓ from (4.2), thus Mℓ is not contained in SL. We can find k1 > 0 small enough satisfying Cb k1 is contained in SL. Next we look for k ∈ (0, kℓ) satisfying L is in the domain restricted by Cb k. By symmetry, we only consider the points of L such that y ≥ 0. By a straightforward calculation, it means that W (y, Y ) ≤ KδpDp, where K = max( 2p , 2mδ+δ−mN p ). Let α̂ = α̂(δ,N) be given by KδpDp = kℓ, then δ −mα̂ = [m(δ −mN) (m+ 1)Kδ2 ] 1−mp+m mp δ −mN δp−3(p− 1)(mδ + δ −mN) . If α > α̂, we can find k2(< kℓ) such that L is contained in the set { (y, Y ) ∈ R2 |W (y, Y ) < k2 } , which has three connected components. Since SL is connected, it is contained in Cb k2 . Hence SL is delimitated by Cb k1 and Cb k2 , it implies SL is bounded and positively invariant. There exist no 26 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 any stationary point in SL, thus it contains a cycle, from the Poincaré-Bendixson theorem (see Fig. 2). The rest of the proof follows similarly as the proof of Theorem 3.4(iii) and (iv) in [5]. □ Remark 4.5. From the numerical studies, in a similar way as the classification of self-similar solutions of singular p-Laplacian equations by Bidaut-Véron [5], here for the polytropic filtration equations we also conjecture that α̂ is unique, and the number of zeros of w is increasing with α ∈ (N, α̂); and moreover there exists α1 = N < α2 < · · · < αn < αn+1 < · · · , such that the regular solutions have n zeros for any α ∈ (αn, αn+1), with limr→∞ rαw = Lr ̸= 0, and n+1 zeros for α = αn+1, with limy→∞ rδwm = ±ℓ. 4.4. Subcase mα ≤ δ ≤ mN . In this case, system (2.14) admits a unique stationary point (0, 0), and N ≥ 2. -3 -2 -1 1 2 3 y -3 -2 -1 1 2 3 Y -3 -2 -1 1 2 3 y -3 -2 -1 1 2 3 Y α = −2, N = 6 α = 2, N = 6 Figure 3. Phase planes with m = 0.8 and p = 1.8 Theorem 4.6. Assume −∞ < mα ≤ δ ≤ mN and α ̸= N . Then the regular solutions, defined on (0,∞), have a constant sign, and the positive ones satisfy limr→∞ rαw = L > 0 if mα ̸= δ, or (3.12) holds if mα = δ. All the other solutions have a reduced domain (Rw > 0). Among them, the solutions satisfy one of the following properties: (1) w is positive, limr→∞ r η mw = c ̸= 0 if δ < mN , or limr→∞ rN (ln r) 1+mN 1+m w = ρ̃ defined at (3.13) if δ = mN . (2) w is positive, limr→∞ rαw = L > 0 if mα ̸= δ, of (3.12) holds if mα = δ. (3) w has one zero, such that limr→∞ rαw = L ̸= 0 if mα ̸= δ, or (3.12) holds if mα = δ. Up to symmetry, all the solutions are described. The proof is similarly as that of [5, Theorem 3.5]. See the illustration in Figure 3. 4.5. Subcase mN ≤ δ ≤ mα. Theorem 4.7. Suppose mN ≤ δ ≤ mα and α ̸= N . Then (i) There is a cycle surrounding (0, 0) in system (2.14), thus the corresponding solution has a changing sign such that rδ|w|m−1w is periodic in ln r. All the other solutions w, in par- ticular the regular ones, are oscillating near ∞, and rδ|w|m−1w is asymptotically periodic in ln r. Equation (Ew) admits solutions w such that limr→0 r η mw = c ̸= 0 if 2 ≤ N < δ m and (3.13) holds if mN = δ or (3.14) holds if N = 1. (ii) Equation (Ew) admits solutions such that Rw > 0 or limr→0 r αw = L ̸= 0 if mα ̸= δ, or (3.12) holds if mα = δ. The proof is similarly as that of [5, Theorem 5.1]. See the illustration in Figure 4. EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 27 -100 -50 50 100 y -100 -50 50 100 Y -40 -20 20 40 y -40 -20 20 40 Y α = 6, N = 2 α = 6, N = 5 Figure 4. Phase planes with m = 0.8 and p = 1.8 -30 -20 -10 10 20 30 y -30 -20 -10 10 20 30 Y Figure 5. Phase plane with α = N = 5, m = 0.8, and p = 1.6 Theorem 4.8. Suppose mα = δ = mN . Then the regular solutions, given by (2.6), have a constant sign. For any k ∈ R, w(r) = |k| 1 m−1kr−N is a solution. There are solutions satisfying one of the following properties: (i) w is positive, limr→0 r Nw = c1 > 0 and limr→∞ rNw = c2 > 0(c1 < c2); (ii) w has one zero, limr→0 r Nw = c1 > 0 and limr→∞ rNw = c2 < 0; (iii) w is positive, Rw > 0, and limr→0 r Nw = c ̸= 0; (iv) w has one zero, Rw > 0,and limr→∞ rNw = c ̸= 0. Up to a symmetry, all the solutions are described. The proof is similar to that of [5, Theorem 5.2 ]. See the illustration Figure 5. Note that for δ = mN , (2.5) is equivalent to Y ≡ y1/m − C, from (2.17). For any k ∈ R, we find (y, Y ) ≡ (k, |mNk|p−2mNk), on the curve M, is a solution of system (2.14), 4.6. Subcase δ < min(mα,mN). In this case, the system admits three stationary points, (0, 0), is a saddle point, and Mℓ,M ′ ℓ are sinks when δ ≤ mN 1+m or δ > mN 1+m and α < α∗, and sources when δ > mN 1+m and α > α∗, and node points whenever δ > mN 1+m − 2 √ mp′(mN−δ) 1+m and α ≤ α1, or δ > mN 1+m + 2 √ mp′(mN−δ) 1+m and α ≥ α2, where α1, α2 are defined at (3.19). From Remark 3.15, α1 can be greater or less than η m . A very interesting point is that there exist two types of periodic 28 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 trajectories in system (2.14), either surrounding (0, 0), coinciding with changing sign solutions, or located in Q1 or Q3, coinciding with constant sign solutions. We know that δ < mN implies δ < mN < η from (2.1). And mN 1+m < δ implies η m < α∗ from (2.26). In a similar way as the classification of self-similar solutions of singular p-Laplace equations by Bidaut-Véron [5], here for the polytropic filtration equations we also show some general properties of the phase plane firstly. Remark 4.9. (i) The trajectory Tr starts in Q1. Because (0, 0) is a saddle point, system (2.14) admits a unique trajectory Ts converging to (0, 0), in Q1 for large τ , and limr→∞ r η mw = c > 0, from Proposition 3.7 and 3.8. Furthermore, if Tr does not stay in Q1, then Ts stays in Q1, and Ts is bounded and in the interior of the domain delimitated by Q1 ∪ Tr, from Remark ??. If Tr is homoclinic, it stays in Q1. (ii) Any trajectory T , where y is not monotone near ∞, is bounded near ∞ from Proposition 3.11. According to the Poincaré-Bendixson theorem, and trajectory, which is T bounded at ±∞, either converges to (0, 0) or ±Mℓ, or the limit set Γ± at ±∞ is cycle, or Γ± is homoclinic hence T = Tr, Γ± = Tr. (iii) If system (2.14) admits a limit cycle surrounding (0, 0), then it also surrounds the points ±Mℓ from (3.15). First consider the case mα ≤ η, where there exists no cycle or no homoclinic orbit in Q1, from Theorem 3.17. -3 -2 -1 1 2 3 y -3 -2 -1 1 2 3 Y -2 -1.5 -1 -0.5 0.5 1 1.5 2 y -2 -1.5 -1 -0.5 0.5 1 1.5 2 Y α = 5, N = 4 α = 10, N = 4 Figure 6. Phase planes with m = 0.5, p = 1.5, and η = 5 Theorem 4.10. Suppose δ < min(mα,mN), and mα ≤ η. Then the regular solutions have a constant sign, and limr→∞ rδ|w|m−1w = ℓ. And w(r) = ℓ1/mr−δ/m is a solution. There are the solutions satisfying one of the following properties: (i) If mα < η, there exist solutions such that (1) w is positive, limr→0 r αw = L and limr→∞ rδ|w|m−1w = ℓ; (2) w is positive, Rw > 0, and limr→∞ r η mw = c > 0; (3) w is positive, Rw > 0, and limr→∞ rδ|w|m−1w = ℓ; (4) w has one zero, Rw > 0, and limr→∞ rδ|w|m−1w = ℓ; (ii) If mα = η, then w = Cr−η/m is a solution. The equation (Ew) admits the solutions of type (4), but not of type (2) or (3). Proof. (i) For the case when mα < η, the proof follows similarly as that of [5, Theorem 5.4]. See the illustration in Figure 6. (ii) For the case when mα = η (see Figure 6), there exists no positive solution satisfying Rw > 0, thus no solution of type (2) or (3). Indeed all the trajectories stay under Ts, and Ts is defined by the equation ζ ≡ η, that means w ≡ cr−η/m for (2.30), or Ts is equivalent to EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 29 Yη ≡ c, Y ′ η ≡ 0. Consider any trajectory T[P ] running through some point P = (φ, 0), φ > 0, and the solution (y, Y ) starting from P at time 0. Then Yη(0) = 0, and Yη < 0 on (−∞, 0) thus Y ′ η = 1 me 1−mp+m m (δ−η)y 1−m m η (ηy − |Y | 2−p p−1 ) > 0 in Q4. Suppose it satisfies Rw = 0. Then T[P ] starts from Q3 with limτ→−∞ ζ = mα = η and limτ→−∞ yη = −L < 0, then limτ→−∞ Yη = −(mα|L|)p−1. From (2.12), a straightforward calculation shows y′′η = ηy′η − e(p+η(p−1)− η m )τ m(p− 1) |y′η − ηyη|2−p|yη| 1 m−1y′η Then y′′η < 0 near −∞, which is contradictory. Thus Rw > 0 and w is of type (4). □ We now pay more attention to the interesting case, where η < mα. Lemma 4.11. Assume δ < min(mα,mN) and η < mα. If δ > mN 1+m and α < α∗ and Ts stays in Q1, then it has a limit cycle at −∞ in Q1, or it is homoclinic. The proof is to that of [5, Lemma 5.5]. Here we omit i. -30 -20 -10 10 20 30 y -30 -20 -10 10 20 30 Y -10 -5 5 10 y -10 -5 5 10 Y α = 6.6, mN < η < mα α = 6.9, mN < η < mα -15 -10 -5 5 10 15 y -15 -10 -5 5 10 15 Y -15 -10 -5 5 10 15 y -15 -10 -5 5 10 15 Y α = 7 < α∗ = 7.083 α = 7.2 > α∗ = 7.083 Figure 7. Phase planes with m = 0.8, p = 1.8, N = 6, andη = 5.25 Theorem 4.12. Suppose mN 1+m ≤ δ < min(mα,mN) and 1 < p < 2. Then w(r) = ℓ1/mr−δ/m is still a solution. Moreover (i) There exists a (maximal) critical value αcrit of α, such that max( η m ,α1) < αcrit < α∗, 30 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 and the regular trajectory is homoclinic: all the regular solutions have a constant sign and satisfy limr→∞ r η mw = c ̸= 0. (ii) For any α ∈ (αcrit, α ∗), there does exist a unique cycle in Q1, which implies that the equation (Ew) admits positive solutions w such thatrδ|w|m−1w is asymptotically periodic in ln r near 0 and limr→∞ rδ|w|m−1w = ℓ ̸= 0. The equation (Ew) also has positive solutions such that rδ|w|m−1w is asymptotically periodic in ln r near 0 and limr→∞ r η mw = c > 0. (iii) For any α ≥ α∗, there does not exist such a cycle in Q1, but the equation (Ew) admits positive solutions such that limr→0 r δ|w|m−1w = ℓ and limr→∞ r η mw = c > 0. (iv) For any α > αcrit, there exists also a cycle around (0, 0) and ±Mℓ, thus rδ|w|m−1w is changing sign and periodic in ln r. The regular solutions, are changing sign and oscillating at ∞, and rδ|w|m−1w is asymptotically periodic in ln r. There are solutions in the equation (Ew) satisfying Rw > 0 or limr→0 r αw = L ̸= 0, and oscillating at ∞, and rδ|w|m−1w is asymptotically periodic in ln r. The proof is similarly to that of [5, Theorem 5.6 ]. See the illustration Figure 7. Note that for non-Newtonian polytropic filtration equation (1.1), both m and p exert critical influence on the asymptotic behavior. The result in Theorem 3.14 is valid under the condition 1 < p < 2. And when applying Theorem 3.17, we should pay attention to the fact that its conclusion holds exclusively under the conditions 0 < m < 1, 1 < p < 2 and (p−2)(mα− δ)(mα− δ+2mδ)+m(1−m)δ2 < 0. Remark 4.13. An open question is the uniqueness of αcrit. It can be shown that if there exist two critical values α1 crit > α2 crit, then the first orbit is contained in the second one. In the case δ ≤ mN 1+m , there exists no cycle in R2 from Proposition 3.10, and we obtain the following result. Theorem 4.14. Suppose δ ≤ mN 1+m and δ < mα. Then the regular solutions have a constant sign, and limr→∞ rδ|w|m−1w = ℓ. All the solutions have a finite number of zeros. And w(r) = ℓ1/mr−δ/m is a solution. Moreover, if mα ≤ η, the results were shown in Theorem 4.10. If η ≤ mα, there exist at least one zero in all the other solutions. There exist solutions, such that limr→∞ r η mw = c ̸= 0, with a number n of zeros, for n ≥ 1. All the other solutions satisfy limr→∞ rδ|w|m−1w = ℓ, with n or n+ 1 zeros, for n ≥ 1. The iss similarly to that of [5, Theorem 5.7 ]. Acknowledgments. S. Ji was supported by the NSFC Grant Nos. 12271178 and 12171166, by the Guangzhou Basic and Applied Basic Research Foundation No. 2024A04J2022, and by the TCL Young Scholar (2024-2027). J. Yin was supported by the NSFC Grant No. 12171166, by the Natural Science Foundation of Guangdong Province No. 2025A1515012026, and by Jiangxi Provincial Natural Science Foundation No. 20243BCE51015. The data that supporting the findings of this study are available from the corresponding author upon request. References [1] J. Aguirre, M. Escobedo, E. Zuazua; Self-similar solutions of a convection diffusion equation and related semilinear elliptic problems, Communications in Partial Differential Equations, 1990, 15(2): 139-157. [2] G. Akagi; Stability of non-isolated asymptotic profiles for fast diffusion, Communications in Mathematical Physics, 2016, 345(1): 77-100. [3] A. Audrito, J.L. Vázquez; The Fisher-KPP problem with doubly nonlinear diffusion, Journal of Differential Equations, 2017, 263(11): 7647-7708. [4] G. I. Barenblatt; On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh, 1952, 16: 679-698. [5] M. F. Bidaut-Véron; Self-similar solutions of the p-Laplace heat equation: the fast diffusion case, Pacific Journal of Mathematics, 2006, 227(2): 201-269. [6] M. F. Bidaut-Véron; Self-similar solutions of the p-Laplace heat equation: the case when p > 2, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 2009, 139(1): 1-43. [7] M.F. Bidaut-Véron; The p-Laplace heat equation with a source term: self-similar solutions revisited, Advanced Nonlinear Studies, 2006, 6(1): 69-108. EJDE-2025/63 SINGULAR POLYTROPIC FILTRATION EQUATIONS 31 [8] A. Bouzelmate, A. Gmira, G. Reyes; Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation, Electronic Journal of Differential Equations, 2007, 67: 1-21 [9] A. Bouzelmate, M. El Hathout; Existence and global behavior of decaying solutions of a nonlinear elliptic equation, WSEAS Transactions on Mathematics, 2020, 19: 662-675. [10] E. Chasseigne, J. L. Vázquez; Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from Singularities, Archive for Rational Mechanics and Analysis, 2002, 164(2): 133-187. [11] E. Di Benedetto, M. A. Herrero; Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy-problem when 1 < p < 2, Archive for Rational Mechanics and Analysis, 1990, 111(3): 225-290. [12] E. Di Benedetto; Degenerate parabolic equations, Springer Science & Business Media, 2012. [13] M. Escobedo, E. Zuazua; Self-similar solutions for a convection-diffusion equation with absorption in RN , Israel Journal of Mathematics, 1991, 74(1): 47-64. [14] R. K. Giri, J. Kinnunen, K. Moring; Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case, Nonlinear Differential Equations and Applications, 2021, 28: 1-21. [15] J. K. Hale, H. Koaçk; Dynamics and bifurcations, Springer Science & Business Media, 2012. [16] Y. Z. Han, W. J. Gao; Extinction and non-extinction for a polytropic filtration equation with a nonlocal source, Applicable Analysis, 2013, 92(3): 636-650. [17] J. H. Hubbard, B. H. West; Differential equations: a dynamical systems approach: higher-dimensional systems, Springer Science & Business Media, 2012. [18] K. M. Hui, J. Park; Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space, Discrete and Continuous Dynamical Systems, 2021, 41(11): 5473-5508. [19] K. M. Hui, S. Kim; Vanishing time behavior of solutions to the fast diffusion equation, Communications in Analysis and Geometry, 2023, 31(2): 243-290. [20] R. G. Iagar; Some results on regularity, self-similarity and asymptotic behavior for nonlinear difusion equations, Universidad Autónoma de Madrid, 2010. [21] R. G. Iagar, M. Latorre, A. Sánchez; Eternal solutions in exponential self-similar form for a quasilinear reaction- diffusion equation with critical singular potential, Discrete and Continuous Dynamical Systems, 2024, 44(5): 1329-1353. [22] C. H. Jin, J. X. Yin; Critical exponents and non-extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources, Nonlinear Analysis: Theory, Methods & Applications, 2007, 67(7): 2217-2223. [23] C.H. Jin, J.X. Yin, Y.Y. Ke, Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources, Proceedings of the Edinburgh Mathematical Society, 2009, 52(2), 419-444. [24] A. S. Kalashnikov; Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Mathematical Surveys, 1987, 42(2): 169-222. [25] L. S. Leibenson; General problem of the movement of a compressible fluid in a porous medium, Izvestia Akad. Nauk SSSR, 1983, 9(1): 7-10. [26] Z.P. Li, C.L. Mu, Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux, Journal of Mathematical Analysis and Applications, 2008, 346(1): 55-64. [27] Z. P. Liu, S. M. Ji; Classification of self-similar solutions of the degenerate polytropic filtration equations, Acta Mathematica Scientia, 2025, 45B(2): 1-21. [28] M. M. Porzio, V. Vespri; Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, Differential Equations, 1993, 103(1): 146-178. [29] D. Stan, J. L. Vázquez; Asymptotic behaviour of the doubly nonlinear diffusion equation ut = ∆pum on bounded domains, Nonlinear Analysis: Theory, Methods & Applications, 2013, 77: 1-32 [30] J. L. Vázquez, L. Véron; Different kinds of singular solutions of nonlinear parabolic equations, Nonlinear Problems in Applied Mathematics (SIAM), 1996: 240-249. [31] J. L. Vázquez; Smoothing and decay estimates for nonlinear diffusion equations: equations of porous medium type, OUP Oxford, 2006. [32] Y. F. Wang, J. X. Yin; Critical extinction exponents for a polytropic filtration equation with absorption and source, Mathematical Methods in the Applied Sciences, 2013, 36(12): 1591-1597. [33] T. P. Witelski, A. J. Bernoff; Self-similar asymptotics for linear and nonlinear diffusion equations, Studies in Applied Mathematics, 1998, 100(2): 153-193. [34] Z. Q. Wu, J. N. Zhao, J. X. Yin, H. L. Li; Nonlinear diffusion equations, World Scientific, 2001. [35] T. Y. Xu, S. M. Ji, M. Mei, J. X. Yin; Convergence to sharp traveling waves of solutions for Burgers-Fisher-KPP equations with degenerate diffusion, J. Nonlinear Sci., 34(44), 2024, doi:10.1007/s00332-024-10021-x. [36] H. L. Ye, J. X. Yin’ Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection, Journal of Mathematical Analysis and Applications, 2015, 421(2): 1225-1237. [37] H. J. Yuan, S. Z. Lian, C. L. Cao, W.J. Gao, X. J. Xu; Extinction and positivity for a doubly nonlinear degenerate parabolic equation, Acta Mathematica Sinica, 2007, 23(10): 1751-1756. [38] P. Zheng, C. L. Mu; Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption, Mathematical Methods in the Applied Sciences, 2013, 6(36): 730-743. [39] J. Zhou, C. L. Mu; Global existence and blow-up for a non-Newton polytropic filtration system with nonlocal source, The ANZIAM Journal, 2008, 50(1): 13-29. 32 Y. ZHENG, J. YIN, S. JI EJDE-2025/63 Yanzhi Zheng School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631, China Email address: zhengyanzhi51@163.com Jingxue Yin School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631, China Email address: yjx@scnu.edu.cn Shanming Ji (corresponding author) School of Mathematics, South China University of Technology, Guangzhou, Guangdong, 510641, China Email address: jism@scut.edu.cn 1. Introduction 2. Main results and related systems 2.1. Explicit solutions 2.2. Main results 2.3. Different formulations of the problem 2.4. Stationary points and energy functionals of system (2.14) 2.5. Other systems for positive solutions 3. Global solutions 3.1. Existence of solutions of equation (Ew) 3.2. Sign properties 3.3. Behaviour of w near 0 or 3.4. Behaviour of w near Rw>0 3.5. More information on the stationary points 4. Classification of self-similar solutions 4.1. General properties 4.2. Subcase mmN< 4.3. Subcase mN