GLOBAL EXISTENCE AND ASYMPTOTIC PROFILE FOR A DAMPED WAVE EQUATION WITH VARIABLE-COEFFICIENT DIFFUSION

. We considered a Cauchy problem of a one-dimensional semilin-ear wave equation with variable-coeﬃcient diﬀusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic proﬁle are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct.


Introduction
We investigate the asymptotic profile and lifespan estimate of solutions to a one-dimensional semilinear wave equation with variable-coefficient diffusion, timedependent damping, and perturbations for t > 0, x ∈ R, u(0, x) = εu 0 (x), ∂ t u(0, x) = εu 1 (x), x ∈ R, where ε is a small parameter describing the smallness of the initial data, the diffusion coefficient a(x) is Lipshitz continuous and possesses a positive lower bound, the coefficients b(t), c(t), d(t) are smooth, b(t) ∼ (1 + t) −β , β ∈ [−1, 1), c(t)∂ x u, d(t)u can be regarded as small perturbations and the nonlinear term and the precise assumptions regarding these terms and the initial data will be provided in Section 2. The primary objective is to establish the global well-posedness and asymptotic profile of the solutions to (1.1) under the following conditions The second objective is to determine the lower bound estimate of the lifespan of the solutions when Equation (1.1) typically arises from gene and population dynamics models in Biology, where ∂ x (a(x)∂ x ) is called the diffusion operator, c.f. [4].The spatial distribution of individuals is described by Brownian motion, so the population densities are solutions to the corresponding reaction diffusion equations.However, when n = 1, the process is often substituted with damped wave equations, and the disturbance terms in (1.1) stem from the non-uniformity of the medium.The derivation of (1.1) and its physical background can be found in [2], [6], [10], [14], [20].
Todorova and Yordanov [15] demonstrated the existence of a critical exponent, denoted as p F (n) = 1 + 2 n , which plays a pivotal role in distinguishing between global existence and non-existence of solutions to the equation (1.4) More precisely, when p F (n) < p ≤ n n−2 , n ≥ 3 or p F (n) < p < ∞, n = 1, 2, (1.4) has a global solution, however when 1 < p < p F (n), for all n ∈ N, the solution blows up in finite time even for small initial values.Subsequently, Zhang [21] proved that the case of p = p F (n) belongs to the blow-up scenario.It should be noted that p F (n) is known as the Fujita exponent, and it serves as the critical index for the corresponding Cauchy problems of the heat equation (see [3]).Furthermore, the lifespan of the solution to (1.4) can be estimated as follows , 1 < p < p F (n), e Cε −(p−1) , p = p F (n), ∞, p > p F (n). (1.5) More details can be found in Ikeda-Ogawa [7], Ikeda-Wakasugi [9], Lai-Zhou [11], Li [12], Li-Zhou [13].
Wirth [17,18,19] studied systematically the influence of the index γ on the behavior of the solutions to the linear wave equations with time-dependent damping where Γ(t) = (1 + t) −γ , γ ∈ R.He classified the behavior as follows: when γ < −1, the damping term is referred to as "over-damping", in which case the solution does not decay to zero as t → ∞; for −1 ≤ γ < 1, the damping term is called "effective" because the solution behaves similarly to the corresponding heat equation, and the asymptotic profile of the solution is described by the scaled Gaussian; when γ > 1, the damping term is labeled "non-effective", indicating that the solution behaves similarly to the corresponding wave equation.From this perspective, the assumption regarding the index β in (1.1) is reasonable.Gallay and Raugel [4] conducted a comprehensive study on the asymptotic expansions for the damped wave equation (1.6) Among their notable findings, they successfully obtained the first-order asymptotic profile by employing scaling variables y = x √ t + t 0 , s = log(t + t 0 ), t 0 > 0 is fixed, (1.7) and the methodology primarily relied on energy-based approaches.Subsequently, Wakasugi [16] explored a similar problem of the equation (1.9) When the motion occurs within an inhomogeneous medium, the diffusion coefficients in (1.8) depend on the space variable x, as highlighted in [4].This naturally prompts the question: how will the solution behave when considering variable diffusion coefficients in (1.8)?In this article, we focus our attention on this intriguing problem.By employing the scaling variables given in (1.9), the equation (1.1) can be transformed into a first-order differential system, and the abstract theory of operator semigroup can be used to obtain the local well-posedness.Building upon the foundational work of [4] and [16], we employ spectral decomposition and the energy method to investigate global existence and asymptotic behavior.However, as the diffusion coefficient of (1.1) is no longer constant, the energy functional utilized by Wakasugi in [16] becomes inapplicable.To overcome this issue, some modifications on the energy functionals have to be made such that an a prior estimate on the blowup quantity can be obtained, and as a result, the global existence and asymptotic behavior can be achieved.
This article is organized as follows.In Section 2, we present a set of assumptions on the coefficients and the nonlinear term, and outline our main results.Section 3 is dedicated to establishing local well-posedness using the semigroup method.In Section 4, we prove the global well-poesdness and the asymptotic profile.The lower bound estimate of the lifespan for (1.1) is derived in Section 5. Notation.f g (f g) means there exists a constant C > 0 such that f ≤ Cg (f ≥ Cg), and f ∼ g when g f g.H k,m (R) is the weighted Sobolev space equipped with the norm and in situations where no ambiguity arises, we sometimes omit R in the norm H k,m (R).C k I; X denotes the space of k-times continuously differentiable mapping from I to X with respect to the topology in X.Moreover, the positive constant C varies from line to line in this paper.
(A3) The coefficient functions c(t), d(t) satisfy In addition, to ensure the existence of local-in-time solutions, we assume ) is regarded as a sufficiently large number when Moreover, for each µ > −1/2, we assume that , and varepsilon > 0, then a + = a − = 1 and it is easy to verify that a 0 (x Definition 2.3.For a fixed ε > 0, the lifespan of the mild solution to (1.1) is defined as : there exists a unique mild solution to (1.1) on [0, T )}.
Our main results are the following three theorems.
By introducing scaling variables, (1.1) can be transformed into an abstract evolution equation, and the abstract semigroup theory on semilinear evolution equations can be applied to prove Theorem 2.4.Utilizing a standard a priori energy estimate, we can demonstrate the existence of a solution at any given time for sufficiently small initial data.This is achieved through the application of Banach's fixed point theorem.
We will make the spectral decomposition on the unknown functions, namely decompose these functions into the leading terms and the remainder terms, respectively.By employing an a priori estimate (Proposition 4.3) based on the energy argument, we can establish the proof of Theorem 2.5.Instead of (2.3), we require that which is equivalent to (2.10) Theorem 2.6 (Lower bound estimate of lifespan).Under the assumptions (A1)-(A5), there exist ε 2 > 0, C > 0, such that for each ε ∈ (0, ε 2 ], where B(t) is given by (2.7).
This theorem will be proved by employing a similar argument as those in Theorem 2.5.Here we remark that the lower bound on the lifespan is sharp in some situations.However, as indicated in Remark 5.2, we are currently unable to established an upper bound of the lifespan.and v(s, y) = e s/2 u(t(s), e s/2 y), w(s, y) = b(t(s))e 3s/2 u t (t(s), e s/2 y). Then where t(s) = B −1 (e s − 1) (B −1 denotes the inverse function of B).Equation (1.1) is transformed into (3.4)16]).We have Lemma 3.2 ( [16]).Under assumption (A2), the following estimates hold For completeness, we recall the following results on the existence of solutions to semilinear evolution equations in abstract Banach spaces, see Proposition 4.3.3,Theorem 4.3.4 and Proposition 4.3.9 in [1] for details.

Lemma 3.4 ([1]
).Let T > 0, X be a Banach space, A be a m-dissipative operator in X with dense domain D(A).For any x ∈ X and a local Lipschitz mapping f : X → X, consider the semilinear problem and the associated integral equation where (T (t)) t≥0 is the contraction semigroup generated by A, then the following results hold: (i) Let M > 0 and let x ∈ X be such that x ≤ M , then there exits a unique solution u ∈ C([0, T M ]; X) to (3.7).(ii) Assume that X is reflexive.Let T > 0, x ∈ X, and let u ∈ C([0, T ]; X) be a solution to (3.7).Then, if x ∈ D(A), u is the solution to problem (3.6).(iii) There exits a function T : X → (0, ∞] with the following properties: for all x ∈ X, there exists u ∈ C([0, T (x)); X) such that for all 0 < T < T (x), u is the unique solution to (3.7).In addition, we have the following alternatives: 3.2.Proof of Theorem 2.4.For using Lemma 3.4, we introduce Then where where .
i.e., A is skew-adjoint, so A is an m-dissipative operator and D(A) is dense in X (c.f.[1, Corollary 2.4.9]).Therefore, A generates a contraction semigroup e tA .Consider the integral form of (3.10), By Lemma 3.4, it suffices to verify that N (U) is local Lipschitz.Indeed, so ) is a solution to (3.9), and u is a strong solution to (1.1).In addition, if the lifespan We now prove that the solution u exists at any given time T 0 > 0. Indeed, consider the non-homogeneous linear equation of (3.9) , there exists a unique distribution solution which has the standard energy estimate [5] sup (i) L(K) ⊂ K.There are three cases according to (2.3).
and by Sobolev's embedding theorem, It follows from (3.16)-(3.17) that where Then by (2.4), Applying Sobolev's embedding theorem, we have when ε is sufficiently small.It is easy to see that there exits a small ε 0 > 0, such that for each 0 < ε ≤ ε 0 , (3.18) and (3.20) are valid.Therefore by Banach's fixed point theorem, there exists a U ∈ K such that LU = U , i.e., U is the unique solution to (3.9).Moreover, if

Proof of Theorem 2.5
We begin with the definition of solutions to system (3.3).
In particular, for each given S 0 > 0, there exists a ε * 0 > 0, such that for each ε ∈ [0, ε * 0 ), the solution to (3.3) will exist on [0, S 0 ], i.e., S 0 < S(ε).In the sequel, to simplify calculations, applying a simple density argument and using the continuous dependence on initial data guaranteed by Theorem 2.4, we can assume that the initial data (v 0 , w 0 ) ∈ H 2,1 ×H 1,1 , and (v, w) is the strong solution to (3.3).We can establish the following crucial priori estimate for the unique mild solution to (3.3), which plays a fundamental role in the proof of Theorem 2.5.Proposition 4.3.Under assumptions (A1)-(A5), there exist s 0 > 0, ε0 > 0, and To prove Proposition 4.3, we will decompose v and w into the leading terms and the remainder terms.Subsequently, we will employ the energy argument to derive the decay estimates for the remainder terms.
where r(s, y) is given by (3.4).
Substituting the three identities above into (4.21),we have We define µ 0 = min(0, µ), Proof.In view of the definition of E 4 , it suffices to estimate E 0 , E 1 and E 2 .By Lemma 3.2, there exists a s 1 > 0 such that for each s ≥ s 1 , e −s b 2 (t(s)) ≤ 1 8 .As for the terms containing F G and f g in E 0 , E 1 and E 2 , for s ≥ s 1 , Young's inequality shows that Hence To obtain the lower bound estimates for E 1 and E 2 , it suffices to estimate α(s) R a 0 (ye s/2 )ϕ 0 f y dy and α(s) R y 2 a 0 (ye s/2 )ϕ 0 f y dy.
Lemma 4.10.We have where then there exists a s 0 ≥ s 2 (as defined in Lemma 4.9), such that for each s ≥ s 0 , we have the following estimates Here, 1 1+β and −2βp3 1+β are regarded as sufficiently large numbers when β = −1 and p 3 = 0, η is any positive constants, R 4 (s) and R 5 (s) are given in Lemmas 4.6 and 4.10, respectively.
Proof of Theorem 2.5.For the proof we use Corollary 4.2 and Proposition 4.3.
Proof.There are two cases according to the value of β.Sobolev's inequality shows that v(s) So by Hölder's inequality, e 3s/2 N e −s/2 v, e −s v y , b −1 (t(s))e −3s/2 w .
From (4.28), (4.29), and (4.31), it follows that  (5.6) However, because of the variable-coefficient diffusion, we still lack information about the sharpness of estimate (5.6).We have not yet established an upper bound estimate for (1.1).
holds for s ≥ s 1 , where L 4 (s) and s 1 are given by Lemmas 4.6 and 4.7, respectively.