Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 108, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.108 COMPLICATED ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO DOUBLY NONLINEAR DIFFUSION EQUATION IN UNBOUNDED SPACES CAN LU, LIANG-WEI WANG, JING-XUE YIN, MEI-LING ZHOU Abstract. In this article, we study the complicated asymptotic behavior of doubly nonlinear diffusion equations in unbounded spaces. We find a workspace in which is unbounded and can exhibit complicated asymptotic behavior of the solution to the Cauchy problem. To overcome the difficulties caused by the nonlinearity of the equations and the unbounded solutions, we establish propagation estimates, growth estimates, and Weighted L1-L∞ estimates for the solutions. 1. Introduction In this article, we study the complicated asymptotic behavior of solutions to the Cauchy problem for doubly nonlinear diffusion equations, ut − div(|∇um|p−2∇um) = 0 in S = RN × (0,∞), (1.1) u(x, 0) = u0(x) in RN , (1.2) where m > 0, p > 1, m(p− 1) > 1 and the nonnegative initial values satisfy u0 ∈ Yσ(RN ) = {φ ∈ C(RN ) : lim |x|→∞ (1 + |x|2)−σ/2φ(x) = 0}. In recent decades, many authors have shown a great interest in studying a number of evolution equations, especially concerning the complicated asymptotic behavior of the solutions[5, 2, 6, 18, 19, 3, 10]. Among these studies, we mention an important work by Kamin et al. [5, 6], for the equation (1.1) with p = 2. They showed that the solution of the porous medium equation converges uniformly to the Barenblatt solution of the equation for porous media with the same mass as u0 if the initial value u0 ∈ L1 +(RN ) = {φ ∈ L1(RN ;φ(x) ≥ 0)}. Subsequently, it was shown by Zhao and Yuan [20] that for the doubly nonlinear diffusion equation (1.1), the solution of the multidimensional Cauchy-problem (1.1) converges to the Barenblatt solution of the porous media equation of the same mass as u0 if the initial value u0 ∈ L1 +(RN ) = {φ ∈ L1(RN ;φ(x) ≥ 0)}. For the solutions of certain evolution equations, the occurrence of complicated asymptotic behavior primarily depends on the workspace to which the initial data belongs, see [9, 8, 17, 1]. Note that in the aforementioned work, the problem concerning the complicated asymptotic behavior of the solutions is only considered in some bounded spaces. Based on the existence theory of doubly nonlinear diffusion equations, it is concluded that the solutions to the problems (1.1)-(1.2) are global, even if the initial data belongs to some unbounded spaces [21, 12, 4, 11]. Thus, the complicated asymptotic behavior of solutions to the doubly nonlinear diffusion equation may also appear in unbounded spaces. Inspired by this, we focus here on the complicated asymptotic behavior of problems (1.1)-(1.2) in the unbounded spaces Yσ(RN ) with p p−1 ≤ σ < p m(p−1)−1 . Next, we consider that the solution u(x, t) of problem (1.1)-(1.2) with initial data u0 ∈ Yσ(RN ) may be unbounded solutions. It is worth noting that the doubly nonlinear diffusion equation has nonlinearity and degeneracy not 2020 Mathematics Subject Classification. 35B40, 35K55. Key words and phrases. Doubly nonlinear diffusion equation; asymptotic behavior; unbounded function; weighted L1 − L∞ estimates. ©2025. This work is licensed under a CC BY 4.0 license. Submitted June 6, 2025. Published November 17, 2025. 1 2 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 found in the heat equation [13, 14]. To overcome these difficulties, we establish the propagation speed estimation and the growth estimation in a given weighted spaces, and extend the weighted L1−L∞ estimates of Zhao and Xu [21] to the unbounded spaces Yσ(RN ) and apply them to obtain the existence of a weak solution of problem (1.1)-(1.2). Finally, we investigate the asymptotic behavior of solutions for the problem (1.1)-(1.2) and give the fact that the spaces Yσ(RN ) with p p−1 ≤ σ < p m(p−1)−1 can provide the work spaces where complexity causing the asymptotic behavior of solutions. Specifically, using the properties of solutions in unbounded spaces, we can obtain that there exists a function ϕ ∈ Yσ(RN ) such that the set of ω-limits set ωµ,β σ (u0) = Y + σ (RN ) for some µ, β, where ωµ,β σ (u0) = {f ∈ Yσ(RN ) : ∃tn → ∞ s.t. tµ/2n u(tβnx, tn) tn→∞−−−−→f in Yσ(RN )}, Y + σ (RN ) = {φ ∈ Yσ(RN ) : φ ≥ 0 and φ(0) = 0}, and u(x, t) is the solution of problem (1.1)-(1.2) with the initial data u0(x) = ϕ(x). So the unbounded spaces Yσ(RN ) can provide the work spaces where complexity occurs in the asymptotic behavior of solution of the Cauchy-problem (1.1)-(1.2), according to Vazquez and Zuazua [15]. The main points of the rest of this paper are as follows. In the next section, we introduce some essential definitions and properties to provide theoretical support. subsequently, we present the propagation estimates, growth estimates, and weighted L1-L∞ estimates for the solutions of problem (1.1)-(1.2) with initial data u0 ∈ Yσ(RN ). Finally, we study the complicated asymptotic behavior of the solution of problem (1.1)-(1.2) in the unbounded spaces Yσ(RN ) given by p p−1 ≤ σ < p m(p−1)−1 . 2. Preliminaries Definition 2.1. For r > 0, f ∈ L1 loc(RN ), let |||f |||r = sup R≥r R−N[m(p−1)−1]+p m(p−1)−1 ∫ x≤R |f(x)|dx. The space X is defined as X = {f ∈ L1 loc(RN ); |||f |∥1 <∞} with the norm ||| · |||1. Thus X is a Banach space. For any norm ||| · |||r (r > 0) is an equivalent norm, taking f ∈ X, we will define it as follows ℓ(f) = lim |r|→∞ |||f |∥r. Taking a subspace X0(RN ) of the function space X, for the sake of convenience, simply denoted as X0 = {φ ∈ X; ℓ(f) = 0}. If the initial value u0 ∈ X0, then the existence and uniqueness of a global weak solution to problem (1.1)-(1.2) is proved in [21, 12, 4, 11]. To prove our result, this solution also needs to satisfy the following proposition as support. Proposition 2.2 ( [11]). If u0 ∈ X0, the Cauchy problem for the doubly nonlinear diffusion equation (1.1)-(1.2) generates a continuous bounded semigroup in X0, S(t) : u0 → u(x, t), i.e. u(x, t) = S(t)u0 ∈ C([0,∞);X0). Note that the semigroup is bounded in Lp(RN ) for all p ≥ 1. Definition 2.3. Let 0 ≤ σ < ∞, ρσ(x) = (1 + |x|2)−σ/2 , the weighted space Lp(ρσ) was defined as Lp(ρσ) = {φ ∈ L1 loc(RN ) : φρσ ∈ Lp(RN )} and ∥φ∥Lp(ρσ) = ∥φρσ∥Lp(RN ). Also the space Yσ(RN ) is defined as Yσ(RN ) = {φ(x) ∈ C(RN ) : lim |x|→∞ φ(x)ρσ(x) = 0} EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 3 with the norm ∥φ∥Yσ(RN ) = ∥φρσ∥L∞(RN ). This section verifies that both Yσ(RN ) and Lp(ρσ) are Banach spaces. Remark 2.4. From the above definitions, we can see that Yσ(RN ) ⊂ L∞(ρσ) for every σ ≥ 0, and Yσ(RN ) ⊂ X0 for p p−1 ≤ σ < p m(p−1)−1 , and Y0(RN ) = C0(RN ), and Lp(ρ0) = Lp(RN ) for σ = 0. Proposition 2.5 ([12, 21]). For any u0 ∈ X, there exists a constant T = T (u0) and a weak solution u(x, t) to problems (1.1)-(1.2) in QT = RN × [0, T ), where T (u0) = { Cℓ(u0) 1−m(p−1) if ℓ(u0) > 0, ∞ if ℓ(u0) = 0. Furthermore, if u0 ∈ C(RN ), then u(x, t) ∈ C(RN × [0, T )). Definition 2.6. A nonnegative function u(x, t) is called a solution of (1.1)-(1.2) if u satisfies (i) u ∈ C(ST ), um ∈ lploc(0, T ;W 1,p loc (R N )), ut is a regular measure on ST of locally bounded variation; (ii) ∫ ST [u(x, t)φt(x, t)− |Dum|p−2Dum ·Dφ]dxdτ = 0, ∀φ ∈ C1 0 (ST ); (iii) lim t→0 ∫ |x| 0, where ST = RN × (0, T ) and 0 ≤ T < ∞. As a way of measuring the growth of a function f ∈ L1 loc(RN ) as |x| → ∞, we let |||f |∥rρ = sup ρ≥r ρ− λ m(p−1)−1 ∫ Bρ |f(x)|ρσdx, where r > 0, λ = N [m(p− 1)− 1] + p, Bρ = {x ∈ RN : |x| < ρ}. Definition 2.7. Let p p−1 ≤ σ < p m(p−1)−1 , µ, β > 0, and u0 ∈ Yσ(RN ). Then the ω-limit is as follows ωµ,β σ (u0) = {f ∈ Yσ(RN ) : ∃tn → ∞ s.t. Dµ,β√ tn [S(tn)u0] tn→∞−−−−→ f in Yσ(RN )}. For φ ∈ L1 loc(RN ) and λ > 0 there are Dµ,β λ φ(x) = λµφ(λ2βx), S(t) is the semigroup operator given in Proposition2.2. In [3, 16], the following exchange relation between semigroup operators and dilation operators is obtained, Dµ,β λ [S(λ2t)u0] = S(λ2−2pβ−µ[m(p−1)−1]t)[Dµ,β λ u0]. For λ > 0, the proof will not be repeated here. Definition 2.8. Let d(x) = sup{R : u0(y) = 0 a.e. in BR(x)} be the distance from x to the support point of u0. We define Ω(t) = {x ∈ RN : u(x, t) > 0}. We also define the ρ-neighborhood of Ω(t) as Ωρ(t) = {x ∈ RN : d(x,Ω(t)) < ρ}, where d(x,Ω(t)) is the distance from x to Ω(t). 3. Some estimates In this section, we provide estimates for the solution of problem (1.1)-(1.2) using initial data u0 ∈ L1(ρσ). To accomplish this, we need to use the following lemma given in [16], with particular emphasis in [21]. 4 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 Lemma 3.1. Given a non-negative weak solution u(x, t) of problem (1.1)-(1.2). For every x ∈ RN , if B(x) = sup R≥0 R−N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x) u0(y)dy <∞, where BR(x) = {y : |x− y| < R}, for all 0 < t ≤ C(p,N)B(x)−[m(p−1)−1], then u(x, t) = 0. Next, we prove the propagation estimates of solutions to problem (1.1)–(1.2) when the initial value u0 belongs to L1(ρσ) and L ∞(ρσ). Theorem 3.2 (Propagation estimates). Let p p−1 ≤ σ < p m(p−1)−1 and assume that u0 ∈ L1(ρσ). Then for all 0 ≤ t1 ≤ t2 <∞, Ω(t2) ⊂ Ωρ(t2−t1)(t1), where ρ(t2 − t1) = Cmax { ∥u0∥ m(p−1)−1 N[m(p−1)−1]+p L1(ρσ) (t2 − t1) 1 N[m(p−1)−1]+p , ∥u0∥ − m(p−1)−1 p+(N−σ)[m(p−1)−1] L1(ρσ) (t2 − t1) 1 p+(N−σ)[m(p−1)−1] }. Proof. We consider only the case of t1 = 0, assuming that x0 ∈ RN and BR(x0) > 0, if R < d(x0), then ∫ BR(x0) u0(y)dy = 0. If R ≥ d(x0), then R−N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x0) u0(y)dy = R−N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x0) u0(y)ρσ(x)(1 + |x|2)σ/2dy ≤ R−N[m(p−1)−1]+p m(p−1)−1 (1 +R2)σ/2 ∫ BR(x0) u0(y)ρσ(x)dy ≤ R−N[m(p−1)−1]+p m(p−1)−1 (1 +R2)σ/2∥u0∥L1(ρσ) ≤ 2σ/2∥u0∥L1(ρσ) max{R−N[m(p−1)−1]+p m(p−1)−1 , R− (N−σ)[m(p−1)−1]+p m(p−1)−1 } ≤ 2σ/2∥u0∥L1(ρσ) max{d(x0)− N[m(p−1)−1]+p m(p−1)−1 , d(x0) − (N−σ)[m(p−1)−1]+p m(p−1)−1 }. (3.1) So, from the above two equations it follows that B(x0) = sup R≥d(x0) R−N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x0) u0(y)dy ≤ C∥u0∥L1(ρσ) max{d(x0)− N[m(p−1)−1]+p m(p−1)−1 , d(x0) − (N−σ)[m(p−1)−1]+p m(p−1)−1 }. (3.2) By Lemma 3.1 u(x0, t) = 0, for 0 < t ≤ C∥u0∥−[m(p−1)−1] L1(ρσ) min{d(x0)N [m(p−1)−1]+p , d(x0) (N−σ)[m(p−1)−1]+p}. Then Ω(t) ⊂ Ωρ(t)(0), where ρ(t) = Cmax { ∥u0∥ m(p−1)−1 N[m(p−1)−1]+p L1(ρσ) t 1 N[m(p−1)−1]+p , ∥u0∥ − m(p−1)−1 p+(N−σ)[m(p−1)−1] L1(ρσ) t 1 p+(N−σ)[m(p−1)−1] } . The proof is complete. □ Theorem 3.3 (Propagation estimates). Let p p−1 ≤ σ < p m(p−1)−1 and assume that u0 ∈ L∞(ρσ). (3.3) Then for any 0 ≤ t1 ≤ t2 < ∞, we have Ω(t2) ⊂ Ωρ(t2−t1)(t1). Note that Ωρ(t2−t1)(t1) is a ρ-neighborhood of Ω(t2) and ρ((t2 − t1)) = Cmax { ∥u0∥ m(p−1)−1 p L∞(ρσ) (t2 − t1) 1/p, ∥u0∥ −[m(p−1)−1] p−N[m(p−1)−1] L∞(ρσ) (t2 − t1) 1 p−N[m(p−1)−1] } . EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 5 Proof. As above, we only consider the case t1 = 0. Suppose that x0 ∈ RN and BR(x0) > 0. Then, for R < d(x0), it holds ∫ BR(x0) u0(y)dy = 0. As for R ≥ d(x0), using Definition 2.3, it can be estimated as R−N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x0) u0(y)dy = R−N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x0) u0(y)ρσ(x)(1 + |x|2)σ/2dy ≤ R−N[m(p−1)−1]+p m(p−1)−1 ∥u0∥L∞(ρσ)(1 +R2)σ/2 ∫ BR(x0) dy ≤ R− p m(p−1)−1 ∥u0∥L∞(ρσ)(1 +R2)σ/2 ≤ 2σ/2∥u0∥L∞(ρσ) max{R− p m(p−1)−1 , R− p m(p−1)−1 +σ} ≤ 2σ/2∥u0∥L∞(ρσ) max{d(x0)− p m(p−1)−1 , d(x0) − p m(p−1)−1 +σ}; (3.4) therefore, B(x0) = sup { R ≥ d(x0)R −N[m(p−1)−1]+p m(p−1)−1 ∫ BR(x0) u0(y)dy } ≤ C∥u0∥L∞(ρσ) max { d(x0) − p m(p−1)−1 , d(x0) − p m(p−1)−1 +σ} . (3.5) Using Lemma3.1 it follows that u(x0, t) = 0, for every 0 < t ≤ C∥u0∥L∞(ρσ) min { d(x0) − p m(p−1)−1 , d(x0) − p m(p−1)−1 +σ} . Hence Ω(t) ⊂ Ωρ(t)(0), where ρ(t) = Cmax { ∥u0∥ m(p−1)−1 p L∞(ρσ) t1/p, ∥u0∥ −[m(p−1)−1] p−N[m(p−1)−1] L∞(ρσ) t 1 p−N[m(p−1)−1] } . We have thus completed the proof. □ In the following theorem, we examine the properties of the solutions to problem (1.1)-(1.2) when the initial value satisfies u0 ∈ L∞(ρσ). Theorem 3.4 (Growth estimates). Let p p−1 ≤ σ < p m(p−1)−1 , if 0 ≤ u0 ∈ L∞(ρσ). Then there exists a constant C such that 0 ≤ S(t)u0(x) ≤ C((1 + t) 2 p−σ[m(p−1)−1] + |x|2)σ/2, thereby, ∥S(t)u0(x)∥L∞(ρσ) ≤ C(1 + t) σ p−σ[m(p−1)−1] . Furthermore, if 0 ≤ u0 ∈ Yσ(RN ), for any t ≥ 0, we have S(t)u0(x) ∈ Yσ(RN ). Proof. We consider the problem ut −∆um = 0 in S, u(x, 0) = v0(x) =M |x|σ in RN . (3.6) For λ > 0, let λ1 = λ p−σ[m(p−1)−1] 2p , µ = 2σ p− σ[m(p− 1)− 1] , β = 1 p− σ[m(p− 1)− 1] and satisfying the equation 2− µ[m(p− 1)− 1]− 2pβ = 0. 6 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 Applying the exchange relation given in Definition 2.7, it follows that λ− σ p [S(λ1− σ[m(p−1)−1] p t)v0](λ 1/px) = λµ1 [S(λ 2 1t)v0](λ 2β 1 x) = S(t)[λµ1v0(λ 2β 1 ·)](x) = S(t)v0(x). (3.7) In the above equation, setting t = 1, s = λ p−σ[m(p−1)−1] p and g(x) = S(1)v0(x), we obtain S(s)v0(x) = s p p−σ[m(p−1)−1] g(s− 1 p−σ[m(p−1)−1]x). (3.8) Indeed, it follows from regularity theory that for t > 0, 0 ≤ S(t)v0 ∈ C∞((0,∞)× RN ) ∩ C([0,∞)× RN ), for |x| = 1, taking the limit in both sides of the above equation simultaneously with respect to s→ 0 as follows that S(s)v0(x) → v0(x) =M |x|σ =M, and taking y = s− 1 p−σ[m(p−1)−1]x; note that |y| → ∞ as s→ 0. Then upon |y|−σg(y)−M → 0 as |y| → ∞, (3.9) there exists a non-negative constant C such that g(x) ≤ C(1 + |x|2)σ/2. We deduce from (3.8) that 0 ≤ S(s)v0(x) ≤ C(s 2 p−σ[m(p−1)−1] + |x|2) σ/2 . This implies S(t)g(x) = S(t)[S(1)v0](x) = S(t+ 1)v0(x) ≤ C((1 + t) 2 p−σ[m(p−1)−1] + |x|2)σ/2. Putting φ(x) =M(1 + |x|2)σ/2, we have S(t)φ(x) ≤ C((1 + t) 2 p−σ[m(p−1)−1] + |x|2)σ/2. If we let M = ∥u0∥L∞(ρσ) in question (3.6). we have S(t)u0(x) ≤ C((1 + t) 2 p−σ[m(p−1)−1] + |x|2)σ/2. Next we prove the second part of the theory, owing to 0 ≤ u0 ∈ Yσ(RN ) ⊂ L∞(ρσ). For every t > 0, R > 0, let R(t) = R+ 1 + Cmax{∥u0∥ m(p−1)−1 p L∞(ρσ) t1/p, ∥u0∥ −[m(p−1)−1] p−N[m(p−1)−1] L∞(ρσ) t 1 p−N[m(p−1)−1] }, and taking χR+1(x) to be a cut-off function defined on BR+1 associated with BR, then χR+1(x) ∈ C∞ 0 (RN ), 0 ≤ χR+1(x) ≤ 1, such that χR+1(x) = { 1 for x ∈ BR, 0 for x /∈ BR+1. We can deduce from Lemma3.1 that supp[S(t)(χR+1u0)] ⊂ {x ∈ RN : |x| ≤ R(t)}. It means that for t > 0, R > 0,, the value of S(t)u0(x) in RN \ BR(t) depends only on the initial value u0(x) in RN \BR(t), in other words, if |x| > R(t), then S(t)[(1− χR+1)u0](x) = S(t)u0(x). (3.10) EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 7 For any ε > 0, and the following assumptions are satisfied 0 ≤ u0 ∈ Yσ(RN ), then there exists a constant R1 > 1 > 0, It follows that for |x| > R1, (1 + |x|2)−σ/2 u0(x) < ε 2 . At this point lettin R = R1 and we obtain (1− χR+1)u0 < ε|x|σ. Let M = ε in equation (3.6), and then use equation (3.9) and the Comparison Principle to show that there exists a constant R2 and the following inequality holds when |x| > R2, S(1)[(1− χR+1)u0](x) ≤ g(x) ≤ 2ε|x|σ. (3.11) If |x|t− 1 p−σ[m(p−1)−1] > R2, then S(t)[(1− χR+1)u0](x) ≤ t σ p−σ[m(p−1)−1] g(t− 1 p−σ[m(p−1)−1]x) ≤ 2ε|x|σ. (3.12) Then combining (3.10) with the above equation, if |x| > max{R1(t), t 1 p−σ[m(p−1)−1]R2}, then, according to (3.11), S(t)u0(x) ≤ 2ε|x|σ. (3.13) Clearly, using the first part of the theorem, for ∀t > 0, S(t)u0(x) ∈ L∞(ρσ) and S(t)u0(x) ∈ C(RN ). By (3.13), it follows that for ∀t > 0, S(t)u0(x) ∈ Yσ(RN ). (3.14) The proof of all statements of Theorem3.4 is complet. □ The ideas of the proof of the following theorem come from [21, 4]. Theorem 3.5 (Weighted L1-L∞ estimates). Let p p−1 ≤ σ < p m(p−1)−1 , if u0 ∈ L1(ρσ), then for all t > 0, u(t) = S(t)u0 ∈ L∞(ρσ). (3.15) Moreover, if 0 < t ≤ γ∥u0∥−[m(p−1)−1] L1(ρσ) , then ∥u(·, t)∥L∞(ρσ) ≤ γt−N/λ∥u0∥p/λL1(ρσ) . (3.16) Proof. We consider the Cauchy problem ut − div(|∇um|p−2∇um) = 0 in ST , (3.17) u(·, 0) = u0(·) in RN , (3.18) where u0(x) ∈ L1(ρσ), by the results of [20], equations (3.17) and (3.18) have a nonnegative solution u satisfying u ∈ C([0, T ] : L1(RN ) ∩ L∞(ST ) ∩ C(ST ), u m ∩ Lp loc(0, T :W 1,p(RN )). Such regularity will suffice to justify the calculations to follow. We will denote by γ a generic positive constant depending only on N , m, p. The proof of Theorem3.5 requires several steps. Let T > 0, ρ > 0 be fixed and consider the sequences Tj = T 2 − T 2j+2 , ρj = ρ+ ρj 2 , ρ̄j = 1 2 (ρj + ρj+1) = ρ+ 3ρ 2j+2 . Set Bj = Bρj , B̄j = Bρ̄j , Qj = Bj × (Tj , T ), Q̄j = B̄j × (Tj+1, T ), kj = k − k 2j+1 , k > 0, and let ξj(x, t) be a piecewise smooth cut-off function in Qj satisfying ξj(x, t) = 1, for (x, t) ∈ Q̄j , |Dξj | ≤ 2j+2 ρ , for 0 ≤ ξjt ≤ 2j+2 T , 8 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 Finally, we define φ(t) = φr(t) = sup τ∈(0,t) τN/λ sup ρ>r ∥u(·, τ)∥L∞(ρσ),Bρ ρ p m(p−1)−1 , (3.19) where r > 0, λ = N [m(p− 1)− 1] + p, and ψ(t) = sup τ∈(0,t) |||u(·, τ)|||rρ . Since u0(x) ∈ L1(ρσ), u ∈ L∞(ST ) and φ(t), ψ(t) are well defined. Lemma 3.6. There exists a constant γ(N,m, p), such that for any t > 0, ∥u(·, t)∥L∞(ρσ),Bρ ≤ γ[K(t)] N+p γ (∫ t t 4 ∫ B2ρ (uρσ) mpdxdτ )p/β , (3.20) where β = N(m(p− 1)− 1) +mp2, and K(t) = t −N(m(p−1)−1) λ φm(p−1)−1(t) + t−1. (3.21) Proof. Let χj(x) be a segmented smooth cut-off function in B̄j that satisfies the following condi- tions: x ∈ Bj+1, χj(x) = 1, and |Dχj(x)| ≤ 2j+2ρ−1. Set wj = (uρσ − kj) m(2p−1)−1 p + . (3.22) Recall the the Gagliardo-Nirenberg inequality, for t ∈ (0, T ), ∥wjχj∥q,B̄j(t) ≤ γ∥D(wjχj)∥ηp,B̄j(t) ∥wjχj∥1−η s,B̄j(t) , (3.23) where η = (1 s − 1 q )( 1 N − 1 p + 1 s )−1 and γ = γ(N,m, p). Choose η = p q , s = mp2 m(2p−1)−1 , and q = p(1+ s N ). An application of q-powers to the integration of (Tj+1, T ) results in the following:∫∫ Qj+1 w p(1+ s N ) j+1 dxdτ ≤ ∫∫ Q̄j (wj+1χj) qdxdτ ≤ γ (∫∫ Q̄j |Dwj+1|qdxdτ + 2jp ρp ∫∫ Q̄j wp j+1dxdτ )( sup Tj+1≤t≤T ∫ B̄j(t) ws j+1dx )p/N . (3.24) Next, we estimate the right-hand side of the aforementioned equation. We use ξj as defined in (3.24). We multiply (3.17) by the test function φ(x, t) = ξpj (uρσ − kj+1) mp−1 + = max{0, ξpj (uρσ − kj+1) mp−1} (3.25) and integrate over Qj to obtain sup Tj+1≤t≤T ∫ B̄j(t) ws j+1dx+ ∫∫ Q̄j |Dwj+1|qdxdτ ≤ 2jpγ ρp ∫∫ Qj∩{uρσ≥kj+1} (uρσ) (m−1)(p−1)(uρσ − kj) mp+p−2dxdτ + γ2j T ∫∫ Qj (uρσ − kj) mp + dxdτ . (3.26) For p > 2, we have (uρσ) (m−1)(p−1)(uρσ − kj) p−2 + ρ−p ≤ γt −N(m(p−1)−1) λ φm(p−1)−1(t). For p < 2, if uρσ 2 ≥ kj such that (uρσ) (m−1)(p−1)(uρσ − kj) p−2 ≤ (uρσ) (m−1)(p−1)( uρσ 2 )p−2 ≤ C(uρσ) m(p−1)−1, EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 9 and if kj+1 ≤ uρσ ≤ 2kj such that (uρσ) (m−1)(p−1)(uρσ − kj) p−2 ≤ (uρσ) (m−1)(p−1)(kj+1 − kj) p−2 ≤ C2(2−p)j(uρσ) m(p−1)−1. Under the above conditions, (uρσ) (m−1)(p−1)(uρσ − kj) p−2ρ−p ≤ γt −N(m(p−1)−1) λ φm(p−1)−1(t) in {uρσ ≥ kj+1}. Thus, (3.26) can be written as sup Tj+1≤t≤T ∫ B̄j(t) ws j+1dx+ ∫∫ Q̄j |Dwj+1|qdxdτ ≤ γ2j(p+1)K(T ) ∫∫ Qj ws jdxdτ , (3.27) Again, by the above definition of wj , it follow that ρ−p ∫∫ Q̄j wp j+1dxdτ ≤ γK(T ) ∫∫ Qj ws jdxdτ . (3.28) By applying equation (3.27) to equation (3.24) and estimating the right-hand side of (3.27), the following result is obtained:∫∫ Qj+1 wq j+1dxdτ ≤ γ ( 2(p+1)jK(T ) )N+p N (∫∫ Qj ws jdxdτ )N+p N . (3.29) By Hölder inequality,∫∫ Qj (uρσ − kj+1) mp + dxdτ ≤ (∫∫ Qj+1 wq j+1dxdτ )s/q |Aj+1|1− s q , (3.30) where Aj = {(x, t) ∈ Qj : u(x, t)ρσ > kj}, j = 0, 1, 2, . . . and |Aj | = measAj . Thus∫∫ Qj (uρσ − kj+1) mp + dxdτ ≥ ∫∫ Qj∩[uρσ>kn+1] (uρσ − kj+1) mpdxdτ ≥ |kj+1 − kj |p|Aj+1| ≥ kmp 2mp(j+2) |Aj+1|, (3.31) we deduce from (3.29) and (3.30) that∫∫ Qj (uρσ − kj+1) mp + dxdτ ≤ γbj [K(T )] (N+p)s Nq k−mp(1− s q ) (∫∫ Qj (uρσ − kj) mp + dxdτ )1+ ps Nq , (3.32) it is worth noting that b = 2mp+[(1+ p N )(p+1)−mp] s q . By using [7, Lemma 5.6] it is known that∫∫ Qj (uρσ − kj) mp + dxdτ → ∫∫ Q∞ (uρσ − k)mp + dxdτ = 0, provided ∫∫ Q0 (uρσ) mpdxdτ ≤ γ[K(T )] N+p p kmN( q s−1). In this scenario, if the value of k is selected and satisfies the requisite conditions k = γ[K(T )] N+p β (∫∫ Q0 (uρσ) mpdxdτ )p/β , β = N(m(p− 1)− 1) +mp2, (3.33) then supQ∞ uρσ ≤ k. Since T > 0 is arbitrary, the lemma follows. □ Lemma 3.7. For all t > 0 it holds φ(t) ≤ γ ∫ t 0 τ− N[m(p−1)−1] λ [φ(τ)]m(p−1)dτ + γψ(t) p/λ , λ = N(m(p− 1)− 1) + p. (3.34) 10 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 Proof. Multiply both sides of (3.20) by ρ p m(p−1)−1 and τ N λ , τ ∈ ( t2 , t) simultaneously. It is necessary to recall the definition of K(t) as presented in equation (3.21), we have that for all τ ∈ ( t2 , t), and all t > 0, τ N λ ∥u(·, τ)∥L∞(ρσ),Bρ ρ− p m(p−1)−1 ≤ γτ N λ −N(N+p)[m(p−1)−1] λβ [φ(τ)] (N+p)[m(p−1)−1] β (∫ t t 4 ∫ B2ρ ρ− β m(p−1)−1 (uρσ) mpdxdτ )p/β + γτ N λ −N+p λβ (∫ t t 4 ∫ B2ρ ρ− β m(p−1)−1 (uρσ) mpdxdτ )p/β = E1 + E2. (3.35) Subsequently, the values of E1 and E2 are estimated: E1 ≤ γ1[φ(τ)] 1− p β [ ∫ t t 4 τ− N[m(p−1)−1] λ ( τ N λ ∥u(·, τ)∥L∞(ρσ),B2ρ (2ρ) p m(p−1)−1 )mp dτ ]p/β ≤ γ1[φ(t)] 1− p β (∫ t 0 τ− N[m(p−1)−1] λ [φ(τ)]m(p−1) dτ )p/β ≤ 1 4 φ(t) + γ2 ∫ t 0 τ− N[m(p−1)−1] λ [φ(τ)]m(p−1) dτ, (3.36) and E2 ≤ γ (1 t ∫ t t 4 τ− N[m(p−1)−1] λ (∥u(·, τ)∥L∞(ρσ),B2ρ (2ρ) p m(p−1)−1 )mp−1 ∫ B2ρ u(x, τ)ρσdx (2ρ) λ m(p−1)−1 dτ )p/β ≤ γ[φ(t)] p(mp−1) β (1 t ∫ t 0 |||u(·, τ)|||rρdτ )p/β ≤ γ[φ(t)] p(mp−1) β [ψ(t)]p/β ≤ 1 4 φ(t) + γ3[ψ(t)] p/λ. (3.37) The above conclusions are all based on Cauchy inequality scaling estimates. Now, we carry these estimates in (3.35) and take the supremum first over all p ≥ r > 0 and then over all τ ∈ (0, t), for all t > 0. Recalling the definition (3.19) of φ(t), the lemma follows. □ Lemma 3.8. Let us consider a piecewise smooth cutoff function, designated as χ(x), within the context of B2ρ, such that in Bρ satisfying χ = 1 and |Dχ| ≤ 2 ρ . Let ρ > r > 0, for ∀t > 0 such that ∫ t 0 ∫ B2ρ |Dum|p−1χp−1ρσ dx dτ ≤ γρ1+ λ m(p−1)−1 {∫ t 0 τ p+1 λ [φ(τ)] [m(p−1)−1](p+1) p ψ(τ) dτ + ∫ t 0 τ 1 λ−1[φ(τ)] m(p−1)−1 p ψ(τ) dτ } p−1 p (∫ t 0 τ 1 λ−1[φ(τ)] m(p−1)−1 p ψ(τ) dτ ) 1 p . (3.38) Proof. The initial requirement is that u be strictly positive. This can be made strict by substituting u+ ε for u. By Hölder inequality,∫ t 0 ∫ B2ρ |Dum|p−1χp−1ρσdxdτ ≤ ∫ t 0 ∫ B2ρ ( τ m(p−1)−1 p2 |Dum|p−1(uρσ) (h−m)(p−1) p χp−1ρ m(p−1) p σ ) × ( τ −m(p−1)−1 p2 (uρσ) − (h−m)(p−1) p ρ p−m(p−1) p σ ) dxdτ ≤ (∫ t 0 ∫ B2ρ τ1/p|Dum|p−1(uρσ) h−mχpρmσ dxdτ ) p−1 p EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 11 × ( ∫ t 0 ∫ B2ρ τ− p−1 p (uρσ) −(p−1)(h−m)ρp−m(p−1) σ dxdτ)1/p ≤ ( ∫ t 0 ∫ B2ρ τ1/p|Dum|p−1(uρσ) h−mχpρmσ dxdτ) p−1 p × ( ∫ t 0 ∫ B2ρ τ− p−1 p (uρσ) −(p−1)(h−m)dxdτ )1/p = [F1(t)] p−1 p [F2(t)] 1/p, and∫ t 0 ∫ B2ρ |Dum|p−1χp−1ρσdxdτ ≤ ∫ t 0 ∫ B2ρ ( τ m(p−1)−1 p2 |Dum|p−1(uρσ) (h−m)(p−1) p χp−1ρ m(p−1) p σ ) × ( τ −m(p−1)−1 p2 (uρσ) − (h−m)(p−1) p ρ p−m(p−1) p σ ) dxdτ ≤ (∫ t 0 ∫ B2ρ τ1/p|Dum|p−1(uρσ) h−mχpρmσ dxdτ ) p−1 p × (∫ t 0 ∫ B2ρ τ− p−1 p (uρσ) −(p−1)(h−m)ρp−m(p−1) σ dxdτ )1/p ≤ (∫ t 0 ∫ B2ρ τ1/p|Dum|p−1(uρσ) h−mχpρmσ dxdτ ) p−1 p × (∫ t 0 ∫ B2ρ τ− p−1 p (uρσ) −(p−1)(h−m)dxdτ )1/p = [F1(t)] p−1 p [F2(t)] 1/p, where h = ( m(p−1)−1 ) /p, for estimating J1(t), we take φ = τ1/p(uρσ) h χp in the weak formulation of (3.17) and obtain F1(t) ≤ γ ρp ∫ t 0 ∫ B2ρ τ1/p(uρσ) h+m(p−1) dxdτ + γ ∫ t 0 ∫ B2ρ τ 1 p−1(uρσ) h+1 dxdτ = V1 + V2. (3.39) Estimating V1, V2, separately, we have V1 ≤ γρ1+ λ m(p−1)−1 ∫ t 0 τ p+1 λ [φ(t)] (p+1)[m(p−1)−1] p ψ(t)dτ, V2 ≤ γρ1+ λ m(p−1)−1 ∫ t 0 τ 1 λ−1[φ(t)] m(p−1)−1 p ψ(t)dτ . (3.40) Note that F2(t) = V2, and combing (3.39)-(3.40) the conclusion follows. □ Lemma 3.9. It holds ψ(t) ≤ γ|||u0|||rρ + γ ∫ t 0 τ p+1 λ −1[φ(τ)] (p+1)[m(p−1)−1] p ψ(τ)dτ + ∫ t 0 τ 1 λ−1[φ(τ)] m(p−1)−1 p ψ(τ)dτ . (3.41) Proof. It is necessary to take the cutoff function χ(x) introduced in Lemma 3.8. By simultaneously multiplying both sides of Equation (3.20) by φ(x) = χp(x)ρσ and utilizing distributional integrals and standard calculations, we obtain∫ Bρ u(x, t)ρσdx ≤ ∫ B2ρ u0ρσdx+ γ ρ ∫ t 0 ∫ B2ρ |Dum|p−1χp−1ρσdxdτ , (3.42) 12 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 valid for all t > 0 and all p > r > 0. Multiply both sides by ρ− λ m(p−1)−1 , estimate the last term by Lemma 3.8, and take the supremum over all p > r > 0. □ Lemma 3.10. For t positive, let φ(t) and ψ(t) be two continuous, non-decreasing functions sat- isfying the following conditions: φ(t) ≤ γ ∫ t 0 τ− N[m(p−1)−1] λ [φ(τ)]m(p−1)dτ + γ[ψ(t)]p/λ, (3.43) ψ(t) ≤ γρ− λ m(p−)−1 ∥u(·, τ)∥L1(ρσ),Bρ + γ ∫ t 0 τ p+1 λ −1[φ(τ)] (p+1)[m(p−1)−1] p ψ(τ)dτ + γ ∫ t 0 τ 1 λ−1[φ(τ)] m(p−1)−1 p ψ(τ)dτ . (3.44) Furthermore, there exist constants γ0, γ1, γ2, which are dependent on the variables γ, m, p and N , and which satisfy φ(t) ≤ γ1|||u0|||p/λrρ , (3.45) ψ(t) ≤ γ2|||u0|||rρ , (3.46) for all 0 < t < γ0|||u0|||−[m(p−1)−1] rρ . Proof. It is established that the function t → ψ(t) is non-decreasing. For all t∗ > 0, from (3.43) we have φ(t) ≤ γ ∫ t 0 τ− N(m(p−1)−1) λ [φ(τ)]m(p−1)dτ + γ[ψ(t)]p/λ, (3.47) and satisfy 0 < t ≤ t∗. For the constant t∗ will be chosen later, at this stage is an arbitrary positive value of t. As a consequence of the preceding argument, it follows from (3.47) that φ(t) can be displayed using the solution to G′(t) = γt− N[m(p−1)−1] λ G(t) p−1 , G(0) = γψ(t∗) p/λ . (3.48) Solving explicitly φ(t) ≤ G(t) = γψ(t∗) p/λ{1− γ[m(p− 1)− 1][tψ(t∗)m(p−1)−1]p/λ}− 1 m(p−1) . (3.49) For any given t ∈ [0, t∗], it is guaranteed that the value within the brackets is positive. Provided that the bracket is of a positive nature. In this instance, the aforementioned estimate can be expressed as t = t∗. Given that t∗ is an arbitrary positive number, it can be inferred that φ(t) ≤ γψ(t) p/λ{1− γ[m(p− 1)− 1][tψ(t)m(p−1)−1]p/λ}− 1 m(p−1) , (3.50) for all t > 0 for which the bracket is positive. If on t we impose γ(tψ(t) m(p−1)−1 )p/λ ≤ 2−[m(p−1)−1] m(p− 1)− 1 [2m(p−1)−1 − 1]. (3.51) The following statement will be obtained: There exist two constants γ̄1 and γ̄0, and are dependent only on N , m, and p, such that φ(t) ≤ γ̄1ψ(t) p/λ , (3.52) for all t satisfying [tψ(t)m(p−1)−1]p/λ ≤ γ̄0. (3.53) Substituting inequality (3.52) into (3.44) yields ψ(t) ≤ γ|||u0|||rρ + γ ∫ t 0 τ 1 λ−1ψ(t) m(p−1)−1 p +1 dτ , (3.54) EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 13 for all t for which (3.53) holds and for a new constant γ = γ(N,m, p). It can be reasonably deduced that the solution to ψ(·) is H ′(t) = γt 1 λ−1H(t) m(p−1)−1 p +1 , H(0) = γ|||u0|||rρ , (3.55) whence ψ(t) ≤ H(t) = |||u0|||rρ [ 1− γ(t|||u0|||m(p−1)−1 rρ )p/λ ]− λ m(p−1)−1 , (3.56) ensure that the inside of the brackets are positive. The constant γ being quantitatively determined a prior, we can find a constant ¯̄γ0 such that ψ(t) ≤ γ2|||u0|||rρ , (3.57) as long as t is so small as to satisfy (3.53) and 0 < t < ¯̄γ0|||u0|||−[m(p−1)−1] rρ . (3.58) Putting together (3.53), (3.57), and (3.58), we deduce that there exists a constant γ0 = γ0(N,m, p) such that if 0 < t < γ0|||u0|||−[m(p−1)−1] rρ , (3.53) and (3.58) holds. □ Now combining (3.52) and (3.57) leads to the following inference. Corollary 3.11. There exist constants γ0, γ1, γ2 depending only on N,m, p such that ρ ≥ r > 0, ∥u(, t)∥∞,Bρ ≤ γ1t −N/λρ p m(p−1)−1 |||u0|||−[m(p−1)−1] rρ . (3.59) for all 0 < t < γ0|||u0|||−[m(p−1)−1] rρ . Corollary 3.12. There exist constants γ, depending only on N,m, p such that ∥u(, t)∥L∞(ρσ) ≤ γt−N/λ∥u0∥−[m(p−1)−1] L1(ρσ) , (3.60) for all 0 < t < γ∥u0∥−[m(p−1)−1] L1(ρσ) . Setting T = γ∥u0∥−[m(p−1)−1] L1(ρσ) , we duce from (3.60) that S(t)u0 ∈ L∞(ρσ). (3.61) Hence, for T ≤ t, by Theorem3.5, we have S(t)u0 = S(T − t)[S(T )u0] ∈ L∞(ρσ), (3.62) and the proof is complete. Remark 3.13. The L1−L∞estimates for ball Bρ have been reflected in paper [20]. To obtain our results, we generalize them to the weighted space Lp(ρσ) and provide the weights L1−L∞estimates. □ 4. Complicated asymptotic behavior To complete the body of this article, we introduce the work spaces Yσ(RN ) with p p−1 ≤ σ < p m(p−1)−1 and study the complicated asymptotic behavior of the solutions to problems (1.1)-(1.2) in these spaces. We define a closed subset of Yσ(RN ) as Y + σ (RN ) = {φ ∈ Yσ(RN ) : φ(x) ≥ 0 and φ(0) = 0}. (4.1) For convenience, let γ = N [m(p− 1)− 1] + p, η = 1 p− σ[m(p− 1)− 1] . Our main task is to study the complicated asymptotic behavior of the solution of problem (1.1)- (1.2). 14 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 Theorem 4.1. Let p p− 1 ≤ σ < p m(p− 1)− 1 , (4.2) suppose there exist µ > 0 and β > 0, and the following inequalities are satisfied β > max( 2− µ[m(p− 1)− 1] 2p , η), (4.3) 0 < µ+ 2pβσ γ < 2N γ . (4.4) Then for any f ∈ Y + σ (RN )∩L1(ρσ), there exists an initial value u0 ∈ Y + σ (RN ), and a subsequence {tn} converging to ∞ as n→ ∞, such that Dµ,β√ tn [S(tn)u0] n→∞−−−−→ f in Yσ(RN ), (4.5) in other words, f ∈ ωµ,β σ (u0). Proof. For every f ∈ Y + σ (RN ) ∩ L1(ρσ), without loss of generality, we assume that ∥f∥L∞(ρσ) ≥ 1 and ∥f∥L1(ρσ) ≥ 1. (4.6) We define ℓ = max{∥f∥L∞(ρσ), ∥f∥ η L∞(ρσ) , ∥f∥ηL1(ρσ) }, (4.7) η = max{η, 1 γ , 1 γ − σ[m(p− 1)− 1] , 1 2 } ≥ 1 2 . (4.8) From (4.3) and (4.4) it follows that β > η ≥ 1 2 and 0 < 2N − µγ − 2pβσ < 2pβN − pµ − 2pβσ. Next we define a sequence λj → ∞ as j → ∞ through λ1 = 2, and for any j > 1, λj = max{2 j µλj−1, (2 j+1Cℓ+ 22jλ2βj−1) 1 2β−2η , j γ+2 2N−µγ−2pβσ λ 2pβN−pµ−2pβσ 2N−µγ−2pβσ j−1 , 2 j β λj−1}. (4.9) Below we define the initial value u0(x) as u0(x) = ∞∑ j=1 λ−µ j χj ( x λ2βj ) f(x/λ2βj ) = ∞∑ j=1 Dµ,β λ−1 j [χjf(x)]. (4.10) In this context, χj(x) represents the cut-off function, defined within set Φj = {2−j < |x| < 2j} with respect to set Ψj = {2−(j−1) < |x| < 2j−1}, and χj(x) ∈ C∞ 0 (RN ), 0 ≤ χj(x) ≤ 1, such that χj(x) = { 1 for x ∈ Φj , 0 for x /∈ Ψj . (4.11) So that u0(x) ≥ 0, u0(0) = 0. (4.12) Next we consider 0 < λ ≤ 1, consequently ∥f∥Yσ(RN ) = sup x∈RN |(1 + |x|2)−σ/2f(λx)| ≤ sup x∈RN ∣∣∣f(λx)(1 + |x|2)−σ/2 (1 + |λx|2 1 + |x|2 )σ/2∣∣∣ ≤ sup x∈RN ∣∣∣f(λx)(1 + |λx|2 )−σ/2∣∣∣ = ∥f∥Yσ(RN ). (4.13) Therefore, using (4.10) we have ∥u0∥Yσ(RN ) ≤ ∞∑ j=1 2−j∥f(x)∥Yσ(RN ) ≤ ℓ. (4.14) EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 15 Consequently, the series (4.10) is convergent within Yσ(RN ). Additionally, equation (4.12) allows for the conclusion that u0 ∈ Y + σ (RN ). (4.15) Now se conduct a detailed analysis of the initial value u0. This will be broken down into three parts. un(x) = n−1∑ j=1 λ−µ j χj ( x λ2βj ) f(x/λ2βj ) = n−1∑ j=1 Dµ,β λ−1 j [χj(x)f(x)], (4.16) vn(x) = Dµ,β λ−1 n [χn(x)f(x)], (4.17) wn(x) = ∞∑ j=n+1 λ−µ j χj ( x λ2βj ) f(x/λ2βj ) = n+1∑ j=1 Dµ,β λ−1 j [χj(x)f(x)]. (4.18) From Theorems 3.3 and 3.4, we have supp [S(λ2n)un(x)] ⊂ {x : |x| ≤ 2n−1λ2βn−1 + Cλ2ηn ℓ}, (4.19) supp [S(λ2n)vn(x)] ⊂ {x : 2−nλ2βn − Cλ2ηn ℓ ≤ |x| ≤ 2nλ2βn + Cλ2ηn ℓ}, (4.20) supp [S(λ2n)wn(x)] ⊂ {x : |x| ≥ 2−n−1λ2βn+1 − Cλ2ηn ℓ}. (4.21) From (4.9), it is apparent that (2j+1Cℓ+ 22jλ2βj−1) 1 2β−2η ≤ λj , so that for j > 1, 2j−1λ2βj−1 + 2Cλ2ηj ℓ < 2j−1λ2βj−1λ 2η j + 2Cλ2ηj ℓ ≤ 2−jλ2βj , hence 2n−1λ2βn−1 + Cλ2ηn ℓ < 2−nλ2βn − Cλ2ηn ℓ < 2nλ2βn + Cλ2ηn ℓ < 2−n−1λ2βn+1 − Cλ2ηn ℓ. It can be demonstrated from equations (4.19), (4.20) and (4.21) that supp [S(λ2n)un(x)] ∩ supp [S(λ2n)vn(x)] = ∅, supp [S(λ2n)vn(x)] ∩ supp [S(λ2n)wn(x)] = ∅. Thus S(λ2n)u0 = S(λ2n)un(x) + S(λ2n)vn(x) + S(λ2n)wn(x). (4.22) Applying the exchange relationship in Definition2.6 to (4.22), we have Dµ,β λn S(λ2n)u0 = Dµ,β λn [ S(λ2n)un(x) ] +Dµ,β λn [ S(λ2n)vn(x) ] +Dµ,β λn [ S(λ2n)wn(x) ] = S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn un(x)] + S(λ2−2pβ−µ[m(p−1)−1] n ) × [S(λ2n)vn(x)] + S ( λ2−2pβ−µ[m(p−1)−1] n )[ S(λ2n)wn(x) ] . (4.23) Now we consider the case of λ ≥ 1. ∥f(λ·)∥L1(ρσ) = ∫ RN ∣∣(1 + |x|2)−σ/2f(λx) ∣∣dx ≤ ∫ RN ∣∣∣f(λx)(1 + |x|2)−σ/2 (1 + |λx|2 1 + |x|2 )σ/2∣∣∣dx ≤ λσ (∫ RN |f(λx)(1 + |λx|2)−σ/2|dx ) = λσ−N∥f∥L1(ρσ). (4.24) It can be deduced from (4.9) and (4.16) that ∥Dµ,β λn un(x)∥L1(ρσ) ≤ n( λn λn−1 )µ−2βN+2βσ∥f∥L1(ρσ) <∞, (4.25) because Dµ,β λn un(x) ∈ L1(ρσ); furthermore, the results in section (4.3) show that 2− µ[m(p− 1)− 1]− 2pβ < 0. It follows again from (4.9) that there exists an integer N , such that if n > N , then 0 < λ2−µ[m(p−1)−1]−2pβ n < C∥Dµ,β λn un(x)∥−[m(p−1)−1] L1(ρσ) . (4.26) 16 C. LU, L.-W. WANG, J.-X. YIN, M.-L. ZHOU EJDE-2025/108 An application of the weighted L1-L∞ estimates, as proposed in Theorem3.5, to Equation (4.25), yields ∥S(λ2−2pβ−µ[m(p−1)−1] n )Dµ,β λn un(x)∥Yσ(RN ) ≤ Cn p γ λ µ+ 2pβσ−2N γ n λ 2pβN−µp−2pβσ γ n−1 . (4.27) From (4.9) we conclude that ∥S ( λ2−2pβ−µ[m(p−1)−1] n ) Dµ,β λn un(x)∥Yσ(RN ) ≤ Cn−1ℓ→ 0 as n→ ∞. (4.28) To arrive at the solution for (4.18), we combine equations (4.9) and (4.13), for n > 1, ∥Dµ,β λn wn(x)∥Yσ(RN ) ≤ ∥ ∞∑ j=n+1 ( λn λj )µf(( λn λj )2β ·)∥Yσ(RN ) ≤ ∞∑ j=n+1 ( λn λj )µ∥f∥Yσ(RN ) ≤ ∥f∥L1(ρσ) <∞, (4.29) thereby Dµ,β λn wn(x) ∈ Yσ(RN ). Applying Theorem3.4, we obtain S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn wn(x)] ∈ Yσ(RN ). (4.30) As a consequence, for all n ≥ 1 and any ε > 0, using Comparison Principle, there exists a constant M > 0 such that |x| > M . Then |(1 + |x|2)−σ/2 S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn wn(x)](x)| < ε 2 . (4.31) Since Dµ,β λn wn(x) ⊂ { x : |x| ≥ 2−n−1( λn+1 λn )2β } , (4.32) from Theorem3.3, it follows that supp { S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn wn(x)] } ⊂ { x : |x| ≥ 2−n−1 (λn+1 λn )2β − Cλη[2−2pβ−µ(m(p−1)−1)] n ℓ } . (4.33) Note that λ η[2−2pβ−µ(m(p−1)−1)] n < 1, Since (4.9) implies 22n ≤ (λn+1 λn )2β , the existence of an integer N , such that if n > N , then 2−n−1( λn+1 λn )2β − Cλη2−2pβ−µ[m(p−1)−1] n ℓ > M. (4.34) The preceding equation can be expressed as follows: if n > N , then ∥S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn wn(x)]∥Yσ(RN ) < ε. (4.35) Subsequently, we can show that ∥S ( λ2−2pβ−µ[m(p−1)−1] n ) Dµ,β λn vn(x)− f∥Yσ(RN ) n→∞−−−−→ 0, (4.36) because Dµ,β λn vn(x) implies Dµ,β λn vn(x) = f(x) for x ∈ Φj , (4.37) and Dµ,β λn vn(x) ≤ f(x) for x ∈ RN . (4.38) We conclude from Theorem3.5 that S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn vn(x)] ∈ Yσ(RN ). (4.39) Note that f ∈ Yσ(RN ). As a result, for all n ≥ 1, ε > 0, using comparison principle, there exists a constant M > 0 such that sup |x|>M (1 + |x|2)−σ/2∣∣S(λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn vn](x)| < ε 3 , (4.40) EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 17 and sup |x| > M(1 + |x|2)−σ/2|f(x)| < ε 3 . (4.41) Then f ∈ C(RN ) implies that S(t)f ∈ C([0,∞) × RN ). Note that λ 2−2pβ−µ(m(p−1)−1) n → 0 as n→ ∞. Thereupon, for any ε > 0, there exists a integer N1, such that if n > N1, then sup |x|>M (1 + |x|2)−σ/2| [ S ( λ2−2pβ−µ[m(p−1)−1] n ) f ] (x)− f(x) ∣∣ < ε 18 . (4.42) It follows from (4.9) that there exists a integerN2, if n > N2, so that 2n−1 > M, (4.43) based on the assumption f(0) = 0 that there exists a integerN3, if n > N3, such that sup |x|>2−N3+1 ( 1 + |x|2 )−σ/2|f(x)| < ε 18 . (4.44) By (4.43), if n > N1, then sup |x|>2−N3+1 ( 1 + |x|2 )−σ/2∣∣[S(λ2−2pβ−µ[m(p−1)−1] n ) f ] (x) ∣∣ < ε 9 . (4.45) By combining equations (4.38) and (4.45) with the comparison principle, if n > N1, we have sup |x|>2−N3+1 (1 + |x|2)−σ/2|[S(λ2−2pβ−µ[m(p−1)−1] n )Dµ,β λn vn(x)](x)| < ε 9 . (4.46) Combine equations (4.37), (4.42), (4.43), (4.45) and (4.46) that if n > max{N1, N2, N3}, the reason why sup |x|≤M ( 1 + |x|2 )−σ/2∣∣S(λ2−2pβ−µ[m(p−1)−1] n )[ Dµ,β λn vn ] (x)− f(x) ∣∣ ≤ sup |x|≤M ( 1 + |x|2 )−σ/2∣∣[S(λ2−2pβ−µ[m(p−1)−1] n ) f ] (x)− f(x) ∣∣ + sup 2−n+1≤|x|≤M ( 1 + |x|2 )−σ/2∣∣S(λ2−2pβ−µ[m(p−1)−1] n ) × [Dµ,β λn vn](x)− S ( λ2−2pβ−µ[m(p−1)−1] n ) f(x) ∣∣ + sup |x|≤2−n+1 (1 + |x|2)−σ/2 ∣∣S(λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn vn](x) ∣∣ + sup |x|≤2−n+1 (1 + |x|2)−σ/2 ∣∣[S(λ2−2pβ−µ[m(p−1)−1] n ) f ] (x) ∣∣ < ε 3 . (4.47) It follows from (4.40), (4.41) and (4.47) that (4.36) holds. Finally, it follows from (4.23), (4.28), (4.35), and (4.36) that ∥S ( λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn vn]− f∥Yσ(RN ) ≤ sup |x|≤M (1 + |x|2)−σ/2 ∣∣S(λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn vn](x)− f(x) ∣∣ + sup |x|>M (1 + |x|2)−σ/2 ∣∣S(λ2−2pβ−µ[m(p−1)−1] n ) [Dµ,β λn vn](x) ∣∣ + sup |x|>M (1 + |x|2)−σ/2|f(x)| < ε. 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Can Lu College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China Email address: lc002535@163.com Liang-Wei Wang (corresponding author) College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China Email address: wanglw08@163.com Jing-Xue Yin School of Mathematical Science, South China Normal University, Guangzhou 510631, China Email address: yjx@scnu.edu.cn EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 19 Mei-Ling Zhou College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China Email address: zml253944@163.com 1. Introduction 2. Preliminaries 3. Some estimates 4. Complicated asymptotic behavior Acknowledgments References