Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 08, pp. 1–26. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.08 BEHAVIOR NEAR THE EXTINCTION TIME FOR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH SUBLINEAR DISSIPATION TERMS LUAN HOANG Abstract. This article focuses on the behavior near the extinction time of solutions to systems of ordinary differential equations with a sublinear dissi- pation term. Suppose the dissipation term is a product of a linear mapping A and a positively homogeneous scalar function H of a negative degree −α. Then any solution with an extinction time T∗ behaves like (T∗ − t)1/αξ∗ as time t → T− ∗ , where ξ∗ is an eigenvector of A. The result allows the higher order terms to be general and the nonlinear function H to take very com- plicated forms. As a demonstration, our theoretical study is applied to an inhomogeneous population model. 1. Introduction This article continues our investigations of exact asymptotic behaviors of solu- tions of systems of nonlinear ordinary differential equations (ODE), see [12, 13, 26, 27], and the Navier–Stokes equations (NSE), see, e.g., [10, 28, 30, 31]. However, in contrast to the cited work above, the current paper studies the asymptotic behavior near a finite extinction time, instead of time infinity, for equations with sublinear dissipation terms, instead of superlinear in [27], or linear in the others. The phe- nomena of extinction are widely studied in mathematics, biology and physics, see e.g. [14, 15, 16, 17, 32, 36, 37]. In biology, the extinction time reflects the time when the species’ populations go extinct, while, for fast diffusive fluid flows in porous me- dia, it indicates the time when the pressure or density vanishes everywhere. For the precise description of the solutions near a finite extinction time, the mathematical results in [14, 16, 36] are only for very specific equations. Instead, our aim is to establish the results for general systems of ODEs. It turns out that we can achieve this by modifying and improving some techniques by Foias and Saut in [18] for the NSE. For that reason, we review [18] and the related literature here. The NSE with potential body forces can be written in the following functional form, which holds in a certain weak sense in a suitable functional space, u′ +Au+B(u, u) = 0, (1.1) 2020 Mathematics Subject Classification. 34D05, 41A60. Key words and phrases. Sublinear dissipation; extinction time; extinction profile; vanish in finite time; asymptotic behavior; asymptotic approximation. ©2025. This work is licensed under a CC BY 4.0 license. Submitted April 26, 2024. Published January 17, 2025. 1 2 L. HOANG EJDE-2025/08 where A is the (linear) Stokes operator which has positive eigenvalues and B(·, ·) is a bilinear form. It is proved in [18] that any nontrivial solution u(t) of (1.1) has the following asymptotic behavior eΛtu(t) → ξ∗ in any Cm-norm as t → ∞, (1.2) where Λ is an eigenvalue of A and ξ∗ is an eigenfunction of A associated with Λ. For finer asymptotic behaviors than (1.2), Foias and Saut develop a theory of asymptotic expansions for the solutions of the NSE (1.1) in [19, 20]. On the one hand, the result (1.2) and its proof are extended to abstract differ- ential inequalities in [22, 21]. On the other hand, the asymptotic expansion theory is developed further for both ODE and partial differential equations (PDE). See [13, 27, 29, 31, 35, 38] for systems without forcing functions, [10, 11, 12, 26, 28, 30] for systems with forcing functions, and [25] for the Lagrangian trajectories for vis- cous incompressible fluids. Both techniques from [18] and [19] are combined in [13] to deal with nonsmooth ODE systems. Originally, the asymptotic expansions can be obtained independently from the limit (1.2). In [13], however, they are obtained only after the first asymptotic approximation (1.2) is established. In the original NSE in [18] as well as the systems in the extended work cited above, except for [27], the ODE or PDE have linear dissipation terms. On contrary, the author recently studied in [27] the following ODE system in Rn y′ = −H(y)Ay +G(t, y), (1.3) where A is a constant n×nmatrix with positive eigenvalues,H is a positive function, and G represents a higher order term. In [27], H is additionally assumed to be a positively homogeneous function of a positive degree α. Note that the dissipation term H(y)Ay in (1.3) is nonlinear compared with, say, the linear dissipation term Au in (1.1), and the higher order term G is not required to be bilinear like B(u, u). It is proved that any nonzero, decaying solution of (1.3) behaves like t−1/αξ∗ as t → ∞, where ξ∗ is an eigenvector of A. The current paper considers the opposite scenario when the function H in (1.3) has a negative degree −α. In this case, many solutions start out with nonzero values and then become zero at a finite time. Such time is called the extinction time. Our goal is to describe the behavior of these solutions near this extinction time. The main result can be briefly described as follows. Under appropriate assumptions, any solution y(t) of (1.3) with the extinction time T∗ behaves exactly like (T∗ − t)1/αξ∗ as time t → T− ∗ , where ξ∗ is an eigenvector of A. It is worth mentioning that the existence of the extinction time is guaranteed under the small nonzero initial data condition, see Theorem 2.5 below. Our proof will make use and adapt the techniques from [18, 27]. In particular, the recent perturbation method in [27] will be utilized. This method is needed to deal with the nonlinear dissipation in our problem. It will be implemented successfully in this paper for the study of the asymptotic behavior near the finite extinction time, instead of at time infinity as in [27]. The obtained result, in addition to its merits for ODE, also gives hints to a type of results that may be expected for general nonlinear PDE of the similar structure. This article is organized as follows. Section 2 contains the main results. While the condition for the matrix A is the natural Assumption 2.1, the more technical conditions for the function H are specified in Assumption 2.8. A key requirement of H, namely, property (HC) is introduced in Definition 2.6. Theorem 2.5 states that EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 3 under appropriate conditions on A, H and G, for any sufficiently small nonzero initial condition, there exists a solution of (1.3) that will become zero at finite time. Solutions with a finite extinction time of this type are the objects of our investigation in this paper. The asymptotic behavior of the solutions to a more general equation (2.11) near the extinction time is established in Theorem 2.9. Its counterpart for equation (1.3) is Theorem 2.10. The proof of Theorem 2.5 is given in Section 3. Section 4 prepares for the proof of Theorem 2.9. Preliminary estimates for the solutions are obtained in Lemma 4.1. Although they provide only a rough description of y(t), the upper and lower bounds with the same rate 1/α obtained in (4.1) are important in our further analysis. Section 5 proves Theorem 2.9 for a special case in the form of equation (5.2). This will also serve as the basis for the perturbation argument for the general case in Section 7. In Section 6, we obtain essential properties of the solutions of (2.11) when the matrix A is symmetric. In particular, we establish an eigenvalue Λ as the limit of the quotient λ(t), see (6.1), in Proposition 6.1, and a unit vector v∗ as the limit of y(t)/|y(t)| in Propositions 6.2 and 6.3. In Section 7, we give proof to Theorem 2.9 first. It combines all the previous preparations with the perturbation method mentioned earlier, see equation (7.2). This equation is a reduction of equation (2.11) of y(t) to the simple form (5.2), but for the projection RΛy(t) and with a frozen coefficient ΛH(v∗). The proof of Theorem 2.10 is then quickly provided. Section 8 contains some examples for the function H in subsection 8.1, and an application to an inhomogeneous population model in subsection 8.2. Notation. Throughout this article, n ∈ N = {1, 2, 3, . . .} is the spatial dimension. For any vector x ∈ Rn, we denote by |x| its Euclidean norm. For an n × n real matrix A = (aij)1≤i,j≤n, its Euclidean norm is ∥A∥ = ( n∑ i=1 n∑ j=1 a2ij )1/2 . The unit sphere in Rn is Sn−1 = {x ∈ Rn : |x| = 1}. 2. Main results Assumption 2.1. Matrix A is a (real) diagonalizable n× n matrix with positive eigenvalues. Under this assumption, the matrix A has n positive eigenvalues (counting their multiplicities) Λ1 ≤ Λ2 ≤ Λ3 ≤ . . . ≤ Λn, and there exists an invertible n×n (real) matrix S such that A = S−1A0S, where A0 = diag[Λ1,Λ2, . . . ,Λn]. (2.1) In the case A is symmetric, the matrix S is orthogonal, i.e., S−1 = ST, and Λ1|x|2 ≤ x ·Ax ≤ Λn|x|2 for all x ∈ Rn. (2.2) More specifically, the distinct eigenvalues of A are denoted by λj , with 1 ≤ j ≤ d for some integer d ∈ [1, n], and are arranged to be (strictly) increasing in j, i.e., 0 < λ1 = Λ1 < λ2 < . . . < λd = Λn. The spectrum of A is σ(A) = {Λk : 1 ≤ k ≤ n} = {λj : 1 ≤ j ≤ d}. 4 L. HOANG EJDE-2025/08 For 1 ≤ k, ℓ ≤ n, let Ekℓ be the elementary n× n matrix (δkiδℓj)1≤i,j≤n, where δki and δℓj are the Kronecker delta symbols. For Λ ∈ σ(A), define R̂Λ = ∑ 1≤i≤n,Λi=Λ Eii and RΛ = S−1R̂ΛS. Then one immediately has In = d∑ j=1 Rλj , RλiRλj = δijRλj , ARλj = RλjA = λjRλj . (2.3) Thanks to (2.3), each RΛ is a projection, and RΛ(Rn) is the eigenspace of A asso- ciated with the eigenvalue Λ. In the case A is symmetric, RΛ is the orthogonal projection from Rn to the eigenspace of A associated with Λ, and, hence, |RΛx| ≤ |x| for all x ∈ Rn. (2.4) The functionH will be assumed to have some type of homogeneity which is specified in the next definition. Definition 2.2. Let X and Y be two (real) linear spaces, and β < 0 be a given number. A function F : X \ {0} → Y is positively homogeneous of degree β if F (tx) = tβF (x) for any x ∈ X \ {0} and t > 0. We define Hβ(X,Y ) to be the set of functions from X \{0} to Y that are positively homogeneous of degree β. If F ∈ Hβ(X,Y ) and F is not the zero function, then the degree β is unique. Assumption 2.3. The function H is in H−α(Rn,R) for some α > 0, and in C(Rn \ {0}, (0,∞)). For a function H in Assumption 2.3, it is positive and continuous on Sn−1. Hence, we have 0 < c1 = min |x|=1 H(x) ≤ max |x|=1 H(x) = c2 < ∞. (2.5) By writing H(x) = |x|−αH(x/|x|) for any x ∈ Rn \ {0} and using (2.5), we derive c1|x|−α ≤ H(x) ≤ c2|x|−α for all x ∈ Rn \ {0}. (2.6) Regarding the function G in equation (1.3), we have the following assumption. Assumption 2.4. Let t0 be any given number in [0,∞). We assume that the function G(t, x) is continuous on [t0,∞) × (Rn \ {0}), and there exist positive numbers c∗, r∗, δ such that |G(t, x)| ≤ c∗|x|1−α+δ for all t ≥ t0, and all x ∈ Rn with 0 < |x| ≤ r∗. (2.7) In our first theorem below, we show that the extinction time exists for, at least, certain small solutions of (1.3). Theorem 2.5. Under Assumptions 2.1, 2.3 and 2.4, there exists a number r0 > 0 such that for any y0 ∈ Rn \ {0} with |y0| ≤ r0, there are a number T∗ > t0 and a function y ∈ C1([t0, T∗),Rn) such that y(t0) = y0, y(t) ̸= 0 for all t ∈ [t0, T∗), (2.8) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 5 lim t→T− ∗ y(t) = 0, (2.9) and y(t) satisfies equation (1.3) for all t ∈ (t0, T∗). In other words, y(t) is a solution of (1.3) on (t0, T∗) with the initial data y0 at time t0 and has the extinction time T∗. In fact, y(t) is any solution with the maximal interval [t0, Tmax) in the existence theorem 3.1 below, and T∗ = Tmax. Moreover, if additional conditions are imposed to guarantee the uniqueness of solutions to the initial value problem for equation (1.3), then, in the above Theorem 2.5, any solution with sufficiently small nonzero initial data must have a finite extinction time. The proof of Theorem 2.5 is given in Section 3. For the behavior of a solution near an extinction time, the function H is required to have an extra property. Definition 2.6. Let E be a nonempty subset of Rn and F be a function from E to R. We say F has property (HC) on E if, for any x0 ∈ E, there exist numbers r, C, γ > 0 such that |F (x)− F (x0)| ≤ C|x− x0|γ (2.10) for each x ∈ E with |x− x0| < r. The following are elementary properties of the functions in Definitions 2.2 and 2.6. Lemma 2.7. Let F ∈ H−α(Rn,R) for some α > 0. (i) If F > 0 on Sn−1, then F > 0 on Rn \ {0}. (ii) If F is continuous on Sn−1, then it is continuous on Rn \ {0}. Assume F has property (HC) on Sn−1 in (iii)–(v) below. (iii) Then F has property (HC) on Rn \ {0}. (iv) If φ is a function from Rn \ {0} to Rn \ {0} that has property (HC) on Rn \ {0}. Then F ◦ φ has property (HC) on Rn \ {0}. (v) If K is an invertible n×n matrix, then the function x ∈ Rn \{0} 7→ F (Kx) has property (HC) on Rn \ {0}. The proof of the above lemma is given in the Appendix. Assumption 2.8. The function H belongs to H−α(Rn,R) for some α > 0, has property (HC) on the unit sphere Sn−1, and H > 0 on Sn−1. If the function H satisfies Assumption 2.8, it is obvious that H is continuous on Sn−1. Therefore, thanks to parts (i) and (ii) of Lemma 2.7, it is continuous and positive on Rn \ {0}. Consequently, H satisfies the conditions in Assumption 2.3. Some examples for the function H will be given in subsection 8.1. The next result deals with a more general equation than (1.3), namely, equation (2.11) below. Theorem 2.9 (Main Theorem I). Let Assumptions 2.1 and 2.8 hold. Let t0, T∗ ∈ R be two given numbers with T∗ > t0 ≥ 0. Assume y ∈ C1([t0, T∗),Rn) satisfies (2.8), (2.9) and y′ = −H(y)Ay + f(t) for all t ∈ (t0, T∗), (2.11) where f is a continuous function from [t0, T∗) to Rn such that |f(t)| ≤ M |y(t)|1−α+δ, for all t ∈ [t0, T∗) and some constants M, δ > 0. (2.12) 6 L. HOANG EJDE-2025/08 Then there exist an eigenvalue Λ of A and an eigenvector ξ∗ of A associated with Λ such that |y(t)− (T∗ − t)1/αξ∗| = O((T∗ − t)1/α+ε) as t → T− ∗ for some ε > 0. (2.13) More specifically, |(In −RΛ)y(t)| = O((T∗ − t)1/α+ε) as t → T− ∗ for some ε > 0, (2.14) |RΛy(t)− (T∗ − t)1/αξ∗| = O((T∗ − t)1/α+ε) as t → T− ∗ for some ε > 0, (2.15) αΛH(ξ∗) = 1. (2.16) With a solution y(t) as in Theorem 2.9, we define, for the sake of convenience, y(T∗) = 0, and have y ∈ C([t0, T∗],Rn). (2.17) From Theorem 2.9, we can derive a corresponding result for equation (1.3). Theorem 2.10 (Main Theorem II). Let Assumptions 2.1, 2.4 (with a number t0 ≥ 0) and 2.8 hold. Given a number T∗ > t0. Assume y ∈ C1([t0, T∗),Rn) has properties (2.8), (2.9), and satisfies equation (1.3) for all t ∈ (t0, T∗). Then there exist an eigenvalue Λ and an associated eigenvector ξ∗ of A such that (2.13)–(2.16) hold true. The proofs of Theorems 2.9 and 2.10 are given in Section 7. 3. Existence of the extinction time We prove Theorem 2.5 in this section. First, we present a standard existence theorem for the solutions. Theorem 3.1. Let Assumptions 2.1, 2.3 hold and assume G is a continuous func- tion from [t0,∞) × Rn \ {0} to Rn, for some number t0 ∈ R. Let y0 ∈ Rn \ {0}. Then there exist an interval [t0, Tmax), with t0 < Tmax ≤ ∞, and a function y ∈ C1([t0, Tmax),Rn \ {0}) such that y(t) that satisfies (1.3) on (t0, Tmax), y(t0) = y0, (3.1) and either (a) Tmax = ∞, or (b) Tmax < ∞, and for any ε > 0, y cannot be extended to a function of class C1([t0, Tmax+ε),Rn \{0}) that satisties (1.3) on the interval (t0, Tmax+ε). Moreover, in the case (b), it holds, for any compact set U ⊂ Rn \ {0}, that y(t) ̸∈ U when t ∈ [t0, Tmax) is near Tmax. (3.2) Proof. By Peano’s Existence Theorem [24, Chapter II, Theorems 2.1, page 10] and the continuity of the function G(t, x), there exist a number δ > 0 and a function y ∈ C1([t0, t0+ δ],Rn \{0}) such y(t) that satisfies equation (1.3) on [t0, t0+ δ] and y(t0) = y0. By the Extension Theorem, see [24, Chapter II, Theorem 3.1, page 12] or [23, Chapter I, Theorem 2.1, page 17], applied to this solution y and the open set D = {(t, x) : t > t0, x ∈ Rn \ {0}} ⊂ Rn+1, (3.3) the current solution y can be extended to a solution y ∈ C1([t0, Tmax),Rn \ {0}), for some t0 + δ ≤ Tmax ≤ ∞, that has the properties (3.1), with either (a) or (b), and, additionally, in the case (b), one has, for any compact set V ⊂ D, (t, y(t)) ̸∈ V when t ∈ [t0, Tmax) is near Tmax. (3.4) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 7 Now, consider case (b) and a compact set U ⊂ Rn\{0}. Let V = [T1, T2]×U with T1 = (t0 +Tmax)/2 and T2 = Tmax +1. If t ∈ [t0, Tmax) is sufficiently close to Tmax, then t ∈ [T1, T2]. Therefore, the desired statement (3.2) follows from (3.4). □ Note that such a solution y(t) in Theorem 3.1 may not be unique. Next we prove Theorem 2.5. Proof of Theorem 2.5. Let y(t) be a solution of (1.3) as in Theorem 3.1 with the maximal interval [t0, Tmax). Then y(t) ̸= 0 for all t ∈ [t0, Tmax). (3.5) On (t0, Tmax), we have d dt (|y|α) = α|y|α−2y′ · y = α ( −|y|α−2H(y)(Ay) · y + |y|α−2G(t, y) · y ) . (3.6) Step 1. Consider A is symmetric first. Take r0 > 0 such that 2r0 ≤ r∗ and c∗(2r0) δ ≤ a0 := c1Λ1/2. For t > t0 sufficiently close to t0, we have |y(t)| < 2r0. Let [t0, T ) be the maximal interval in [t0, Tmax) on which |y(t)| < 2r0. Suppose T < Tmax. On the one hand, it must hold that |y(T )| = 2r0. (3.7) On the other hand, combining (3.6) with (2.2), (2.6) and (2.7), we have, for t ∈ (t0, T ), that d dt (|y|α) ≤ α(−c1Λ1 + c∗|y|δ) ≤ α(−c1Λ1 + c∗(2r0) δ) ≤ −αa0 < 0. (3.8) Thus, |y(T )|α ≤ |y0|α, which implies |y(T )| ≤ |y0| ≤ r0. This contradicts (3.7). Therefore, T = Tmax. For t ∈ (t0, Tmax), integrating (3.8) from t0 to t gives |y(t)|α ≤ |y0|α − αa0(t− t0) for all t ∈ [t0, Tmax). (3.9) Step 2. Consider the general matrix A. Using the equivalence (2.1), we set z(t) = Sy(t) and z0 = z(t0) = Sy0. (3.10) Then z′ = −H̃(z)A0z + G̃(t, z) for t ∈ (t0, Tmax), (3.11) where H̃(z) = H(S−1z), G̃(t, z) = SG(t, S−1z) for t ∈ [t0, Tmax) and z ∈ Rn \ {0}. Note that ∥S−1∥−1 · |x| ≤ |Sx| ≤ ∥S∥ · |x| for all x ∈ Rn. (3.12) Clearly, H̃ satisfies the same condition as H in Assumption 2.3. Moreover, G̃(t, z) is continuous on [t0,∞) × (Rn \ {0}). For t ∈ [t0, Tmax) and 0 < |z| ≤ r∗/∥S−1∥, we have 0 < |S−1z| ≤ r∗, and then, by (2.7) and (3.12), |G̃(t, z)| ≤ ∥S∥·c∗|S−1z|1−α+δ ≤ c∗∥S∥· { ∥S−1∥1−α+δ|z|1−α+δ, if 1− α+ δ ≥ 0, ∥S∥−(1−α+δ)|z|1−α+δ, otherwise. 8 L. HOANG EJDE-2025/08 We apply the calculations in Step 1 to the solution z(t) of (3.11). When |y0| > 0 is sufficiently small, we have |z0| > 0 is sufficiently small, and hence, similar to estimate (3.9), |z(t)|α ≤ |z0|α − αã0(t− t0), for all t ∈ [t0, Tmax) and some constant ã0 > 0. Therefore, |y(t)|α ≤ ∥S−1∥α|z(t)|α ≤ ∥S−1∥α(|Sy0|α − αã0(t− t0)) for all t ∈ [t0, Tmax). (3.13) Step 3. If Tmax = ∞, then (3.13) implies that |y(t)|α < 0 for t > t0+ |Sy0|α/(αã0), which is an obvious contradiction. Therefore, Tmax < ∞. As a consequence of (3.13), |y(t)| ≤ R0 on [t0, Tmax), where R0 = ∥S−1∥ · |Sy0| > 0. (3.14) Step 4. Let T∗ = Tmax. Then (3.5) implies (2.8). For any ε > 0, let U = {x ∈ Rn : ε ≤ |x| ≤ 2R0} in (3.2). Taking into account (3.14), one must have |y(t)| < ε when t ∈ [t0, T∗) is near T∗. This proves the zero limit in (2.9). □ We remark that y(t) may be zero for t larger than the above Tmax. However, this is excluded from our consideration of the set D in (3.3). The reason is our sole focus on the finite extinction time and the solution before that time. 4. Preliminary estimates In this section, we prepare for the proof of Theorem 2.9 by obtaining preliminary estimates for the solution y(t) of equation (2.11). They even hold under a weaker condition than Assumption 2.8. Lemma 4.1. Let Assumptions 2.1 and 2.3 hold. Given numbers T∗ > t0 ≥ 0, let y ∈ C1([t0, T∗),Rn) satisfy (2.8), (2.9), (2.11)–(2.12). Then there are positive constants C1 and C2 such that C1(T∗ − t)1/α ≤ |y(t)| ≤ C2(T∗ − t)1/α for all t ∈ [t0, T∗]. (4.1) Proof. We prove (4.1) for the case the matrix A is symmetric first and then for A not symmetric. Case 1. Consider A is symmetric. For t ∈ (t0, T∗), we calculate, similarly to (3.6), d dt (|y|α) = α ( −|y|α−2H(y)(Ay) · y + |y|α−2f(t) · y ) . (4.2) Utilizing (2.2), (2.6), and (2.12), one has α(−c2Λn −M |y|δ) ≤ d dt (|y|α) ≤ α(−c1Λ1 +M |y|δ). Let a1 = c1Λ1/2 and a2 = c2Λn+1. Let r0 > 0 be such thatMrδ0 = min{1, c1Λ1/2}. Thanks to (2.9), there is T ∈ (t0, T∗) such that |y(t)| ≤ r0 on (T, T∗). Hence, −αa2 ≤ d dt (|y|α) ≤ −αa1 on (T, T∗). (4.3) For t ∈ [T, T∗), integrating (4.3) from t to t′ ∈ (t, T∗), passing to the limit t′ → T− ∗ , and using (2.9), we obtain αa1(T∗ − t) ≤ |y(t)|α ≤ αa2(T∗ − t) for all t ∈ [T, T∗]. (4.4) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 9 Above, (4.4) holds for t = T∗ thanks to (2.17). Note also that 0 < a3 := min t∈[t0,T ] (T∗ − t)−1/α|y(t)| ≤ a4 := max t∈[t0,T ] (T∗ − t)−1/α|y(t)| < ∞. (4.5) Combining (4.4) with (4.5), we obtain the desired estimates in (4.1) with C1 = min{(αa1)1/α, a3} and C2 = max{(αa2)1/α, a4}. Case 2. Consider A is not symmetric. Let A = S−1A0S as in (2.1). Same as (3.10), we set z(t) = Sy(t) for t ∈ [t0, T∗]. Then z belongs to C1([t0, T∗),Rn \{0})∩ C([t0, T∗],Rn), z(t) ̸= 0 for all t ∈ [t0, T∗), z(T∗) = 0, and z′ = −H̃(z)A0z + f̃(t) for t ∈ (t0, T∗), (4.6) where H̃(z) = H(S−1z) for z ∈ Rn \ {0}, and f̃(t) = Sf(t) for t ∈ [t0, T∗). (4.7) Thanks to Assumption 2.3, we can verify that H̃ belongs to H−α(Rn,R) and is positive and continuous on Rn \ {0}. Moreover, it is clear that the function f̃ is continuous on [t0, T∗). Thanks to (2.12) and (3.12), it satisfies, for t ∈ [t0, T∗), |f̃(t)| ≤ ∥S∥ · |f(t)| ≤ ∥S∥M |y(t)|1−α+δ ≤ M̃ |z(t)|1−α+δ, (4.8) where M̃ = { M∥S∥ · ∥S−1∥1−α+δ, if 1− α+ δ ≥ 0, M∥S∥ · ∥S∥−(1−α+δ) = M∥S∥α−δ, otherwise. Therefore, we can apply the result in Case 1 to the solution z(t) and equation (4.6). Then there exist two positive constants C ′ 1 and C ′ 2 such that C ′ 1(T∗ − t)1/α ≤ |z(t)| ≤ C ′ 2(T∗ − t)1/α for all t ∈ [t0, T∗]. (4.9) Combining (4.9) with the relations in (3.12), we obtain the estimates in (4.1) for y(t). □ The following are two immediate consequences of Lemma 4.1. (a) By (2.6) and (4.1), we have, for all t ∈ [t0, T∗), C3(T∗ − t)−1 ≤ H(y(t)) ≤ C4(T∗ − t)−1, (4.10) where C3 = c1C −α 2 and C4 = c2C −α 1 . (b) We also observe from (2.12) and (4.1) that, for all t ∈ [t0, T∗), |f(t)| ≤ M(T∗ − t)1/α−1+δ/α · { C1−α+δ 2 , if 1− α+ δ ≥ 0, C1−α+δ 1 , otherwise. (4.11) 5. Proof for a special case Let a be an arbitrarily positive number. Consider equation (2.11) in the case A = In and H(x) = a|x|−α, (5.1) that is, equation (2.11) becomes y′ = −a|y|−αy + f(t) for t ∈ (t0, T∗). (5.2) Theorem 2.9 for this particular case is simply the following. 10 L. HOANG EJDE-2025/08 Theorem 5.1. Given numbers T∗ > t0 ≥ 0. Let y ∈ C1([t0, T∗),Rn) and f ∈ C([t0, T∗),Rn) satisfy (2.8), (2.9), (2.12) and (5.2). Then there exists a vector ξ∗ ∈ Rn such that |ξ∗| = (αa)1/α, (5.3) and, as t → T− ∗ , |y(t)− (T∗ − t)1/αξ∗| = O((T∗ − t)1/α+ε) for some ε > 0. (5.4) Proof. Regarding the function f that satisfies (2.12), we observe, for any number δ′ ∈ (0, δ), that |f(t)| ≤ M ′|y(t)|1−α+δ′ for all t ∈ [t0, T∗), (5.5) where M ′ = M max t∈[t0,T∗] |y(t)|δ−δ′ ∈ (0,∞). (5.6) Note that we used (2.17) in (5.6). Because of property (5.5), we can assume that δ < α in (2.12). With the matrix A and function H in (5.1), they certainly satisfy Assumptions 2.1 and 2.3. Then Lemma 4.1 applies and the estimates from above and below for |y(t)| in (4.1), and estimate (4.11) for |f(t)| hold. For t ∈ (t0, T∗), from (4.2) we have d dt (|y|α) = −αa+ α|y|α−2f(t) · y. (5.7) Integrating equation (5.7) from t to t′ ∈ (t, T∗), passing to the limit t′ → T− ∗ , and using (2.9) give |y(t)|α = αa(T∗ − t) + g(t), where g(t) = −α ∫ T∗ t |y(τ)|α−2f(τ) · y(τ)dτ. (5.8) Hence, for all t ∈ [t0, T∗), one has αa(T∗ − t) + g(t) > 0. Using the Cauchy-Schwarz inequality, (2.12) and the upper bound of |y(t)| in (4.1), we estimate |g(t)| ≤ α ∫ T∗ t |y(τ)|α−1|f(τ)|dτ ≤ αM ∫ T∗ t |y(τ)|δdτ ≤ αMCδ 2 ∫ T∗ t (T∗ − τ)δ/αdτ. We obtain |g(t)| ≤ C3(T∗ − t)1+δ/α for all t ∈ [t0, T∗), where C3 = αMCδ 2 1 + δ/α . (5.9) We consider equation (5.2) as a linear equation of y with time-dependent coeffi- cient −a|y(t)|−α and forcing function f(t). By the variation of constants formula, we solve for y(t) explicitly as y(t) = e−J(t) ( y0 + ∫ t t0 eJ(τ)f(τ)dτ ) for t ∈ [t0, T∗), where J(t) = a ∫ t t0 |y(τ)|−αdτ. (5.10) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 11 Using (5.8) in (5.10), we rewrite J(t) as J(t) = ∫ t t0 a aα(T∗ − τ) + g(τ) dτ = J1(t) + J2(t), where J1(t) = ∫ t t0 1 α(T∗ − τ) dτ and J2(t) = ∫ t t0 h(τ)dτ, with h(τ) = −g(τ) α(T∗ − τ)(aα(T∗ − τ) + g(τ)) . Clearly, J1(t) = − 1 α ln(T∗ − t) + 1 α ln(T∗ − t0). Therefore, y(t) = (T∗ − t)1/α (T∗ − t0)1/α e−J2(t) ( y0 + (T∗ − t0) 1/α ∫ t t0 eJ2(τ) (T∗ − τ)1/α f(τ)dτ ) . (5.11) Consider the integrand h(τ) of J2(t). Taking into account the estimate of |g(τ)| in (5.9), we assert that, as τ → T− ∗ , |h(τ)| = O(|g(τ)|(T∗ − τ)−2) = O((T∗ − τ)−1+δ/α). (5.12) Thus, lim t→T− ∗ J2(t) = ∫ T∗ t0 h(τ)dτ = J∗ ∈ R, (5.13) J2(t) = J∗ − h1(t), where h1(t) = ∫ T∗ t h(τ)dτ ∈ R. (5.14) From estimate (5.12), it follows that |h1(t)| = O((T∗ − t)δ/α) as t → T− ∗ . (5.15) Regarding the integral in formula (5.11), we have, thanks to estimate (4.11) of |f(t)|, that |f(t)| (T∗ − t)1/α = O((T∗ − t)−1+δ/α) as t → T− ∗ . (5.16) Hence, lim t→T− ∗ ∫ t t0 eJ2(τ) (T∗ − τ)1/α f(τ)dτ = ∫ T∗ t0 eJ2(τ) (T∗ − τ)1/α f(τ)dτ = η∗ ∈ Rn, and ∫ t t0 eJ2(τ) (T∗ − τ)1/α f(τ)dτ = η∗ − η(t), (5.17) where η(t) = ∫ T∗ t eJ2(τ) (T∗ − τ)1/α f(τ)dτ ∈ Rn. It follows from (5.13) and (5.16) that |η(t)| = O((T∗ − t)δ/α) as t → T− ∗ . (5.18) 12 L. HOANG EJDE-2025/08 Combining (5.11), (5.14) and (5.17) gives y(t) = (T∗ − t)1/α (T∗ − t0)1/α e−J∗+h1(t) ( y0 + (T∗ − t0) 1/α(η∗ − η(t)) ) for t ∈ [t0, T∗). Then y(t)− (T∗ − t)1/αe−J∗ ( y0 (T∗ − t0)1/α + η∗ ) = (T∗ − t)1/αe−J∗(eh1(t) − 1) ( y0 (T∗ − t0)1/α + η∗ ) − (T∗ − t)1/αe−J∗+h1(t)η(t). Let ξ∗ = e−J∗((T∗ − t0) −1/αy0 + η∗) ∈ Rn. This expression and properties (5.15), (5.18) imply, as t → T− ∗ , |y(t)− (T∗ − t)1/αξ∗| = O ( (T∗ − t)1/α(|eh1(t) − 1|+ |η(t)|) ) = O ( (T∗ − t)1/α(|h1(t)|+ |η(t)|) ) , thus, |y(t)− (T∗ − t)1/αξ∗| = O((T∗ − t)1/α+δ/α). (5.19) Therefore, we obtain the desired estimate (5.4). Because of the lower bound of |y(t)| in (4.1), the vector ξ∗ in (5.4) must be nonzeo. Now we prove property (5.3). By the triangle inequality and (5.19), one has∣∣(T∗ − t)−1/α|y(t)| − |ξ∗| ∣∣ = O((T∗ − t)δ/α). (5.20) From (5.8), (T∗ − t)−1/α|y(t)| = ( aα+ g(t) T∗ − t )1/α . (5.21) Taking into account estimate (5.9) of |g(t)|, from (5.21) we have that as t → T− ∗ ,∣∣(T∗ − t)−1/α|y(t)| − (aα)1/α ∣∣ = O ( |g(t)| T∗ − t ) = O((T∗ − t)δ/α). (5.22) From the two asymptotic estimates (5.20) and (5.22), one must have |ξ∗| = (aα)1/α, which proves (5.3). The proof is complete. □ Remark 5.2. In the case dimension n = 1, Theorem 5.1 already proves Theorem 2.9 for any positive constant A and positive function H ∈ H−α(R,R). We justify this fact below. Let y(t) be the solution of (2.11) as in Theorem 2.9. With H ∈ H−α(R,R), we have H(x) = { |x|−αH(1), for x > 0, |x|−αH(−1), for x < 0. In general, H(1) ̸= H(−1), hence it appears that we do not have equation (5.2) yet. However, for our continuous solution y(t) ̸= 0 on [t0, T∗), we must have either y(t) > 0 on [t0, T∗) or y(t) < 0 on [t0, T∗). Therefore, y(t), in fact, satisfies (5.2) for all t ∈ (t0, T∗), with a = AH(1) or a = AH(−1). Then Theorem 5.1 applies. (As a side note, because S0 = {−1, 1}, property (HC) on S0 is automatically satisfied.) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 13 6. Solutions when the matrix A is symmetric In this section, we assume Assumptions 2.1 and 2.3 hold and, additionally, the matrix A is symmetric. Let numbers T∗ > t0 ≥ 0 be given, and let function y ∈ C1([t0, T∗),Rn) satisfy (2.8), (2.9), (2.11)–(2.12). For t ∈ [t0, T∗), define λ(t) = y(t) ·Ay(t) |y(t)|2 and v(t) = y(t) |y(t)| . (6.1) (The quotient λ(t) in (6.1) imitates the Dirichlet quotient for the heat equations whenA is the negative Laplacian.) Then λ ∈ C1([t0, T∗),R) and v ∈ C1([t0, T∗),Rn). Moreover, one has, |v(t)| = 1 and, thanks to (2.2), Λ1 ≤ λ(t) ≤ Λn ≤ ∥A∥ for all t ∈ [t0, T∗). (6.2) Proposition 6.1. One has lim t→T− ∗ λ(t) = Λ ∈ σ(A). Proof. For t ∈ (t0, T∗), we have λ′(t) = 2 |y|2 y′ ·Ay − 2(y ·Ay) |y|4 y′ · y = 2 |y|2 y′ · (Ay − λ(t)y). (6.3) By equation (2.11), we write y′ as y′ = −H(y)(Ay − λ(t)y)− λ(t)H(y)y + f(t), and use it in (6.3) to obtain λ′(t) = −2H(y) |y|2 |Ay − λ(t)y|2 − 2λ(t)H(y) |y|2 y · (Ay − λ(t)y) + h(t), where h(t) = 2 |y(t)|2 f(t) · (Ay(t)− λ(t)y(t)). Because y(t) · (Ay(t)− λ(t)y(t)) = 0, it follows that λ′(t) = −2H(y)|Av − λ(t)v|2 + h(t). (6.4) Using (2.12), (6.2), the fact |v(t)| = 1, and then (4.1), we estimate |h(t)| ≤ 4M∥A∥ · |y(t)|−α+δ ≤ C5(T∗ − t)−1+δ/α for all t ∈ [t0, T∗), (6.5) where C5 is 4M∥A∥C−α+δ 2 if δ ≥ α, or 4M∥A∥C−α+δ 1 otherwise. For t, t′ ∈ [t0, T∗) with t′ > t, integrating equation (6.4) from t to t′ gives λ(t′)− λ(t) + 2 ∫ t′ t H(y(τ))|Av(τ)− λ(τ)v(τ)|2dτ = ∫ t′ t h(τ)dτ. (6.6) Thanks to (6.5), the last integral can be estimated as∣∣∣ ∫ t′ t h(τ)dτ ∣∣∣ ≤ αC5 δ (T∗ − t)δ/α. (6.7) By taking the limit superior of (6.6), as t′ → T− ∗ , we derive lim sup t′→T− ∗ λ(t′) ≤ λ(t) + αC5 δ (T∗ − t)δ/α < ∞. (6.8) 14 L. HOANG EJDE-2025/08 Then taking the limit inferior of (6.8), as t → T− ∗ , yields lim sup t′→T− ∗ λ(t′) ≤ lim inf t→T− ∗ λ(t). This and (6.2) imply lim t→T− ∗ λ(t) = Λ ∈ [Λ1,Λn]. (6.9) It remains to prove that Λ is an eigenvalue of A. Using properties (6.7) and (6.9) in (6.6) and by the Cauchy criterion, as t, t′ → T− ∗ , we obtain∫ T∗ t0 H(y(τ))|Av(τ)− λ(τ)v(τ)|2dτ < ∞. (6.10) We claim that ∀ε ∈ (0, T∗ − t0),∃t ∈ [T∗ − ε, T∗) : |Av(t)− λ(t)v(t)| < ε. (6.11) Indeed, suppose the claim (6.11) is not true, then ∃ε0 ∈ (0, T∗ − t0),∀t ∈ [T∗ − ε0, T∗) : |Av(t)− λ(t)v(t)| ≥ ε0. (6.12) Combining (6.12) with property (4.10), we have∫ T∗ T∗−ε0 H(y(τ))|Av(τ)− λ(τ)v(τ)|2dτ ≥ ∫ T∗ T∗−ε0 C3(T∗ − τ)−1ε20dτ = ∞, which contradicts (6.10). Hence, the claim (6.11) is true. Thanks to (6.11), there exists a sequence (tj) ∞ j=1 ⊂ [t0, T∗) such that lim j→∞ tj = T∗ and lim j→∞ |Av(tj)− λ(tj)v(tj)| = 0. (6.13) The first equation in (6.13) and (6.9) imply λ(tj) → Λ as j → ∞. Because v(tj) ∈ Sn−1 for all j, we can extract a subsequence (v(tjk)) ∞ k=1, such that v(tjk) → v̄ ∈ Sn−1 as k → ∞. Combining these limits with the second equation in (6.13) written with j = jk and k → ∞ yields Av̄ = Λv̄. Therefore, Λ is an eigenvalue of A. □ From here to the end of this section, Λ is the eigenvalue in Proposition 6.1. Proposition 6.2. There is ε > 0 such that |(In −RΛ)v(t)| = O((T∗ − t)ε) as t → T− ∗ . (6.14) Proof. If σ(A) = {Λ}, then RΛ = Id and (6.14) is true. Consider the case σ(A) ̸= {Λ}. We calculate v′ = 1 |y| y′ − 1 |y|3 (y′ · y)y = −H(y) |y| Ay + 1 |y| f(t) + H(y)(Ay) · y |y|3 y − f(t) · y |y|3 y. We define the function g : [t0, T∗) → Rn by g(t) = 1 |y(t)| f(t)− f(t) · y(t) |y(t)|3 y(t). Then we have v′ = −H(y)(Av − λ(t)v) + g(t) for all t ∈ (t0, T∗). (6.15) Using property (2.12) of f(t), one can estimate |g(t)| ≤ 2M |y(t)|−α+δ for all t ∈ [t0, T∗). (6.16) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 15 Let λj ∈ σ(A)\{Λ}. Applying Rλj to equation (6.15) and taking the dot product with Rλjv yield 1 2 d dt |Rλjv|2 = −H(y)(λj − λ(t))|Rλjv|2 +Rλjg(t) ·Rλjv. (6.17) Set µ = min{|λj − Λ| : 1 ≤ j ≤ d, λj ̸= Λ} > 0. (6.18) Applying Cauchy–Schwarz’s inequality, inequality (2.4) to |Rλj g(t)|, estimate (6.16) for |g(t)|, and then Cauchy’s inequality, we have |Rλj g(t) ·Rλj v| ≤ 2M |y|−α+δ|Rλj v| ≤ µ 4 H(y)|Rλj v|2 + 4M2|y|−2α+2δ µH(y) . Using the first inequality of (2.6) to estimate the last H(y) gives |Rλjg(t) ·Rλjv| ≤ µ 4 H(y)|Rλjv|2 + 4M2 µc1 |y|−α+2δ. Utilizing the estimates in (4.1) for the norm |y(t)|, we obtain, for t ∈ [t0, T∗), |Rλjg(t) ·Rλjv| ≤ µ 4 H(y)|Rλjv|2 + C6 2 (T∗ − t)−1+2δ/α, (6.19) where C6 = 8M2 µc1 · { C−α+2δ 2 , if δ ≥ α/2, C−α+2δ 1 , otherwise. Below, T ∈ (t0, T∗) is fixed and can be taken sufficiently close to T∗ such that |λ(t)− Λ| ≤ µ 4 for all t ∈ [T, T∗). (6.20) Case λj > Λ. In this case, combining (6.17) and (6.19) yields, for t ∈ (t0, T∗), 1 2 d dt |Rλj v|2 ≤ −(λj − λ(t)− µ 4 )H(y)|Rλj v|2 + C6 2 (T∗ − t)−1+2δ/α. By definition (6.18) of µ and the choice (6.20), one has, for all t ∈ [T, T∗), λj − λ(t)− µ 4 = (λj − Λ) + (Λ− λ(t))− µ 4 ≥ µ− µ 4 − µ 4 = µ 2 . (6.21) Thus, for t ∈ [T, T∗), d dt |Rλj v|2 ≤ −µH(y)|Rλj v|2 + C6(T∗ − t)−1+2δ/α. (6.22) Let t and t̄ be any numbers in [T, T∗) with t > t̄. It follows from (6.22) that |Rλj v(t)|2 ≤ e−µ ∫ t t̄ H(y(τ))dτ |Rλj v(t̄)|2 + C6 ∫ t t̄ e−µ ∫ t τ H(y(s))ds(T∗ − τ)−1+2δ/αdτ. (6.23) With C3 being the positive constant in (4.10), we fix a number θ > 0 such that θ ≤ C3 and θµ < 2δ/α. Then H(y(t)) ≥ θ(T∗ − t)−1 for all t ∈ [t0, T∗). (6.24) Utilizing this estimate in (6.23) gives |Rλj v(t)|2 16 L. HOANG EJDE-2025/08 ≤ e−θµ ∫ t t̄ (T∗−τ)−1dτ |Rλjv(t̄)|2 + C6 ∫ t t̄ e−θµ ∫ t τ (T∗−s)−1ds(T∗ − τ)−1+2δ/αdτ = (T∗ − t)θµ (T∗ − t̄)θµ |Rλjv(t̄)|2 + C6(T∗ − t)θµ ∫ t t̄ (T∗ − τ)−1+2δ/α−θµdτ = (T∗ − t)θµ (T∗ − t̄)θµ |Rλj v(t̄)|2 + C6(T∗ − t)θµ 2δ/α− θµ ( (T∗ − t̄)2δ/α−θµ − (T∗ − t)2δ/α−θµ ) . Therefore, |Rλj v(t)|2 ≤ ( |Rλj v(t̄)|2 (T∗ − t̄)θµ + C6(T∗ − t̄)2δ/α−θµ 2δ/α− θµ ) (T∗ − t)θµ. (6.25) Having t̄ = T in (6.25), we obtain |Rλj v(t)| = O((T∗ − t)θµ/2) as t → T− ∗ . (6.26) Case λj < Λ. Using (6.19) to have a lower bound for the last term in (6.17), we have 1 2 d dt |Rλj v|2 ≥ (λ(t)− λj − µ 4 )H(y)|Rλj v|2 − C6 2 (T∗ − t)−1+2δ/α. Same as (6.21), one has, for t ∈ [T, T∗), λ(t)− λj − µ 4 = (λ(t)− Λ) + (Λ− λj)− µ 4 ≥ −µ 4 + µ− µ 4 = µ 2 . Hence, d dt |Rλj v|2 ≥ µH(y)|Rλj v|2 − C6(T∗ − t)−1+2δ/α. Then, for any t, t̄ ∈ [T, T∗) with t > t̄, one has e−µ ∫ t t̄ H(y(τ))dτ |Rλj v(t)|2 − |Rλj v(t̄)|2 ≥ −C6 ∫ t t̄ e−µ ∫ τ t̄ H(y(s))ds(T∗ − τ)−1+2δ/αdτ. (6.27) Note from (6.24) that ∫ T∗ t̄ H(y(τ))dτ = ∞, and from (2.4) that |Rλj v(t)| ≤ |v(t)| = 1. Then lim t→T− ∗ e−µ ∫ t t̄ H(y(τ))dτ |Rλj v(t)|2 = 0. Letting t → T− ∗ in (6.27) and using (6.24) yield |Rλjv(t̄)|2 ≤ C6 ∫ T∗ t̄ e−µ ∫ τ t̄ H(y(s))ds(T∗ − τ)−1+2δ/αdτ ≤ C6 ∫ T∗ t̄ (T∗ − τ)θµ (T∗ − t̄)θµ (T∗ − τ)−1+2δ/αdτ = C6 θµ+ 2δ/α (T∗ − t̄)2δ/α. Therefore, |Rλj v(t̄)| = O((T∗ − t̄)δ/α) as t̄ → T− ∗ . (6.28) We estimate |(In −RΛ)v(t)| now. We have |(In −RΛ)v(t)| = ∣∣∣ ∑ 1≤j≤d,λj ̸=Λ Rλj v(t) ∣∣∣ ≤ ∑ 1≤j≤d,λj ̸=Λ |Rλj v(t)|. (6.29) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 17 In the last sum in (6.29), we estimate |Rλj v(t)| for all λj > Λ by (6.26), and estimate |Rλjv(t)| for all λj < Λ by (6.28). This results in the desired estimate (6.14) for |(In −RΛ)v|, with ε = min{θµ/2, δ/α} = θµ/2. □ We derive from Proposition 6.2 more specific estimates for y(t). Let ε > 0 be as in Proposition 6.2. On the one hand, we have |(In −RΛ)y(t)| = |y(t)| · |(In −RΛ)v(t)|. Together with (4.1) and (6.14), it yields |(In −RΛ)y(t)| = O((T∗ − t)1/α+ε) as t → T− ∗ . (6.30) On the other hand, by the triangle inequality and (4.1), one has |RΛy(t)| ≤ |y(t)|+ |(In −RΛ)y(t)| ≤ C2(T∗ − t)1/α + |(In −RΛ)y(t)|, |RΛy(t)| ≥ |y(t)| − |(In −RΛ)y(t)| ≥ C1(T∗ − t)1/α − |(In −RΛ)y(t)|. Combining these inequalities with estimate (6.30) for |(In − RΛ)y(t)|, we deduce that there exist numbers T0 ∈ [t0, T∗) and C7, C8 > 0 such that C7(T∗ − t)1/α ≤ |RΛy(t)| ≤ C8(T∗ − t)1/α for all t ∈ [T0, T∗). (6.31) Proposition 6.3. There exists a unit vector v∗ ∈ Rn such that |RΛv(t)− v∗| = O((T∗ − t)ε) as t → T− ∗ for some ε > 0. (6.32) Proof. Let ε0 > 0 be such that (6.14) holds for ε = ε0. Then one has∣∣1− |RΛv(t)| ∣∣ = ∣∣|v(t)| − |RΛv(t)| ∣∣ ≤ |v(t)−RΛv(t)| = O((T∗ − t)ε0). (6.33) Let T0 be as in (6.31). Recall that C4 is the positive constant in (4.10). We fix a number ε1 > 0 such that C4ε1 < δ/α. (6.34) Thanks to Proposition 6.1, there is T ∈ [T0, T∗) such that |λ(t)− Λ| ≤ ε1 for all t ∈ [T, T∗). Note from (6.31) that RΛv(t) ̸= 0 for all t ∈ [T, T∗). Applying RΛ to equation (6.15) yields, for t ∈ (t0, T∗), d dt RΛv = −H(y)(Λ− λ(t))RΛv +RΛg(t). (6.35) Then, for t ∈ [T, T∗), d dt |RΛv| = 1 |RΛv| ( d dt RΛv ) ·RΛv = −H(y)(Λ− λ(t))|RΛv|+ g1(t), (6.36) where g1(t) = RΛg(t) ·RΛv(t) |RΛv(t)| . Solving for solution |RΛv(t)| by the variation of constants formula from the differential equation (6.36) gives, for t̄, t ∈ [T, T∗) with t > t̄, |RΛv(t)| = e− ∫ t t̄ H(y(τ))(Λ−λ(τ))dτ ( |RΛv(t̄)|+ ∫ t t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsg1(τ)dτ ) . 18 L. HOANG EJDE-2025/08 It yields∫ t t̄ H(y(τ))(Λ− λ(τ))dτ = ln ( |RΛv(t̄)|+ ∫ t t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsg1(τ)dτ ) − ln |RΛv(t)|. (6.37) We have from (2.4), (4.1) and (6.16) that |g1(t)| ≤ |RΛg(t)| ≤ |g(t)| ≤ C9(T∗ − t)−1+δ/α for all t ∈ [T, T∗), (6.38) where C9 is 2MC−α+δ 2 if δ ≥ α, and is 2MC−α+δ 1 otherwise. By (6.38) and (4.10), we have, for τ ∈ [t̄, T∗), e ∫ τ t̄ H(y(s))(Λ−λ(s))ds|g1(τ)| ≤ e ∫ τ t̄ C4ε1(T∗−s)−1dsC9(T∗ − τ)−1+δ/α = C9(T∗ − t̄)C4ε1(T∗ − τ)−1+δ/α−C4ε1 . Thanks to this and (6.34), lim t→T− ∗ ∫ t t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsg1(τ)dτ = ∫ T∗ t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsg1(τ)dτ = η(t̄) ∈ R. (6.39) Note that |η(t̄)| ≤ C9(T∗−t̄)C4ε1 ∫ T∗ t̄ (T∗−τ)−1+δ/α−C4ε1dτ = C9 δ/α− C4ε1 (T∗−t̄)δ/α. (6.40) Passing to the limit as t → T− ∗ in (6.37), and using that |RΛv(t)| → 1, thanks to (6.33), we have∫ T∗ t̄ H(y(τ))(Λ− λ(τ))dτ = ln(|RΛv(t̄)|+ η(t̄)) ∈ R. (6.41) By (6.41), we can define, for t ∈ [T, T∗), h(t) = ∫ T∗ t H(y(τ))(Λ− λ(τ))dτ ∈ R. We rewrite (6.41) for t̄ = t as h(t) = ln(|RΛv(t)|+ η(t)) = ln(1 + (|RΛv(t)| − 1) + η(t)). With this expression and properties (6.33) and (6.40), we have, as t → T− ∗ , |h(t)| = O( ∣∣|RΛv(t)| − 1 ∣∣+ |η(t)|) = O((T∗ − t)ε0 + (T∗ − t)δ/α) = O((T∗ − t̄)ε2), (6.42) where ε2 = min{ε0, δ/α}. Solving for RΛv(t) from (6.35) by the variation of constants formula, one has RΛv(t) = e− ∫ t t̄ H(y(τ))(Λ−λ(τ))dτ ( RΛv(t̄) + ∫ t t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsRΛg(τ)dτ ) . (6.43) Using the same arguments as those from (6.38) to (6.40) with RΛg(τ) replacing g1(τ), we obtain, similar to (6.39) and (6.40), that lim t→T− ∗ ∫ t t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsRΛg(τ)dτ = ∫ T∗ t̄ e ∫ τ t̄ H(y(s))(Λ−λ(s))dsRΛg(τ)dτ EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 19 = X(t̄) ∈ Rn for all t̄ ∈ [T, T∗), and |X(t̄)| = O((T∗ − t̄)ε2) as t̄ → T− ∗ . (6.44) Taking t → T− ∗ in (6.43) gives lim t→T− ∗ RΛv(t) = v∗ := e−h(t̄)(RΛv(t̄) +X(t̄)) ∈ Rn. Note that X(t) = ∫ T∗ t eh(t)−h(τ)RΛg(τ)dτ. Using h(t), X(t) and v∗, we rewrite (6.43) as RΛv(t) = eh(t)−h(t̄) ( RΛv(t̄) +X(t̄)− ∫ T∗ t eh(t̄)−h(τ)RΛg(τ)dτ ) = eh(t)v∗ −X(t). Thus, |RΛv(t)− v∗| ≤ |eh(t) − 1| · |v∗|+ |X(t)|. Using (6.42) and (6.44), we deduce, as t → T− ∗ , |RΛv(t)− v∗| = O(|h(t))|+ |X(t)|) = O((T∗ − t)ε2). Therefore, we obtain the desired estimate (6.32). Because of (6.32) and (6.33), we have |v∗| = 1. The proof is complete. □ Some immediate consequences of (6.14) and (6.32) are lim t→T− ∗ RΛv(t) = lim t→T− ∗ v(t) = v∗, (6.45) |v(t)− v∗| = O((T∗ − t)ε) as t → T− ∗ for some ε > 0. (6.46) 7. Proofs of the main theorems We prove Theorems 2.9 and 2.10 in this section. Proof of Theorem 2.9. Since H satisfies Assumption 2.8, it follows that, thanks to Lemma 2.7, H is positive, continuous on Rn \ {0}, and has property (HC) on Rn \ {0}. Case 1. Consider the case A is symmetric first. We use the same notation as in Section 6. The estimate (2.14) already comes from (6.30). We prove (2.15) next. Applying RΛ to equation (2.11), we have (RΛy) ′ = −ΛH(y)RΛy +RΛf(t). (7.1) Let v∗ be the unit vector in Proposition 6.3, and ε0 > 0 be such that (6.14), (6.30), (6.32), and (6.46) hold for ε = ε0. We rewrite H(y) on the right-hand side of (7.1) as H(y) = |y|−αH(v) = |RΛy|−αH(v∗) + g0(t), where g0(t) = |y(t)|−α(H(v(t))−H(v∗)) + (|y(t)|−α − |RΛy(t)|−α)H(v∗) = |y(t)|−α { H(v(t))−H(v∗) + (1− |RΛv(t)|−α)H(v∗) } . Then (RΛy) ′ = −ΛH(v∗)|RΛy|−αRΛy + f0(t), (7.2) 20 L. HOANG EJDE-2025/08 where f0(t) = −Λg0(t)RΛy(t) +RΛf(t). We estimate |g0(t)| first. We combine inequality (2.10) in Definition 2.6 applied to F = H, E = Sn−1, x0 = v∗ and x = v(t) for t sufficiently close to T∗, with estimate (6.46). Then there exists a number γ > 0 such that, as t → T− ∗ , |H(v(t))−H(v∗)| = O(|v(t)− v∗|γ) = O((T∗ − t)γε0). (7.3) It is elementary to see |s−α − 1| = O(|s − 1|) as s → 1. By taking s = |RΛv(t)|, which goes to 1 as t → T− ∗ thanks to (6.45), and using estimate (6.33), we derive∣∣1− |RΛv(t)|−α ∣∣ = O (∣∣1− |RΛv(t)| ∣∣) = O((T∗ − t)ε0) as t → T− ∗ . (7.4) Combining (7.3), (7.4) with (4.1), we obtain |g0(t)| = O(|y(t)|−α((T∗ − t)γε0 + (T∗ − t)ε0)) = O((T∗ − t)−1+ε1) (7.5) as t → T− ∗ , where ε1 = ε0 min{1, γ}. We estimate |f0(t)| now. As t → T− ∗ , we have from (4.1) and (4.11) that |RΛy(t)| = O((T∗ − t)1/α) and |RΛf(t)| = O((T∗ − t)1/α−1+δ/α). (7.6) Combining (7.5) and (7.6) gives, as t → T− ∗ , |f0(t)| = O((T∗ − t)−1+ε1(T∗ − t)1/α +(T∗ − t)1/α−1+δ/α) = O((T∗ − t)1/α−1+ε2/α), where ε2 = min{ε1α, δ}. By the virtue of the lower bound of |RΛy(t)| in (6.31), we actually have |f0(t)| = O(|RΛy(t)|1−α+ε2). Fix a number t′0 ∈ [t0, T∗) such that RΛy(t) ̸= 0 and |f0(t)| ≤ M0|RΛy(t)|1−α+ε2 for all t ∈ [t′0, T∗), where M0 is a positive constant. Of course, one already has RΛy(T∗) = 0. We apply Theorem 5.1 to solution RΛy(t) of equation (7.2) on the interval [t′0, T∗). Specifically, RΛy(t) satisfies equation (5.2) on (t′0, T∗) with constant a = ΛH(v∗) and f = f0. Then there exists a nonzero vector ξ∗ ∈ Rn such that |RΛy(t)− (T∗ − t)1/αξ∗| = O((T∗ − t)1/α+ε3) for some ε3 > 0, (7.7) |ξ∗| = (αΛH(v∗)) 1/α. (7.8) The desired statement (2.15) immediately follows from (7.7). Because ξ∗ = lim t→T− ∗ (T∗ − t)−1/αRΛy(t), by (2.15), and the fact ξ∗ ̸= 0, we have ξ∗ ∈ RΛ(Rn) \ {0}. Hence, ξ∗ is an eigenvector of A associated with Λ. Next, we prove (2.13). Writing y(t)− (T∗ − t)1/αξ∗ = (In −RΛ)y(t) + (RΛy(t)− (T∗ − t)1/αξ∗), and using the estimate (6.30) with ε = ε0, and estimate (7.7) yield |y(t)− (T∗ − t)1/αξ∗| = O((T∗ − t)1/α+ε0 + (T∗ − t)1/α+ε3). This implies (2.13) with ε = min{ε0, ε3}. Finally, we prove (2.16). Let w(t) = (T∗−t)−1/αy(t) and write v(t) = w(t)/|w(t)|. Passing t → T− ∗ and noticing that v(t) → v∗ and w(t) → ξ∗, thanks to (6.45) and (2.13), we obtain v∗ = ξ∗/|ξ∗| (7.9) EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 21 Then it follows from (7.8), the fact H is positively homogeneous of degree −α, and relation (7.9) that 1 = αΛH(v∗)|ξ∗|−α = αΛH(|ξ∗|v∗) = αΛH(ξ∗). Hence, we obtain (2.16). This completes the proof for the case of symmetric matrix A. Case 2. Consider the case A is not symmetric. Let A0 and S be as in (2.1). Same as (3.10), we set z(t) = Sy(t) on [t0, T∗]. Then z(t) satisfies equation (4.6) with H̃ and f̃ defined in (4.7). One can verify the following facts. • z(t) ̸= 0 for t ∈ [t0, T∗) and z(T∗) = 0. • H̃ ∈ H−α(Rn) and, thanks to parts (i) and (v) of Lemma 2.7, H̃ > 0 on Sn−1 and H̃ has property (HC) on Sn−1. • Thanks to (4.8), f̃(t) and z(t) satisfy condition (2.12) with the same num- bers α, δ, t0, T∗, and constant M̃ in place of M . We apply the results already established in Case 1 above to the solution z(t) of equation (4.6). Note that A0 replaces A and R̂λj replaces Rλj . Then there exist an eigenvalue Λ of A0 and an eigenvector ξ0 of A0 associated with Λ such that |(In − R̂Λ)z(t)| = O((T∗ − t)1/α+ε), |R̂Λz(t)− (T∗ − t)1/αξ0| = O((T∗ − t)1/α+ε) (7.10) for some number ε > 0, and αΛH̃(ξ0) = 1. (7.11) Let ξ∗ = S−1ξ0. Then Λ is an eigenvalue of A and ξ∗ is an eigenvector of A associated with Λ. We rewrite (7.10) as |S(In −RΛ)y(t)| = O((T∗ − t)1/α+ε), |S(RΛy(t)− (T∗ − t)−1/αξ∗)| = O((T∗ − t)1/α+ε), which imply (2.14) and (2.15). By (2.14), (2.15) and the triangle inequality, we obtain (2.13) in the same way as in Case 1. Finally, (2.16) follows from (7.11) and the relation H̃(ξ0) = H(ξ∗). The proof of Theorem 2.9 is complete. □ Proof of Theorem 2.10. Set f(t) = G(t, y(t)) for t ∈ [t0, T∗). Thanks to (2.9), the is a number t′0 ∈ [t0, T∗) such that |y(t)| ≤ r∗ for all t ∈ [t′0, T∗). This property and (2.7) imply |f(t)| ≤ c∗|y(t)|1−α+δ for all t ∈ [t′0, T∗). Thus, f(t) satisfies condition (2.12) with t′0 replacing t0. Applying Theorem 2.9 to the interval [t′0, T∗) in place of [t0, T∗), we obtain the statements of Theorem 2.10. □ 22 L. HOANG EJDE-2025/08 8. Examples and applications 8.1. Examples. We give some examples for the function H in Assumptions 2.3 and 2.8. For simplicity, we consider the case dimension n = 2. One can easily generalize them for any higher dimension n. (a) For x = (x1, x2) ∈ R2 \ {0}, let H1(x) = (x4 1 + 5x4 2) −3, H2(x) = [ (3|x1|3/2 + |x2|3/2)1/3 + (2|x1|5/3 + 7|x2|5/3)3/10 ]−1/8 . Then H1 is in H−3/4(R2,R) while H2 is in H−1/16(R2,R). Both functions belong to C1(R2 \ {0}). Hence, they have property (HC) on S1 with the same power γ = 1 in (2.10). (b) Another example is H(x) = √ |x1|+ √ |x2| |x| for x = (x1, x2) ∈ R2 \ {0}. Then H belongs to H−1/2(R2,R), is positive on S1 and has property (HC) on S1 with the same power γ = 1/2 in (2.10). Unlike the previous two examples, this function H is not in C1(R2 \ {0}). In fact, the function H can be very complicated, see similar examples in [27, Ex- ample 5.7] and [13, Section 6]. 8.2. Applications. We give an application to a population model [32] in mathe- matical biology. Consider an inhomogeneous population composed of individuals with different death rates. This population consists of n clones, each has the size yi(t), for i = 1, 2, . . . , n, at time t with the death rate ki > 0, where the constants ki are mutually distinct. The model [32, Equations (3.1) and (3.2)] is y′i = −kiyig(N) for i = 1, 2, . . . , n, (8.1) where N = N(t) := y1(t) + y2(t) + . . .+ yn(t) is the total population size, (8.2) and g(s) : (0,∞) → (0,∞) is an appropriate function. Note that the integral form of N(t) in [32, Equation (3.2)] becomes the finite sum in (8.2). Below, we consider the case g(s) = ks−α for any s > 0, where k > 0 and α > 0 are some constants. (8.3) This generalizes the consideration α = 1 in [32, Equation (4.2)]. Define the matrix A and function H(x), for x = (x1, . . . , xn) ∈ Rn \ {0}, as follows A = k diag[k1, k2, . . . , kn] and H(x) = (|x1|+ |x2|+ . . .+ |xn|)−α. We denote C = {x = (x1, . . . , xn) ∈ Rn : xi ≥ 0 for i = 1, 2, . . . , n}. With y = (y1, . . . , yn), if y(t) ∈ C \ {0} is a solution of the system (8.1), (8.2), (8.3), then it is a solution of system (1.3) with G ≡ 0, i.e., y′ = −H(y)Ay. (8.4) Note that the matrix A is symmetric and satisfies Assumption 2.1. The function H clearly belongs to H−α(Rn,R). EJDE-2025/08 BEHAVIOR NEAR THE EXTINCTION TIME 23 Claim. The function H(x) has property (HC) on Sn−1 with the same power γ = min{1, α} in (2.10). Thus, H satisfies Assumption 2.8. Proof of the Claim. We denote the ℓ1-norm |x|ℓ1 = |x1| + |x2| + . . . + |xn|. Then there exists a constant c0 ≥ 1 such that c−1 0 |x| ≤ |x|ℓ1 ≤ c0|x| for all x ∈ Rn. For x, y ∈ Sn−1, we have both norms |x|ℓ1 and |y|ℓ1 belong to the interval [c−1 0 , c0], and, hence, |H(x)−H(y)| = ∣∣|y|αℓ1 − |x|αℓ1 ∣∣ |x|αℓ1 |y|αℓ1 ≤ c2α0 ∣∣|y|αℓ1 − |x|αℓ1 ∣∣. When 0 < α ≤ 1, utilizing the inequality ∣∣|y|αℓ1 − |x|αℓ1 ∣∣ ≤ ||y|ℓ1 − |x|ℓ1 |α, and applying the triangle inequality, one obtains |H(x)−H(y)| ≤ c2α0 |y − x|αℓ1 ≤ c3α0 |y − x|α. When α > 1, applying the Mean Value Theorem to the function s 7→ sα and values s1 = |y|ℓ1 , s2 = |x|ℓ1 both belonging to the interval [c−1 0 , c0] yields the existence of a number C0 > 0 depending on c0 and α such that ||y|αℓ1 − |x|αℓ1 | ≤ C0 ||y|ℓ1 − |x|ℓ1 | . Then applying the triangle inequality again, we deduce |H(x)−H(y)| ≤ c2α0 C0 ||y|ℓ1 − |x|ℓ1 | ≤ c2α0 C0|y − x|ℓ1 ≤ c2α+1 0 C0|y − x|. Therefore, the Claim is true. □ Theorem 8.1. Let y0 ∈ C \ {0} be sufficiently small, then there exist a number T∗ > 0 and a function y ∈ C1([0, T∗), C \ {0}) such that y(t) satisfies (8.1), (8.2), (8.3) for all t ∈ (0, T∗), y(0) = y0, and (2.9) holds. Proof. Applying Theorem 2.5 to equation (8.4) and t0 = 0, we obtain a number T∗ > 0 and a function y ∈ C1([0, T∗),Rn \ {0}) such that y(t) satisfies equation (8.4) for all t ∈ (0, T∗), y(0) = y0, and (2.9) holds. Since y(t) ̸= 0, for all t ∈ [0, T∗), we have, for each i = 1, 2, . . . , n, yi(t) = yi(0)e −kki ∫ t 0 H(y(τ))dτ . (8.5) Together with the fact y(0) ∈ C \ {0}, this implies y(t) ∈ C \ {0}, and hence y(t) is a solution of (8.1), (8.2), (8.3) for all t ∈ (0, T∗). □ Let {ei : i = 1, 2, . . . , n} denote the standard canonical basis of Rn. Theorem 8.2. Let T∗ > 0 and y ∈ C1([0, T∗), C \ {0}) be such that y(t) satisfies (8.1), (8.2), (8.3) for all t ∈ (0, T∗) and (2.9) holds. Then there is an integer i ∈ [1, n] and a number ε > 0 such that∣∣∣y(t)− (αkki) 1/α(T∗ − t)1/αei ∣∣∣ = O ( (T∗ − t)1/α+ε ) as t → T− ∗ . (8.6) Proof. Since y(t) ∈ C\{0}, the function y in fact is a solution of (8.4) on (0, T∗) with the extinction time T∗. By the virtue of Theorem 2.10 applied to equation (8.4) 24 L. HOANG EJDE-2025/08 and t0 = 0, there exist a number ε > 0, an eigenvalue Λ of A and a corresponding eigenvector ξ∗, such that∣∣∣y(t)− (T∗ − t)1/αξ∗ ∣∣∣ = O ( (T∗ − t)1/α+ε ) as t → T− ∗ . (8.7) Clearly, Λ = kki for some 1 ≤ i ≤ n, and ξ∗ = Kei for some number K ̸= 0. Multiplying estimate (8.7) by (T∗−t)−1/α and using the ith coordinate, one derives K = lim t→T− ∗ yi(t)(T∗ − t)−1/α, which implies K ≥ 0. Moreover, thanks to (2.16), |K|α = αkki. Therefore, K = (αkki) 1/α and we obtain (8.6) from (8.7). □ Note in this demonstration that the system (8.1) is simpler than (1.3) and (2.11) which were theoretically studied in the previous sections. Remark 8.3. The following final statements are in order. (a) There is another totally different approach to the local properties of solu- tions of ODE based on the Poincaré–Dulac normal form [1, 3, 34]. It has been generalized and developed by many and for so long, see the books [8, 9, 33], recent papers such as [4, 5, 6, 7], and references therein. Our ap- proach is relatively new and only recently used to explore different classes of equations and problems in ODE. For more comparisons between the other approach and ours, see [27, Remark 5.8] and [13, Remark 6.14]. (b) It is an open problem whether the solutions of (1.3) admit an asymptotic expansion similar to those in [18, 13] near the extinction time. Taking some indications from [13], we expect, in the case the answer is affirmative, that our result (2.13) will play an important role in its proof. 9. Appendix Proof of Lemma 2.7. We denote E = Rn \ {0}. For any x ∈ E, we can write F (x) = |x|−αF (x/|x|). Hence, part (i) is obvious. For parts (ii)–(iv), the proofs are similar to the proof of [27, Lemma 5.1], and “the verification of Assumption 5.2 for H̃” in the proof of [27, Theorem 5.3]. We present the key arguments here. Let x, ξ ∈ E. Then |F (x)− F (ξ)| = ∣∣|x|−αF (x/|x|)− |ξ|−αF (ξ/|ξ|) ∣∣ ≤ |x|−α |F (x/|x|)− F (ξ/|ξ|)|+ ∣∣|x|−α − |ξ|−α ∣∣ · ∣∣F (ξ/|ξ|) ∣∣ (9.1) Using inequality (9.1) and the fact that functions x ∈ E 7→ x/|x| and x ∈ E 7→ |x|−α are C1-functions, we can prove parts (ii) and (iii). We prove part (iv) now. Suppose F has property (HC) on Sn−1. By part (iii), F has property (HC) on E. Clearly, φ is a continuous function on E. Let ξ be any vector in E. Consider x ∈ E sufficiently close to ξ. As x → ξ, we have φ(x) → φ(ξ). Using inequality (2.10) for function F and x0 := φ(ξ) ∈ E, x := φ(x) ∈ E with constant C and power γ, and then inequality (2.10) again for function φ and x0 := ξ ∈ E, x ∈ E with constant C ′ and power γ′, we have |F (φ(x))− F (φ(ξ))| ≤ C|φ(x)− φ(ξ)|γ ≤ C C ′γ |x− ξ|γγ ′ . 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Proofs of the main theorems 8. Examples and applications 8.1. Examples 8.2. Applications 9. Appendix References