Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 04, pp. 1–31. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.04 GLOBAL EXISTENCE AND ASYMPTOTIC PROFILE FOR A DAMPED WAVE EQUATION WITH VARIABLE-COEFFICIENT DIFFUSION YUEQUN LI, HUI LIU, FEI GUO Abstract. 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. 1. Introduction We investigate the asymptotic profile and lifespan estimate of solutions to a one-dimensional semilinear wave equation with variable-coefficient diffusion, time- dependent damping, and perturbations ∂2t u− ∂x(a(x)∂xu) + b(t)∂tu = c(t)∂xu+ d(t)u+N(u, ∂xu, ∂tu), for t > 0, x ∈ R, u(0, x) = εu0(x), ∂tu(0, x) = εu1(x), x ∈ R, (1.1) 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)∂xu, d(t)u can be regarded as small perturbations and the nonlinear term |N(u, ∂xu, ∂tu)| ≤ C|u|p1 |∂xu|p2 |∂tu|p3 , 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 p1 + 2p2 + ( 3− 2β 1 + β ) p3 > 3, p2 + p3 ≤ 1. (1.2) 2020 Mathematics Subject Classification. 35B44, 35G25. Key words and phrases. Semilinear damped wave equation; asymptotic profile; lifespan; scaling variables; energy method. ©2024. This work is licensed under a CC BY 4.0 license. Submitted August 27, 2023. Published January 8, 2024. 1 2 Y. LI, H. LIU, F. GUO EJDE-2024/04 The second objective is to determine the lower bound estimate of the lifespan of the solutions when p1 + 2p2 + ( 3− 2β 1 + β ) p3 ≤ 3, p2 + p3 ≤ 1. (1.3) 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 pF (n) = 1 + 2 n , which plays a pivotal role in distinguishing between global existence and non-existence of solutions to the equation ∂2t u−∆u+ ∂tu = |u|p, t > 0, x ∈ Rn, u(0, x) = εu0(x), ∂tu(0, x) = εu1(x), x ∈ Rn. (1.4) More precisely, when pF (n) < p ≤ n n−2 , n ≥ 3 or pF (n) < p < ∞, n = 1, 2, (1.4) has a global solution, however when 1 < p < pF (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 = pF (n) belongs to the blow-up scenario. It should be noted that pF (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 Lifespan(u) ∼  Cε(− 1 p−1− n 2 )−1 , 1 < p < pF (n), eCε −(p−1) , p = pF (n), ∞, p > pF (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 ∂2t u−∆u+ Γ(t)∂tu = 0, 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 ex- pansions for the damped wave equation ∂2t u− ∂x(a(x)∂xu) + ut = N(u, ux, ut), x ∈ R, t ≥ 0. (1.6) EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 3 Among their notable findings, they successfully obtained the first-order asymptotic profile by employing scaling variables y = x√ t+ t0 , s = log(t+ t0), t0 > 0 is fixed, (1.7) and the methodology primarily relied on energy-based approaches. Subsequently, Wakasugi [16] explored a similar problem of the equation ∂2t u−∆u+ b(t)∂tu = c(t) · ∇u+ d(t)u+N(u,∇u, ∂tu), t > 0, x ∈ Rn, (1.8) where the scaling variables (1.7) were replaced with the following new variables y = x B(t) + 1 , s = log ( B(t) + 1 ) , B(t) = ∫ t 0 dτ b(τ) . (1.9) When the motion occurs within an inhomogeneous medium, the diffusion coeffi- cients 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 dif- fusion 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 op- erator 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 modifi- cations 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. Hk,m(R) is the weighted Sobolev space equipped with the norm ‖f‖Hk,m(R) = ∑ |α|≤k ‖〈x〉mDα xf‖L2(R), where k ∈ Z, m ≥ 0, 〈x〉 = (1 + |x|2)1/2, and in situations where no ambiguity arises, we sometimes omit R in the norm Hk,m(R). Ck ( 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. 2. Main results We use the following assumptions: (A1) The initial data (u0, u1) ∈ H1,1(R)×H0,1(R). (A2) The coefficient of the damping term b(t) satisfies C−1(1 + t)−β ≤ b(t) ≤ C(1 + t)−β , ∣∣db(t) dt ∣∣ ≤ C(1 + t)−1b(t), (2.1) where β ∈ [−1, 1). 4 Y. LI, H. LIU, F. GUO EJDE-2024/04 (A3) The coefficient functions c(t), d(t) satisfy |c(t)| ≤ C(1 + t)−γ , |d(t)| ≤ C(1 + t)−ν (2.2) for some γ > 1 + β/2, ν > 1 + β. (A4) The nonlinear term N satisfies |N(z)| ≤ C|z1|p1 |z2|p2 |z3|p3 , pi ≥ 1 or pi = 0, p1 > 1, p2 + p3 ≤ 1, p1 + 2p2 + (3− 2β 1 + β )p3 > 3. (2.3) In addition, to ensure the existence of local-in-time solutions, we assume |N(z)−N(w)| ≤ C|z1 − w1|(|z1|+ |w1|)p1−1, (2.4) where z = (z1, z2, z3), w = (w1, w2, w3), zi, wi ∈ R (i = 1, 2, 3), and − 2β 1+β is regarded as a sufficiently large number when β = −1. (A5) a : R 7→ R is Lipschitz continuous and a(x) ≥ a > 0. Set a(x) = ã(x) + a0(x), where ã(x) = limx→±∞ a(x) satifies ã(x) = { a+, if x > 0, a−, if x < 0. (2.5) Moreover, for each µ > −1/2, we assume that (1 + |x|)µa0(x) ∈ L2(R) ∩ L∞(R). (2.6) Remark 2.1. Assumption (2.6) is reasonable. Indeed, if a(x) = 1 (1+|x|)1+µ+ε + 1, µ > − 1 2 , and varepsilon > 0, then a+ = a− = 1 and it is easy to verify that a0(x)(1 + |x|)µ ∈ L2(R) ∩ L∞(R). Let B(t) = ∫ t 0 1 b(τ) dτ, G(t, x) = 2√ 4πt (√ a+ + √ a− )e− x2 4tã(x) , (2.7) Equation (2.1) shows that B(t) is strictly increasing and limt→+∞B(t) = +∞. Definition 2.2. u is a mild solution to (1.1) on the interval [0, T ) if (1.1) holds in the sense of distributions and u ∈ C([0, T );H1,1(R)) ∩ C1([0, T );H0,1(R)). More- over, if u ∈ C([0, T );H2,1(R))∩C1([0, T );H1,1(R))∩C2([0, T );H0,1(R)), then u is called a strong solution. Definition 2.3. For a fixed ε > 0, the lifespan of the mild solution to (1.1) is defined as T (ε) := sup{T ∈ (0,∞) : there exists a unique mild solution to (1.1) on [0, T )}. Our main results are the following three theorems. Theorem 2.4 (Local well-posedness). Under assumptions (A1)–(A5), there exists a T = T (ε‖(u0, u1)‖H1,1×H0,1) > 0, such that (1.1) has a unique mild solution u on [0, T ). Furthermore, if (u0, u1) ∈ H2,1(R)×H1,1(R), u becomes a strong solution. In addition, if T = T (ε) <∞, then lim t→T (ε) ‖(u, ut)‖H1,1(R)×H0,1(R) =∞. (2.8) In particular, for each T0 > 0, there exists a ε0 > 0, such that for each ε ∈ (0, ε0], we have T0 < T (ε) (which is the lifespan of (1.1)), i.e., the solution u will exist on [0, T0]. EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 5 By introducing scaling variables, (1.1) can be transformed into an abstract evolu- tion 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. Theorem 2.5 (Global well-posedness and asymptotic profile). Under assumptions (A1)-(A5), there exists a ε1 > 0, such that for each ε ∈ (0, ε1], (1.1) has a unique mild solution u ∈ C([0,∞);H1,1(R)) ∩ C1([0,∞);H0,1(R)). In addition, the limit α∗ = limt→∞ ∫ R u(t, x)dx exists and ‖u(t, ·)−α∗G(B(t)+1, ·)‖L2(R) . ε 2(B(t)+1)− 1 4−λ‖(u0, u1)‖H1,1(R)×H0,1(R), (2.9) where λ = min{ 14 + µ0 2 , λ0, λ1}, µ0 = min{0, µ}, λ0 = min{1− β 1 + β , γ 1 + β − 1 2 , ν 1 + β − 1}, λ1 = 1 2 {p1 + 2p2 + (3− 2β 1 + β )p3 − 3}, in which if β = −1, p3 6= 0, then 1 1+β and −2βp31+β are regarded as sufficiently large numbers. We will make the spectral decomposition on the unknown functions, namely decompose these functions into the leading terms and the remainder terms, respec- tively. 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 p1 + 2p2 + ( 3− 2β 1 + β ) p3 < 3, β ∈ (−1, 1), p1 > 1, pi ≥ 1 or pi = 0 (i = 2, 3), p2 + p3 ≤ 1, which is equivalent to β ∈ (−1, 1), 1 < p1 < 3, p2 = p3 = 0. (2.10) Theorem 2.6 (Lower bound estimate of lifespan). Under the assumptions (A1)– (A5), there exist ε2 > 0, C > 0, such that for each ε ∈ (0, ε2], B(T (ε)) + 1 ≥ Cε− 2(p1−1) 3−p1 , (2.11) where B(t) is given by (2.7). This theorem will be proved by employing a similar argument as those in The- orem 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 estab- lished an upper bound of the lifespan. 3. Proof of Theorem 2.4 3.1. Preliminaries. Let s = log(B(t) + 1), y = (B(t) + 1)−1/2x (3.1) and v(s, y) = es/2u(t(s), es/2y), w(s, y) = b(t(s))e3s/2ut(t(s), e s/2y). 6 Y. LI, H. LIU, F. GUO EJDE-2024/04 Then u(t, x) = ( B(t) + 1 )−1/2 v ( log(B(t) + 1), (B(t) + 1)−1/2x ) , ut(t, x) = b−1(t) ( B(t) + 1 )−3/2 w ( log(B(t) + 1), (B(t) + 1)−1/2x ) , (3.2) where t(s) = B−1(es − 1) (B−1 denotes the inverse function of B). Equation (1.1) is transformed into vs − y 2 vy − 1 2 v = w, s > 0, y ∈ R, e−s b(t(s)) (ws − y 2 wy − 3 2 w) + w = (a(es/2y)vy)y + r(s, y), s > 0, y ∈ R, v(0, y) = εv0(y) = εu0(x), y ∈ R, w(0, y) = εw0(y) = εb(0)u1(x), y ∈ R, (3.3) where r(s, y) = 1 b(t(s))2 db(t(s)) dt w + es/2c(t(s))vy + esd(t(s))v + e3s/2N(e−s/2v, e−svy, b −1(t(s))e−3s/2w). (3.4) Lemma 3.1 ([16]). We have db(t(s)) ds = db(t(s)) dt b(t(s))es, d ds ( 1 b2(t(s)) ) = − 2 b2(t(s)) db(t(s)) dt es. (3.5) Lemma 3.2 ([16]). Under assumption (A2), the following estimates hold (i) when β ∈ (−1, 1), b(t(s)) ∼ e− βs 1+β , e−s b2(t(s)) ∼ e− (1−β)s 1+β , 1 b2(t(s)) ∣∣db(t(s)) dt ∣∣ ≤ Ce− (1−β)s 1+β . (ii) when β = −1 , b(t(s)) ∼ exp(es), e−s b2(t(s)) ∼ exp(−2es − s), 1 b2(t(s)) ∣∣db(t(s)) dt ∣∣ ≤ C exp(−2es). Lemma 3.3 (Gagliardo-Nirenberg inequality, [16]). Let 1 < p < ∞ (n = 1, 2), then for each f ∈ H1,0(Rn), ‖f‖L2p ≤ C‖∇f‖σL2‖f‖1−σL2 , where σ = n(p− 1)/(2p). 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 u′(t) = Au(t) + f(u(t)), t ∈ [0, T ], u(0) = x, u ∈ C([0, T ];D(A)) ∩ C1([0, T ];X) (3.6) EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 7 and the associated integral equation u(t) = T (t)x+ ∫ t 0 T (t− s)f(u(s))ds, (3.7) 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, TM ];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: (a) T (x) =∞; (b) T (x) <∞ and limt↑T (x) ‖u(t)‖ =∞. 3.2. Proof of Theorem 2.4. For using Lemma 3.4, we introduce U(t, x) = 〈x〉u, U0(x) = 〈x〉u0, U1(x) = 〈x〉u1. (3.8) Then Ux = 〈x〉−1xu+ 〈x〉ux, (a(x)Ux)x = 〈x〉(a(x)ux)x + 2a(x)〈x〉−1xux + ȧ(x)〈x〉−1xu+ a(x)〈x〉−3(〈x〉2 − |x|2)u, so Utt + b(t)Ut = (a(x)Ux)x + c̃(t, x)Ux + d̃(t, x)U + Ñ(U,Ux, Ut), for t > 0, x ∈ R, U(0, x) = εU0(x), Ut(0, x) = εU1(x), x ∈ R, (3.9) where c̃(t, x) = c(t)− 2a(x)〈x〉−2x, d̃(t, x) = d(t)− c(t)〈x〉−2x− ȧ(x)〈x〉−2x− a(x)〈x〉−4(〈x〉2 − 3|x|2), Ñ(U,Ux, Ut) = 〈x〉N(〈x〉−1U, 〈x〉−1Ux − 〈x〉−3xU, 〈x〉−1Ut). Let U = (U,Ut) T, U0 = (U0, U1)T. Then (3.9) is equivalent to Ut = AU +N(U), U(0) = εU0, (3.10) where A = ( 0 1 ∂x(a(x)∂x) 0 ) , N(U) = ( 0 −bUt + c̃Ux + d̃U + Ñ(U,Ux, Ut) ) . Set X = H1,0(R)× L2(R), D(A) = H2,0(R)×H1,0(R). Note that A∗ = ( 0 −1 −∂x(a(x)∂x) 0 ) = −A, 8 Y. LI, H. LIU, F. GUO EJDE-2024/04 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 etA. Consider the integral form of (3.10), U(t) = εetAU0 + ∫ t 0 eA(t−s)N(U(s))ds. (3.11) By Lemma 3.4, it suffices to verify that N(U) is local Lipschitz. Indeed, N(U)−N(V) = ( 0 −b(t)(Ut − Vt) + c̃(Ux − Vx) + d̃(U − V ) + Ñ(U,Ux, Ut)− Ñ(V, Vx, Vt) ) for M > 0 and U = (U,Ut) T, V = (V, Vt) T in B(0,M) ⊂ X, we have ‖N(U)−N(V)‖H1,0×L2 = ‖ − b(t)(Ut − Vt) + c̃(Ux − Vx) + d̃(U − V ) + Ñ(U,Ux, Ut)− Ñ(V, Vx, Vt)‖L2 . From (2.1)-(2.4), it follows that ‖ − b(t)(Ut − Vt) + c̃(Ux − Vx) + d̃(U − V )‖L2 ≤ C ( ‖U − V ‖H1,0 + ‖Ut − Vt‖L2 ) = C‖U − V‖H1,0×L2 (3.12) and ‖Ñ(U,Ux, Ut)− Ñ(V, Vx, Vt)‖L2 ≤ C‖|U − V |(|U |+ |V |)p1−1‖L2 ≤ C(M)‖U − V‖H1,0×L2 , (3.13) so ‖N (U)−N (V)‖H1,0×L2 ≤ (C(M) + C)‖U − V‖H1,0×L2 . By (i) of Lemma 3.4, (3.11) has a unique solution U ∈ H1,0(R) × L2(R), which means (1.1) has a unique mild solution u. Furthermore, if U0 ∈ H2,0(R)×H1,0(R), then U ∈ C([0, T ];D(A)) ∩ C1([0, T ];X) is a strong solution to (3.10) by Lemma 3.4 (ii), i.e., U ∈ C([0, T ];H2,0(R)) ∩ C1([0, T ];H1,0(R)) ∩ C2([0, T ];L2(R)) is a solution to (3.9), and u is a strong solution to (1.1). In addition, if the lifespan T (ε) <∞, by Lemma 3.4 (iii), lim t→T (ε) ‖(u, ut)(t)‖H1,1(R)×H0,1(R) =∞. We now prove that the solution u exists at any given time T0 > 0. Indeed, consider the non-homogeneous linear equation of (3.9) Utt + b(t)Ut = (a(x)Ux)x + c̃(t, x)Ux + d̃(t, x)U + Ñ(t, x), t > 0, x ∈ R, U(0, x) = εU0(x), Ut(0, x) = εU1(x), x ∈ R. (3.14) For Ñ ∈ L1((0, T0];L2(R)), there exists a unique distribution solution which has the standard energy estimate [5] sup 0 1, p2 = 0, p3 = 1, then |Ñ(t)| ≤ C|〈x〉||〈x〉−1V |p1 |〈x〉−1Vt| ≤ C|V |p1 |Vt|, and by Sobolev’s embedding theorem, ‖Ñ(t)‖L2 ≤ C‖V ‖p1L∞‖Vt‖L2 ≤ C‖V ‖p1H1,0‖Vt‖L2 ≤ C ( 2C(T0)I0ε )p1+1 . (3.16) Case 2: If p1 > 1, p2 = 1, and p3 = 0, then |Ñ(t)| ≤ C|〈x〉| |〈x〉−1V |p1 |〈x〉−1Vx − 〈x〉−3xV | ≤ C(|V |p1+1 + |V |p1 |Vx|) ‖Ñ(t)‖L2 ≤ C(‖V ‖p1L∞‖V ‖L2 + ‖V ‖p1L∞‖Vx‖L2) ≤ C ( 2C(T0)I0ε )p1+1 . Case 3: If p1 > 1, p2 = p3 = 0 and |Ñ(t)| ≤ C|〈x〉| |〈x〉−1V |p1 ≤ C|V |p1 , then the Gagliardo-Nirenberg inequality yields ‖Ñ(t)‖L2 ≤ C‖|V |p1‖L2 = C‖|V |‖p1 L2p1 ≤ C‖Vx‖ p1−1 2 L2 ‖V ‖ p1+1 2 L2 ≤ C‖Vx‖ p1−1 2 H1,0 ‖V ‖ p1+1 2 H1,0 ≤ C(2C(T0)I0ε) p1 . (3.17) It follows from (3.16)-(3.17) that ‖Ñ(t)‖L2 ≤ C(2C(T0)I0ε) p1 if ε < 1 is sufficiently small. Substituting this estimate in (3.15), we have sup 0 0, x ∈ R, U(0, x) = 0, ∂tU(0, x) = 0, x ∈ R, where F (V 1, V 2) = Ñ(V 1, V 1 x , V 1 t )− Ñ(V 2, V 2 x , V 2 t ). Obviously, F is an element of L1 ( (0, T0];L2(R) ) , so sup 0 0, such that for each 0 < ε ≤ ε0, (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 T0 > T (ε), (iii) of Lemma 3.4 shows that sup0 0, such that (3.3) has a unique mild solution (v, w) on [0, S). Furthermore, if (u0, u1) ∈ H2,1(R) ×H1,1(R), then (v, w) becomes a strong solution. Moreover, if the lifespan T = T (ε) of the mild solution to (1.1) is finite, then lim s→S(ε) ‖(v, w)‖H1,1(R)×H0,1(R) =∞. In particular, for each given S0 > 0, there exists a ε∗0 > 0, such that for each ε ∈ [0, ε∗0), the solution to (3.3) will exist on [0, S0], i.e., S0 < 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 (v0, w0) ∈ H2,1×H1,1, 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 s0 > 0, ε̃0 > 0, and C∗ > 0 such that for each ε ∈ [0, ε̃0), if (v, w) is the unique mild solution to (3.3) on [0, S], where S > s0, then ‖v(s)‖2H1,1 + e−s b2(t(s)) ‖w(s)‖2H0,1 ≤ C∗ε2‖(v0, w0)‖2H1,1×H0,1 , ∀s ∈ [s0, S]. (4.1) EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 11 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. 4.1. Proof of Proposition 4.3. Let α(s) = ∫ R v(s, y)dy, α̇(s) = dα(s) ds , (4.2) since v(s) ∈ H1,1(R), for all s ∈ [0, S), α(s) is well defined by Sobolev’s embedding theorem. Lemma 4.4. We have dα(s) ds = ∫ R w(s, y)dy, (4.3) e−s b2(t(s)) d2α(s) ds2 = e−s b2(t(s)) α̇(s)− α̇(s) + ∫ R r(s, y)dy, (4.4) where r(s, y) is given by (3.4). Proof. Note v(s) ∈ C1 ( [0, S);H1,1(R) ) and w(s) ∈ C ( [0, S);H0,1(R) ) , (4.3) follows from (3.3) and integration by parts dα(s) ds = ∫ R (y 2 vy + v 2 + w ) dy = ∫ R ( ( y 2 v)y + w ) dy = ∫ R w(s, y)dy. Differentiating (4.3) and using (3.3), we have e−s b2(t(s)) d2α(s) ds2 = e−s b2(t(s)) ∫ R ws(s, y)dy = e−s b2(t(s)) ∫ R ( 3 2 w + y 2 wy)dy − ∫ R (w − (a(yes/2)vy)y)dy + ∫ R r(s, y)dy = e−s b2(t(s)) ∫ R w(s, y)dy − ∫ R wdy + ∫ R (a(yes/2)vy)ydy + ∫ R r(s, y)dy = e−s b2(t(s)) dα ds (s)− dα ds (s) + ∫ R r(s, y)dy. � We set ϕ0(y) = 1√ 4π 2 √ a+ + √ a− exp(− |y| 2 4ã(y) ), ψ0(y) = (ã(y)ϕ′0(y))y = −y 2 ϕ′0(y)− 1 2 ϕ0, (4.5) where a± and ã(x) are given by (2.5) and it is easy to verify that∫ R ϕ0(y)dy = 1 and ∫ R ψ0(y)dy = 0. (4.6) We decompose (v, w) as v(s, y) = α(s)ϕ0(y) + f(s, y), w(s, y) = α̇(s)ϕ0(y) + α(s)ψ0(y) + g(s, y). (4.7) 12 Y. LI, H. LIU, F. GUO EJDE-2024/04 By Definition 4.1, (f, g) ∈ C ( [0, S);H2,1(R)×H1,1(R) ) ∩ C1 ( [0, S);H1,1(R)×H0,1(R) ) , (4.8) and from (4.5) and (4.7) it follows that∫ R f(s, y)dy = ∫ R ( v(s, y)− α(s)ϕ0(y) ) dy = α(s)− α(s) = 0,∫ R g(s, y)dy = ∫ R ( w(s, y)− α̇(s)ϕ0(y)− α(s)ψ0(y) ) = α̇(s)− α̇(s) = 0. (4.9) Substituting (4.7) into (3.3) gives fs − y 2 fy − 1 2 f = g, s > 0, y ∈ R, e−s b2(t(s)) (gs − 3 2 g − y 2 gy) + g = (a(yes/2)fy)y + α(s)(a0(yes/2)ϕ′0(y))y + h(s, y), s > 0, y ∈ R, f(0, y) = v(0, y)− α(0)ϕ0(y), y ∈ R, g(0, y) = w(0, y)− α̇(0)ϕ0(y)− α(0)ψ0(y), y ∈ R, (4.10) where h(s, y) = e−s b2(t(s)) ( −2α̇(s)ψ0(y) + α(s)( y 2 ψ′0(y) + 3 2 ψ0(y)) ) + r(s, y)− ϕ0(y) ∫ R r(s, y)dy, (4.11) here we have used (4.5) and a(x) = ã(x) + a0(x). From (4.6) and (4.9) we deduce that ∫ R h(s, y)dy = 0. To obtain decay estimates for f and g, we define F (s, y) = ∫ y −∞ f(s, z)dz, G(s, y) = ∫ y −∞ g(s, z)dz. Lemma 4.5 (Hardy-type inequality, [16]). If f ∈ H0,1(R) satisfies ∫ R f(y)dy = 0, then for F (y) = ∫ y −∞ f(z)dz, it holds∫ R F 2(y)dy ≤ 4 ∫ R y2f2(y)dy. (4.12) From (4.8)-(4.10) and Lemma 4.5, (F,G) ∈ C ( [0, S);H3,0(R)×H2,0(R) ) ∩ C1 ( [0, S);H2,0(R)×H1,0(R) ) (4.13) satisfies Fs − 1 2 yFy = G, e−s b2(t(s)) ( Gs − 1 2 yGy −G ) +G = α(s)a0(yes/2)ϕ′0(y) + a(yes/2)Fyy +H, F (0, y) = ∫ y −∞ f(0, z)dz, G(0, y) = ∫ y −∞ g(0, z)dz, (4.14) where s > 0, y ∈ R, and H(s, y) = ∫ y −∞ h(s, z)dz. (4.15) EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 13 We define the following energies E0(s) = 1 2 ∫ R ( F 2 y + e−s b2(t(s)) G2 a(yes/2) ) dy + ∫ R 1 a(yes/2) (1 2 F 2 + e−s b2(t(s)) FG ) dy, E1(s) = 1 2 ∫ R ( a(ye s 2 )f2y + e−s b2(t(s)) g2 ) dy + α(s) ∫ R a0(yes/2)ϕ′0fydy + ∫ R ( f2 + 2 e−s b2(t(s)) fg ) dy, E2(s) = 1 2 ∫ R ( y2a(yes/2)f2y + e−s b2(t(s)) y2g2 ) dy + α(s) ∫ R y2a0(yes/2)ϕ′0fydy + ∫ R (1 2 y2f2 + e−s b2(t(s)) y2fg ) dy, E4(s) = C0E0(s) + C1E1(s) + E2(s) + E3(s), where E3(s) = 1 2 e−s b2(t(s)) (dα(s) ds )2 + C2e −2λsα2(s), 1 � C1 � C0, C2 > 0 and λ > 0 are given by Lemma 4.7 and Lemma 4.11, respectively. We have the following identities which will be used in the proof of Proposition 4.3. Lemma 4.6. (1) d ds E0(s) + 1 2 E0(s) + L0(s) = R0(s), (4.16) where L0(s) = ∫ R (1 2 F 2 y + 1 a(yes/2) G2 ) dy, R0(s) = 3 2 e−s b2(t(s)) ∫ R 1 a(yes/2) G2dy − 1 b2(t(s)) db(t(s)) dt ∫ R 1 a(yes/2) ( G2 + 2FG ) dy + ∫ R 1 a(yes/2) (( α(s)a0(yes/2)ϕ′0 +H ) (F +G) ) dy. (2) d ds E1(s) + 1 2 E1(s) + L1(s) = R1(s), (4.17) where L1(s) = ∫ R ( a(yes/2)f2y + g2 − f2 ) dy, R1(s) = 3 e−s b2(t(s)) ∫ R g2dy + 2 e−s b2(t(s)) ∫ R fg dy − 1 b2(t(s)) db(t(s)) dt ∫ R (g2 + 4fg)dy + ( α̇(s)− α(s) ) ∫ R a0(yes/2)ϕ′0fy dy − α(s) ∫ R y 2 ϕ′′0(y)fya0(yes/2)dy + ∫ R (hg + 2hf)dy. (3) d ds E2(s) + 1 2 E2(s) + L2(s) = R2(s), (4.18) 14 Y. LI, H. LIU, F. GUO EJDE-2024/04 where L2(s) = ∫ R (1 2 y2a(yes/2)f2y + y2g2 ) dy + 2 ∫ R ya(yes/2)fy(f + g)dy, R2(s) = ( α̇(s)− α(s) ) ∫ R y2a0(yes/2)ϕ′0(y)fydy + e−s b2(t(s)) ∫ R 3 2 y2g2dy − α(s) ∫ R y3 2 a0(yes/2)ϕ′′0(y)fy dy − 2α(s) ∫ R ya0(yes/2)ϕ′0(y)(f + g)dy − 1 b2(t(s)) db(t(s)) dt ∫ R y2(g2 + 2fg)dy + ∫ R y2(fh+ gh)dy. (4) d ds E3(s) + 2λE3(s) + (dα(s) ds )2 = R3(s), (4.19) where R3(s) = 1 2 (2λ+ 1) e−s b2(t(s)) (dα(s) ds )2 − 1 b2(t(s)) db(t(s)) dt (dα(s) ds )2 + dα(s) ds ∫ R r(s, y)dy + 2C2e −2λsα(s) dα(s) ds . (5) d ds E4(s) + 2λE4(s) + L4(s) = R4(s), (4.20) where L4(s) = ( 1 2 − 2λ)(C0E0 + C1E1 + E2) + C0L0 + C1L1 + L2 + (dα(s) ds )2 , R4(s) = C0R0(s) + C1R1(s) +R2(s) +R3(s). Proof. Clearly, (4.20) is the sum of (4.16)-(4.19), (4.19) can be derived by differen- tiating E3(s) directly, and (4.17), (4.18) are similar to (4.16), so it suffices to prove (4.16). Differentiating E0(s) gives d ds E0(s) = d ds (1 2 ∫ R e−s b2(t(s)) G2 a(yes/2) dy ) + d ds (∫ R 1 a(yes/2) (1 2 F 2 + e−s b2(t(s)) FG ) dy ) + 1 4 ∫ R F 2 y dy + ∫ R FyGydy, (4.21) where ∫ R FysFydy = ∫ R ( y 2 FyFyy + 1 2 F 2 y + FyGy)dy = ∫ R ((y 4 F 2 y ) y + 1 4 F 2 y + FyGy ) dy = ∫ R 1 4 F 2 y dy + ∫ R FyGydy has been used. For convenience, we define F̃ (s, x) = F (s, y) = F (s, xe−s/2), G̃(s, x) = G(s, y) = G(s, xe−s/2). EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 15 Note dy = e−s/2dx, F̃s(s, x) = Fs − y 2 Fy, G̃s(s, x) = Gs − y 2 Gy. Therefore, d ds (1 2 ∫ R 1 a(yes/2) F 2dy ) = d ds (1 2 ∫ R 1 a(x) F̃ 2e−s/2dx ) = −1 4 ∫ R 1 a(x) F̃ 2e−s/2dx+ ∫ R 1 a(x) F̃ (x, s)F̃s(x, s)e −s/2dx = −1 4 ∫ R 1 a(yes/2) F 2dy + ∫ R 1 a(yes/2) F (−y 2 Fy + Fs)dy = ∫ R 1 a(yes/2) ( FFs − y 2 FFy − 1 4 F 2 ) dy = ∫ R 1 a(yes/2) ( FG− 1 4 F 2 ) dy, d ds (1 2 e−s b2(t(s)) ∫ R 1 a(yes/2) G2dy ) = d ds (1 2 e−s b2(t(s)) ∫ R 1 a(x) G̃2e−s/2dx ) = −3 4 e−s b2(t(s)) ∫ R 1 a(x) G̃2e−s/2dx+ e−s b2(t(s)) ∫ R 1 a(x) G̃G̃se −s/2dx − 1 b2(t(s)) db(t(s)) dt ∫ R 1 a(x) G̃2e−s/2dx = e−s b2(t(s)) ∫ R 1 a(yes/2) ( GGs − y 2 GGy − 3 4 G2dy ) − 1 b2(t(s)) db(t(s)) dt ∫ R 1 a(yes/2) G2dy = e−s b2(t(s)) ∫ R 1 a(yes/2) 1 4 G2dy − ∫ R 1 a(yes/2) G2dy + ∫ R FyyGdy + ∫ R 1 a(yes/2) ( H + α(s)a0(yes/2)ϕ′ ) Gdy − 1 b2(t(s)) db dt (t(s)) ∫ R 1 a(yes/2) G2dy, and d ds ( e−s b2(t(s)) ∫ R 1 a(yes/2) FGdy ) = d ds ( e−s b2(t(s)) ∫ R 1 a(x) F̃ G̃e−s/2dx ) = e−s b2(t(s)) ∫ R 1 a(yes/2) G2dy − 1 2 e−s b2(t(s)) ∫ R FGdy − ∫ R 1 a(yes/2) FGdy − e−s b2(t(s)) ∫ R 1 a(yes/2) FGdy − 2 b2(t(s)) db(t(s)) dt ∫ R FGdy + ∫ R FyyFdy 16 Y. LI, H. LIU, F. GUO EJDE-2024/04 + e−s b2(t(s)) ∫ R 1 a(yes/2) FGdy + ∫ R 1 a(yes/2) ( HF + α(s)a0(yes/2)ϕ′0F ) dy. Substituting the three identities above into (4.21), we have d ds E0(s) = −1 2 E0(s)− ∫ R (1 2 F 2 y + 1 a(yes/2) G2 ) dy − 1 b2(t(s)) db(t(s)) dt ∫ R 1 a(yes/2) ( G2 + 2FG ) dy + ∫ R 1 a(yes/2) ( (F +G)(αa0(yes/2)ϕ′0 +H) ) dy + 3 2 e−s b2(t(s)) ∫ R G2dy = −1 2 E0(s)− L0(s) +R0(s). � We define µ0 = min(0, µ), A = a−1‖|x|µ0a0‖2L2(R)‖ |y| −µ0ϕ′0‖2L∞(R) , B = a−1‖|x|µ0a0‖2L2(R)‖ |y| 1−µ0ϕ′0‖2L∞(R) . Lemma 4.7. If 0 < λ ≤ 1 4 + µ0 2 , C2 ≥ 2(C1A+B), then there exists a s1 > 0 such that for each s ≥ s1, E4(s) ∼ ‖f‖2H1,1 + e−s b2(t(s)) ‖g‖2H0,1 + e−s b2(t(s)) (dα(s) ds )2 + e−2λsα2(s). (4.22) Proof. In view of the definition of E4, it suffices to estimate E0, E1 and E2. By Lemma 3.2, there exists a s1 > 0 such that for each s ≥ s1, e−s b2(t(s)) ≤ 1 8 . As for the terms containing FG and fg in E0, E1 and E2, for s ≥ s1, Young’s inequality shows that∣∣ e−s b2(t(s)) ∫ R 1 a(yes/2) FGdy ∣∣ ≤ e−s b2(t(s)) ∫ R 1 a(yes/2) (1 8 G2 + 2F 2 ) dy ≤ 1 4 ∫ R 1 a(yes/2) F 2dy + 1 8 e−s b2(t(s)) ∫ R 1 a(yes/2) G2dy, ∣∣2 e−s b2(t(s)) ∫ R fgdy ∣∣ ≤ 2 e−s b2(t(s)) ∫ R ( 2f2 + 1 8 g2 ) dy ≤ 1 2 ∫ R f2dy + 1 4 e−s b2(t(s)) ∫ R g2dy and ∣∣ e−s b2(t(s)) ∫ R y2fgdy ∣∣ ≤ e−s b2(t(s)) ∫ R y2 ( 2f2 + 1 8 g2 ) dy ≤ 1 4 ∫ R y2f2dy + 1 8 e−s b2(t(s)) ∫ R y2g2dy. Hence E0(s) ≥ 1 2 ∫ R F 2 y dy + 1 4 ∫ R 1 a(yes/2) F 2dy + 1 4 e−s b2(t(s)) ∫ R 1 a(yes/2) G2dy. (4.23) To obtain the lower bound estimates for E1 and E2, it suffices to estimate α(s) ∫ R a0(yes/2)ϕ′0fydy and α(s) ∫ R y2a0(yes/2)ϕ′0fydy. EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 17 It is easy to see that |x|µ0a0(x) ∈ L2(R) and |y|−µ0ϕ′0(y) ∈ L∞(R) by assumption (A5), so∣∣α(s) ∫ R a0(yes/2)ϕ′0fydy ∣∣ ≤ 1 4 ∫ R a(yes/2)f2ydy + α2(s) ∫ R 1 a(yes/2) ∣∣a0(yes/2)ϕ′0 ∣∣2dy ≤ 1 4 ∫ R a(yes/2)f2ydy + a−1α2(s)‖a0(yes/2)ϕ′0‖2L2(R) and ‖a0(yes/2)ϕ′0‖L2(R) = ‖a0(yes/2)(yes/2)−µ0(yes/2)µ0ϕ′0‖L2 y(R) ≤ ‖y−µ0ϕ′0‖L∞(R)‖a0(x)|x|µ0‖L2 x(R)e −( 1 4+ µ0 2 )s, then∣∣α(s) ∫ R a0(yes/2)ϕ′0fydy ∣∣ ≤ 1 4 ∫ R a(yes/2)f2ydy +Ae−2( 1 4+ µ0 2 )sα2(s). (4.24) Similarly, we have∣∣α(s) ∫ R y2a0(yes/2)ϕ′0fydy ∣∣ ≤ 1 4 ∫ R y2a(yes/2)f2ydy + α2(s) ∫ R 1 a(yes/2) ∣∣ya0(yes/2)ϕ′0 ∣∣2dy ≤ 1 4 ∫ R y2a(yes/2)f2ydy + a−1α2(s)‖ya0(yes/2)ϕ′0‖2L2(R) and ‖ya0(yes/2)ϕ′0‖L2(R) = ‖ya0(yes/2)ϕ′0(yes/2)µ0(yes/2)−µ0‖L2 y(R) ≤ ‖|y|1−µ0ϕ′0‖L∞(R)‖a0(yes/2)(yes/2)µ0(es/2)−µ0‖L2(R) ≤ ‖|y|1−µ0ϕ′0‖L∞(R) (∫ R a0(x)2x2µ0e−( 1 2+µ0)sdx )1/2 ≤ ‖|y|1−µ0ϕ′0‖L∞(R)‖a0(x)|x|µ0‖L2(R)e −( 1 4+ µ0 2 )s, then ∣∣α(s) ∫ R y2a0(yes/2)ϕ′0fydy ∣∣ ≤ 1 4 ∫ R y2a(yes/2)f2ydy +Bα2(s)e−2( 1 4+ µ0 2 )s. (4.25) So E1(s) ≥ 1 2 ∫ R a(yes/2)f2y + e−s b2(t(s)) g2dy + ∫ R f2dy − 1 2 ∫ R f2dy − 1 4 e−s b2(t(s)) ∫ R g2dy + α(s) ∫ R a0(yes/2)ϕ′0fydy ≥ 1 4 ∫ R a(yes/2)f2ydy + 1 2 ∫ R f2dy + 1 4 e−s b2(t(s)) ∫ R g2dy −Aα2(s)e−2( 1 4+ µ0 2 )s (4.26) 18 Y. LI, H. LIU, F. GUO EJDE-2024/04 and E2(s) ≥ 1 2 ∫ R y2(a(yes/2) + e−s b2(t(s)) g2)dy + 1 4 ∫ R y2f2dy − 1 8 ∫ R e−s b2(t(s)) y2g2dy + α(s) ∫ R y2a0(yes/2)ϕ′0fydy ≥ 1 4 ∫ R y2a(yes/2)f2ydy + 3 8 e−s b2(t(s)) ∫ R y2g2dy + 1 4 ∫ R y2f2dy −Bα2(s)e−2( 1 4+ µ0 2 )s. (4.27) Note C2 ≥ 2(C1A+B) and λ ≤ 1 4 + µ0 2 , it is easy to see that C2e −2λsα2(s)−AC1e −2( 1 4+ µ0 2 )sα2(s)−Be−2( 1 4+ µ0 2 )sα2(s) ≥ 1 2 C2e −2λsα2(s), then by combining (4.23), (4.26) and (4.27), we obtain the lower bound E4(s) ≥ 1 4 min(1, a)‖f‖2H1,1 + 1 4 e−s b2(t(s)) ‖g‖2H0,1 + 1 2 e−s b2(t(s)) (dα(s) ds )2 + 1 2 C2e −2λsα2(s) ≥ C(‖f‖2H1,1 + e−s b2(t(s)) ‖g‖2H0,1 + e−s b2(t(s)) (dα(s) ds )2 + e−2λsα2(s)), where C = 1 4 min(1, a). The upper bound for E0(s), E1(s) and E2(s) are obtained similarly: E0(s) ≤ 1 2 ∫ R F 2 y dy + 3 4a ∫ R ( F 2 + e−s b2(t(s)) G2 ) dy, E1(s) ≤ 3 4 ∫ R a(yes/2)f2ydy + 3 2 ∫ R f2dy + 3 4 e−s b2(t(s)) ∫ R g2dy +Aα2(s)e−2( 1 4+ µ0 2 )s (4.28) and E2(s) ≤ 3 4 ∫ R a(yes/2)y2f2ydy + 3 4 ∫ R y2f2dy + 5 8 e−s b2(t(s)) ∫ R y2g2dy +Bα2(s)e−2( 1 4+ µ0 2 )s. (4.29) Therefore the upper bound for E4(s) can be formulated by Lemma 4.5 as E4(s) = C0E0 + C1E1 + E2 + E3 ≤ C0 (1 2 ∫ R F 2 y (y)dy + 3 4a ∫ R ( F 2 + e−s b2(t(s)) G2 ) dy ) + C1 (3 4 ∫ R a(ye y 2 )f2ydy + 3 4 ∫ R e−s b2(t(s)) g2dy + 3 2 ∫ R f2dy ) + 3 4 ∫ R y2a(yes/2)f2ydy + 3 4 ∫ R y2f2dy + 5 8 ∫ R e−s b2(t(s)) y2g2dy + 1 2 e−s b2(t(s)) (dα(s) ds )2 + e−2λsα2(s) + C1Aα 2(s)e−2( 1 4+ µ0 2 )s +Bα2(s)e−2( 1 4+ µ0 2 )s EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 19 ≤ C ( ‖f‖2H1,1 + e−s b2(t(s)) ‖g‖2H0,1 + e−s b2(t(s)) (dα(s) ds )2 + e−2λsα2(s) ) , where C = C(A,B,C0, C1, C2, ‖a‖L∞ , a). � Lemma 4.8. If 0 < λ ≤ 1 4 + µ0 2 , then L4(s) ≥ C ( ‖f‖2H1,1 + ‖g‖2H0,1 + (dα(s) ds )2 − e−2λsα2(s) ) (4.30) holds for s ≥ s1, where L4(s) and s1 are given by Lemmas 4.6 and 4.7, respectively. Proof. (4.23) shows E0(s) ≥ 0. From (4.26) and (4.27), we have C1E1 + E2 ≥ −C1Aα 2(s)e−2( 1 4+ µ0 2 )s −Bα2(s)e−2( 1 4+ µ0 2 )s ≥ −(C1A+B)α2(s)e−2λs, here we have used 0 < λ ≤ 1 4 + µ0 2 . Note the definition of L4(s), it suffices to estimate C0L0 + C1L1 + L2. In fact, C0L0 + C1L1 + L2 = C0 ∫ R (1 2 F 2 y + G2 a(yes/2) ) dy + C1 ∫ R ( a(yes/2)f2y + g2 − f2 ) dy + ∫ R (1 2 y2a(yes/2)f2y + y2g2 ) dy + 2 ∫ R ya(yes/2)fy(f + g)dy ≥ C0 2 ∫ R f2dy + C1a ∫ R f2ydy + C1 ∫ R g2dy − C1 ∫ R f2dy + a 2 ∫ R y2f2ydy + ∫ R y2g2dy + 2a ∫ R yfy(f + g)dy. Young’s inequality implies 2a ∣∣∫ R yfy(f + g)dy ∣∣ ≤ a 4 ∫ R y2f2ydy + 8a ∫ R (f2 + g2)dy, then C0L0 + C1L1 + L2 ≥ (C0 2 − C1 − 8a ) ∫ R f2dy + C1a ∫ R f2ydy + a 4 ∫ R y2f2ydy + (C1 − 8a) ∫ R g2dy + ∫ R y2g2dy ≥ C ( ‖f‖2H1,1 + ‖g‖2H0,1 ) . � Let E5(s) = E4(s) + 1 2 α2(s) + e−s b2(t(s)) α(s) dα(s) ds . Lemma 4.9. Under the conditions of Lemma 4.8, there exists a s2 ≥ s1 such that for each s ≥ s2, E5(s) ∼ ‖f‖2H1,1 + e−s b2(t(s)) ‖g‖2H0,1 + α2(s) + e−s b2(t(s)) (dα(s) ds )2 . (4.31) Proof. In view of the definition of E5(s), by Lemma 4.7, we only need to estimate e−s b2(t(s))α(s)dα(s)ds . Indeed, by Young’s inequality, for each η > 0,∣∣ e−s b2(t(s)) α(s) dα(s) ds ∣∣ ≤ C(η) e−s b2(t(s)) α2(s) + η e−s b2(t(s)) (dα(s) ds )2 . 20 Y. LI, H. LIU, F. GUO EJDE-2024/04 Choose η sufficiently small such that η e−s b2(t(s)) (dα(s) ds )2 ≤ 1 2 E4(s) for each s ≥ s1. In view of Lemma 3.2, there exists a s2 ≥ s1 such that for each s ≥ s2, C(η) e−s b2(t(s))α 2(s) ≤ 1 4α 2(s). Then E5(s) ≥ 1 2 E4(s) + 1 4 α2(s). This and (4.22) show that ‖f‖2H1,1 + e−s b2(t(s)) ‖g‖2H0,1 + α2(s) + e−s b2(t(s)) (dα(s) ds )2 ≤ CE5(s). The upper bound estimate for E5(s) can be similarly derived. � Lemma 4.10. We have d ds E5(s) + 2λE4(s) + L4(s) = R5(s), (4.32) where R5(s) = R4(s) + e−s b2(t(s))) (dα(s) ds )2 − 2 b2(t(s)) db(t(s)) dt α(s) dα(s) ds + α(s) ∫ R r(s, y)dy, r(s, y) and R4(s) are given by (3.4) and (4.20), respectively. Proof. (4.32) follows from Lemma 4.4, (4.20), and the direct calculation: d ds E5(s) = d ds E4(s) + α(s)α̇(s) + e−s b2(t(s))) (dα(s) ds )2 + e−s b2(t(s))) α(s) d2α(s) ds2 − 2 b2(t(s)) db(t(s)) dt α(s) dα(s) ds − e−s b2(t(s))) α(s) dα(s) ds = d ds E4(s) + e−s b2(t(s))) (dα(s) ds )2 − 2 b2(t(s)) db(t(s)) dt α(s) dα(s) ds + α(s) ∫ R r(s, y)dy. � Lemma 4.11. Let λ = min{ 14 + µ0 2 , λ0, λ1}, where λ0 = min{1− β 1 + β , γ 1 + β − 1 2 , ν 1 + β − 1}, λ1 = 1 2 ( p1 + 2p2 + (3− 2β 1 + β )p3 − 3 ) , (4.33) then there exists a s0 ≥ s2 (as defined in Lemma 4.9), such that for each s ≥ s0, we have the following estimates |R4(s)| ≤ η ( L4(s) + e−2λsα2(s) ) + C(η)e−2λsE5(s) + C(η)e−2λ1sE5(s)p1+p2 ( E5(s)p3 + ( L4(s) + e−2λsα2(s) )p3) , EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 21 |R5(s)| ≤ η ( L4(s) + e−2λsα2(s) ) + C(η)e−λsE5(s) + C(η)e−2λ1sE5(s)p1+p2 ( E5(s)p3 + ( L4(s) + e−2λsα2(s) )p3) + C(η)e−λ1sE5(s) p1+p2+p3+1 2 . Here, 1 1+β and −2βp31+β are regarded as sufficiently large numbers when β = −1 and p3 6= 0, η is any positive constants, R4(s) and R5(s) are given in Lemmas 4.6 and 4.10, respectively. We postpone the proof of Lemma 4.11 and now complete the proof of Proposition 4.3. The combination of Lemmas 4.10 and 4.11 gives d ds E5(s) + L4(s) = R5(s)− 2λE4(s) ≤ η ( L4(s) + e−2λsα2(s) ) + C(η)e−λsE5(s) + C(η)e−2λ1sE5(s)p1+p2 ( E5(s)p3+ ( L4(s) + e−2λsα2(s) )p3) + C(η)e−λ1sE5(s) p1+p2+p3+1 2 . Note that from Lemma 4.9, we know α2(s) can be controlled by E5(s), so choose η = 1/2, for each s ≥ s0, when p3 = 0, we have d ds E5(s) ≤ Ce−λsE5(s) + C(η)e−2λ1sE5(s)p1+p2 + C(η)e−λ1sE5(s) p1+p2+1 2 (4.34) and when p3 = 1, we have d ds E5(s) ≤ Ce−λsE5(s) + C(η)e−2λ1sE5(s)p1+p2 ( E5(s) + L4(s) + e−2λsα2(s) ) + C(η)e−λ1sE5(s) p1+p2+2 2 . (4.35) We set Λ(s) = exp(−C ∫ s s0 e−λτdτ). Obviously, e− Ce−λs0 λ ≤ Λ(s) ≤ 1 for each s ≥ s0 and Λ(s0) = 1, where s0 is given in Lemma 4.11. Multiplying Λ(s) on both sides of (4.34) and integrating on the interval [s0, s], we obtain Λ(s)E5(s) ≤ E5(s0) + C ∫ s s0 Λ(τ) [ e−2λ1τE5(τ)p1+p2 + e−λ1τE5(τ) p1+p2+1 2 ] dτ. (4.36) Let M(s) = sup s0≤τ≤s E5(τ). As λ1 > 0, (4.36) implies that, for each s ≥ s0, M(s) ≤ CM(s0) + C ( M(s)p1+p2 +M(s) p1+p2+1 2 ) . From the proof of Theorem 2.4, there exists a small ε̃0 > 0, such that for each ε ∈ (0, ε̃0], ‖(v(s0), w(s0))‖H1,1×H0,1 < 2C(s0)I0ε, then by Lemma 4.9, we obtain M(s0) ≤ C(s0) ( ‖(f(s0), g(s0))‖2H1,1×H0,1 + α2(s0) + α̇(s0)2 ) ≤ C(s0)‖(v(s0), w(s0))‖2H1,1×H0,1 ≤ C(s0)ε2I20 , (4.37) 22 Y. LI, H. LIU, F. GUO EJDE-2024/04 then M(s) ≤ C3ε 2I20 + C3 ( M(s)p1+p2 +M(s) p1+p2+1 2 ) , ∀ s ≥ s0. (4.38) Multiplying by Λ(s) on both sides of (4.35), similar to (4.36), we have Λ(s)E5(s) ≤ E5(s0) + ∫ s s0 Λ(τ)e−2λ1τE5(τ)p1+p2(L4(τ) + e−2λτα2(τ))dτ + C ∫ s s0 [Λ(τ)e−2λ1τE5(τ)p1+p2+1 + Λ(τ)e−λ1τE5(τ) p1+p2+2 2 ]dτ, then, for each s ≥ s0, M(s) ≤M(s0) + C ∫ s s0 M(τ)p1+p2L4(τ)dτ + C(M(s)p1+p2+1 +M(s) p1+p2+2 2 ), (4.39) so M(s) ≤ C3ε 2I20 + C3M(s)p1+p2 ∫ s s0 L4(τ)dτ + C3 ( M(s)p1+p2+1 +M(s) p1+p2+2 2 ) . (4.40) We take ε̃0 sufficiently small such that 2C3ε 2I20 > C3ε 2I20 + C3[(4C3ε 2I20 )p1+p2 + (4C3ε 2I20 )(p1+p2+1)/2] and 2C3ε 2I20 > C3ε 2I20 + C3(4C3ε 2I20 )p1+p2 ∫ s s0 L4(τ)dτ + C3((4C3ε 2I20 )p1+p2+1 + (4C3ε 2I20 ) p1+p2+2 2 ) hold for each ε ∈ (0, ε̃0]. By the continuous induction method, from (4.37), (4.38) and (4.40), for each s ≥ s0 and ε ∈ (0, ε̃0), it follows that M(s) ≤ 2C3ε 2‖(v0, w0)‖2H1,1×H0,1 , (4.41) by Lemma 4.9, i.e., ‖f‖2H1,1 + e−s b2(t(s)) ‖g‖2H0,1 + α2(s) + e−s b2(t(s)) (dα(s) ds )2 ≤ Cε2‖(v0, w0)‖2H1,1×H0,1 . Consequently, by (4.7), for each s ≥ s0 and ε ∈ (0, ε̃0), we have ‖v(s)‖2H1,1 + e−s b2(t(s)) ‖w(s)‖2H0,1 ≤ Cε2‖(v0, w0)‖2H1,1×H0,1 , which completes the proof of Proposition 4.3. Proof of Theorem 2.5. For the proof we use Corollary 4.2 and Proposition 4.3. Step 1: Global well-posedness. For s0 > 0, as given in Lemma 4.11, it follows from Corollary 4.2 that there exists a ε∗0 > 0 such that for each ε ∈ [0, ε∗0), the mild solu- tion (v, w) uniquely exists on [0, s0] and S(ε) > s0. Let ε1 = min(ε∗0, ε̃0), where ε̃0 is given by Proposition 4.3. It is claimed that for each ε ∈ (0, ε1], S(ε) =∞. Indeed, if there exists a ε∗ ∈ (0, ε1] such that S(ε∗) < ∞, let (v, w) be the corresponding mild solution to (3.3), then by Proposition 4.3, for each s ∈ [s0, S(ε∗)), ‖v(s)‖2H1,1 + e−s b(t(s))2 ‖w(s)‖2H0,1 ≤ C∗ε2∗‖(v0, w0)‖H1,1×H0,1 . (4.42) EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 23 However, it follows from Corollary 4.2 that lims→S(ε∗) ‖(v, w)(s)‖H1,1(R)×H0,1(R) = ∞, which contradicts (4.42). Step 2: Asymptotic profile. Taking η = 1/4 in Lemma 4.11 and using (4.20), we have d ds E4(s) + 2λE4(s) + L4(s) ≤ 1 4 L4(s) + Ce−2λsE5(s) + Ce−2λ1sE5(s)p1+p2+p3 + Ce−2λ1sE5(s)p1+p2L4(s)p3 ≤ 1 4 L4(s) + Ce−2λsE5(s) + Ce−2λ1sE5(s)p1+p2+p3 + C(2C3ε 2I20 )p1+p2L4(s)p3 . We provide the proof only for the case when p3 = 1 because the case of p3 = 0 can be handled similarly. Take a small ε1 such that 1 4 + C(2C3ε 2I0)p1+p2 ≤ 1 2 for each ε ∈ (0, ε1], then d ds E4(s) + 2λE4(s) + 1 2 L4(s) ≤ Ce−2λsE5(s) + Ce−2λ1sE5(s)p1+p2+1 ≤ Ce−2λsε2‖(u0, u1)‖2H1,1×H0,1 , multiplying by e2λs gives d ds (e2λsE4(s)) + e2λs 2 L4(s) ≤ Cε2‖(u0, u1)‖2H1,1×H0,1 , i.e., d ds ( e2λsE4(s) ) + e2λs 2 ( L4(s) + e−2λsα2(s) ) ≤ Cε2‖(u0, u1)‖2H1,1×H0,1 + α2(s) ≤ Cε2‖(u0, u1)‖2H1,1×H0,1 , (4.43) where Lemma 4.9 has been used. Integrating (4.43) on [s0, s], multiplying by e−2λs and using Lemma 4.8, we have E4(s) + 1 2 ∫ s s0 e−2λ(s−τ)(‖f‖2H1,1 + ‖g‖2H0,1 + α̇2(s))dτ ≤ Ce−2λsε2(s− s0)‖(u0, u1)‖2H1,1×H0,1 . Note that when s0 ≤ s̃ ≤ s, |α(s)− α(s̃)|2 = (∫ s s̃ dα dτ (τ)dτ )2 ≤ (∫ s s̃ e−2λτdτ )(∫ s s̃ e2λτ ( dα dτ (τ))2dτ ) ≤ Ce−2λs̃ε2‖(u0, u1)‖2H1,1×H0,1 , so α∗ = lims→+∞ α(s) exists and |α(s) − α∗|2 ≤ Ce−2λsε2‖(u0, u1)‖2H1,1×H0,1 . Therefore, ‖v(s)− α∗ϕ0‖2H1,1 ≤ 2‖f(s)‖2H1,1 + 2|α(s)− α∗|2‖ϕ0‖2H1,1 ≤ Ce−2λsε2‖(u0, u1)‖2H1,1×H0,1 . By (3.2) and G(B(t) + 1, x) = (B(t) + 1)−1/2ϕ0((B(t) + 1)−1/2x), we infer that ‖u(t, ·)− α∗G(B(t) + 1, ·)‖2L2 ≤ Cε2(B(t) + 1)− 1 2−2λ‖(u0, u1)‖2H1,1×H0,1 . 24 Y. LI, H. LIU, F. GUO EJDE-2024/04 Proof of Lemma 4.11. To obtain the upper bound of |R4(s)| and |R5(s)|, in view of their definitions, we need to estimate the H0,1 norms of r(s), h(s) and H(s). Recall that r(s) is defined by (3.4). Lemma 4.12. Under assumptions (2.1) and (2.3), we have∥∥e3s/2N(e−s/2v, e−svy, b−1(t(s))e−3s/2w )∥∥2 H0,1 ≤ Ce−2λ1s ( ‖f‖2H1,1 + α2(s) )p1+p2(‖g‖2H0,1 + α2(s) + (dα(s) ds )2)p3 (4.44) holds for each s ≥ 0, where λ1 is defined by (4.33). Proof. There are two cases according to the value of β. Case 1: β ∈ (−1, 1). By Lemma 3.2 and (2.3), we have (1 + y2)e3sN2 ( e−s/2v, e−svy, b −1(t(s))e−3s/2w ) ≤ C(1 + y2)e3se−p1s|v|2p1e−2p2s|vy|2p2e−(3− 2β 1+β p3)s|w|2p3 = C(1 + y2)e−2λ1s|v|2p1 |vy|2p2 |w|2p3 . Sobolev’s inequality shows that ‖v(s)‖L∞ ≤ C‖v(s)‖H1,0 , then (1 + y2)e−2λ1s|v|2p1 |vy|2p2 |w|2p3 = Ce−2λ1s(1 + y2)1−p2−p3(1 + y2)p2(1 + y2)p3 |v2|p1+p2+p3−1|v2|1−p2−p3 |v2y|p2 |w2|p3 ≤ Ce−2λ1s‖v(s)‖2(p1+p2+p3−1)H1,0 ( (1 + y2)v2 )1−p2−p3( (1 + y2)v2y )p2( (1 + y2)w2 )p3 . So by Hölder’s inequality,∥∥e3s/2N(e−s/2v, e−svy, b−1(t(s))e−3s/2w )∥∥2 H0,1 ≤ Ce−2λ1s‖v(s)‖2(p1+p2+p3−1)H1,0 ‖v(s)‖2(1−p2−p3)H1,1 ‖v(s)‖2p2H1,1‖w(s)‖2p3H0,1 ≤ Ce−2λ1s‖v(s)‖2(p1+p2−1)H1,0 ‖v(s)‖2H1,1‖w(s)‖2p3H0,1 ≤ Ce−2λ1s ( ‖f‖H1,1 + α(s) )2(p1+p2)(‖g‖H0,1 + α(s) + α̇(s) )2p3 ≤ Ce−2λ1s ( ‖f‖2H1,1 + α2(s) )p1+p2(‖g‖2H0,1 + α2(s) + (dα(s) ds )2)p3 . Case 2: β = −1, p3 6= 0. By Lemma 3.2 and (2.3), we obtain (1 + y2)e3sN2 ( e−s/2v, e−svy, b −1(t(s))e−3s/2w ) ≤ C(1 + y2)e(3−p1−2p2−3p3)sb−p3(t(s))|v|2p1 |vy|2p2 |w|2p3 ≤ C(1 + y2)e−λ∗s|v|2p1 |vy|2p2 |w|2p3 ≤ C(1 + y2)e−2λ1s|v|2p1 |vy|2p2 |w|2p3 , here we have used b−1(t(s)) ∼ exp(−es) by Lemma 3.2, and λ∗ = 2 e s s p3 + (p1 + 2p2 + 3p3− 3) > 2λ1 is regarded as a sufficiently large number. Then (4.44) can be derived similarly to Case 1. � EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 25 Lemma 4.13. Under assumptions (A1)–(A4), for each s ≥ 0, we have ‖r(s)‖2H0,1 ≤ Ce−2λ0s ( ‖f‖2H1,1 + ‖g‖2H0,1 + α2(s) + (dα(s) ds )2) + Ce−2λ1s ( ‖f‖2H1,1 + α2(s) )p1+p2(‖g‖2H0,1 + α2(s) + (dα(s) ds )2)p3 , (4.45) where r(s) is given by (3.4), λ0 and λ1 are given by (4.33). Proof. By the definition of r(s), it suffices to estimate 1 b2(t(s)) db(t(s)) dt w + es/2c(t(s))vy + esd(t(s))v. Combining (4.44) with ‖ 1 b2(t(s)) db(t(s)) dt w‖2H0,1 ≤ C ( ‖g‖2H0,1 + α2(s) + (dα(s) ds )2)×{e− 2(1−β)s 1+β , β ∈ (−1, 1), exp(−4es), β = −1, ‖es/2c(t(s))vy‖2H0,1 ≤ C ( ‖f‖2H1,1 + α2(s) ) × { e−( 2γ 1+β−1)s, β ∈ (−1, 1), exp(−2γes + s), β = −1 ‖esd(t(s))v‖2H0,1 ≤ C ( ‖f‖2H1,1 + α2(s) ) × { e−( 2ν 1+β−2)s, β ∈ (−1, 1), exp(−2νes + 2s). β = −1, we obtain(4.45), where Lemma 3.2 and (2.2) have been used. � Lemma 4.14. Under assumptions (A1)–(A4), ‖h(s)‖2H0,1 ≤ Ce−2λ0s ( ‖f‖2H1,1 + ‖g‖2H0,1 + α2(s) + (dα(s) ds )2) + Ce−2λ1s ( ‖f‖2H1,1 + α2(s) )p1+p2(‖g‖2H0,1 + α2(s) + (dα(s) ds )2)p3 (4.46) holds for each s ≥ 0, where h(s), λ0, λ1 are given by (4.11) and (4.33). Proof. Clearly,∥∥∥ e−s b2(t(s)) ( −2 dα(s) ds ψ0(y) + α(s) (y 2 ψ′0(y) + 3 2 ψ0(y) ))∥∥∥2 H0,1 ≤ ∥∥(−2 dα(s) ds ψ0(y) + α(s) (y 2 ψ′0(y) + 3 2 ψ0(y) ))∥∥2 H0,1 × { e−2( 1−β 1+β )s, β ∈ (−1, 1), exp(−2es − s), β = −1 ≤ Ce−2λ0s ( α̇2(s)‖ψ0(y)‖2H0,1 + α2(s)‖ψ0(y)‖2H0,1 ) ≤ Ce−2λ0s ( α̇2(s) + α2(s) ) . 26 Y. LI, H. LIU, F. GUO EJDE-2024/04 Hölder’s inequality gives∣∣ ∫ R r(s, y)dy ∣∣ ≤ (∫ R 1 1 + y2 dy )1/2(∫ R (1 + y2)r2(s, y)dy )1/2 ≤ C‖r(s, ·)‖H0,1 . (4.47) Then in view of the definition of h(s), (4.46) follows from Lemma 4.13. � Lemma 4.15. Under assumptions (A1)–(A4), for each s ≥ 0, we have ‖H(s)‖2H0,1 ≤ Ce−2λ0s ( ‖f‖2H1,1 + ‖g‖2H0,1 + α2(s) + (dα(s) ds )2) + Ce−2λ1s ( ‖f‖2H1,1 + α2(s) )p1+p2(‖g‖2H0,1 + α2(s) + (dα(s) ds )2)p3 , where H(s), λ0, λ1 are given by (4.15) and (4.33). Proof. By Lemma 4.5,∫ R H2(s, y)dy ≤ 4 ∫ R y2h2(s, y)dy ≤ 4‖h(s)‖2H0,1 . A direct calculation and Young’s inequality show that∫ R y2H2(s, y)dy = ∫ |y|≤1 y2H2(s, y)dy + ∫ |y|>1 y2H2(s, y)dy ≤ ∫ |y|≤1 H2(s, y)dy + ∫ |y|>1 y3H2(s, y)dy ≤ 4‖h(s)‖2H0,1 − 6 ∫ |y|>1 y2H(s, y)h(s, y)dy ≤ 4‖h(s)‖2H0,1 + 1 2 ∫ |y|>1 y2H2dy + 72 ∫ |y|>1 y2h2dy, so ∫ R y2H2(s, y)dy ≤ C ∫ R y2h2(s, y)dy ≤ ‖h(s)‖2H0,1 . Then Lemma 4.15 can be deduced by Lemma 4.14. � Proof of Lemma 4.11. Recall that R4(s) = C0R0(s) + C1R1(s) + R2(s) + R3(s), where Ri(s) (i = 0, 1, 2, 3) are defined by (4.16)-(4.19). Lemmas 4.7, 4.7 and Young’s inequality show that there exists a s3 ≥ s2 such that the terms that do not include r(s, y), h(s, y) and H(s, y) can be controlled by η(L4(s) + e−2λsα2(s)) + C(η)e−2λsE5(s). For example,∣∣ ∫ R 1 a(yes/2) (α(s)a0(yes/2)ϕ′0(F +G)dy ∣∣ ≤ 1 a (‖F‖L2 + ‖G‖L2)(|α|‖a0(yes/2)ϕ′0‖L2) ≤ 1 a ( η(‖f‖H1,1 + ‖g‖H0,1) + C(η)e−2( 1 4+ µ0 2 )sα2(s) ) ≤ C ( η(L4(s) + e−2λsα2(s)) + C(η)e−2λsE5(s) ) . (4.48) The other terms can be estimated similarly. The remainder terms consist of∫ R (F +G)Hdy, ∫ R (1 + |y|2)(f + g)hdy, dα(s) ds ∫ R r(s, y)dy. EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 27 Using Lemmas 4.5, 4.8, 4.9, 4.14, 4.15 and Young’s inequality, the first two terms can be controlled by η(L4 + e−2λsα2(s)) + C(η)e−2λsE5(s) + C(η)e−2λ1sE5(s)p1+p2 ( E5(s)p1 + (L4 + e−2λsα2(s))p3 ) . For example,∣∣ ∫ R (F +G)Hdy ∣∣ ≤ C(‖F‖L2 + ‖G‖L2)‖H‖L2 ≤ C(η(‖f‖2H1,1 + ‖g‖2H0,1) + C(η)‖h‖2H0,1) ≤ η(L4 + e−2λsα2(s)) + C(η)e−2λsE5(s) + C(η)e−2λ1sE5(s)p1+p2(E5(s)p3 + (L4 + e−2λsα2(s))p3). (4.49) In addition, Young’s inequality implies∣∣(∫ R r(s, y)dy )dα(s) ds ∣∣ ≤ η(dα(s) ds )2 + C(η)( ∫ R r(s, y)dy)2, then the third term can be estimated by using Lemmas 4.8, 4.9, 4.13, and (4.47). So far, the estimate for R4(s) has been obtained. Note that R5(s) defined in Lemma 4.10 can be rewritten as R5(s) = R̃5(s) + R4(s), where R̃5(s) = e−s b2(t(s)) (dα(s) ds )2 − 2 b2(t(s)) α(s) db(t(s)) dt dα(s) ds + α(s) ∫ R r(s, y)dy, so it suffices to estimate R̃5(s). Obviously, there exists a s4 ≥ s2 such that for each s ≥ s4, e−s b2(t(s)) (dα(s) ds )2 − 2 b2(t(s)) α(s) db(t(s)) dt dα(s) ds ≤ η(L4 + e−2λsα2(s)) + C(η)e−2λsE5(s). Using Lemma 4.13, (4.47) and |α(s)| ≤ CE5(s)1/2 (by Lemma 4.9), there exists a s5 ≥ s2, such that for each s ≥ s5, |α(s) ∫ R r(s, y)dy| ≤ |α(s)|‖r(s)‖H0,1 ≤ CE5(s)1/2[(L4(s) + e−2λsα2(s))1/2 + e−λ0sE5(s)1/2] + Ce−λ1sE5(s) p1+p2+1 2 (E5(s)p3/2 + (L4(s) + e−2λsα2(s))p3/2) ≤ η̃(L4(s) + e−2λsα2(s)) + Ce−λsE5(s) + Ce−λ1sE5(s) p1+p2+1 2 (E5(s)p3/2 + (L4(s) + e−2λsα2(s))p3/2). If p3 = 1 and e−λ1sE5(s) p1+p2+1 2 (L4 + e−2λsα2(s))1/2 ≤ η(L4 + e−2λsα2(s)) + C(η)e−2λ1sE5(s)p1+p2+1, 28 Y. LI, H. LIU, F. GUO EJDE-2024/04 then |α(s) ∫ R r(s, y)dy| ≤ η(L4(s) + e−2λsα2(s)) + Ce−λsE5(s) + Ce−λ1sE5(s) p1+p2+p3+1 2 + C(η)e−2λ1sE5(s)p1+p2+p3 . The case of p3 = 0 can be estimated similarly. Setting s0 = max{s3, s4, s5}, the estimate for R5(s) can be obtained. � 5. Proof of Theorem 2.6 For each s∗ > 0, let M(s) = sups∗≤τ≤sE5(τ), Λ(s) = exp(−C ∫ s s∗ e−λτdτ), where E5(s) is defined in Lemma 4.9. Then (4.31) implies that M(s) ∼ sup s∗≤τ≤s ( ‖v(τ, ·)‖2H1,1 + e−τ b2(t(τ)) ‖w(τ, ·)‖2H0,1 ) . Lemma 5.1. Under assumptions (A1)–(A5), where (2.3) is replaced by (2.10), there exist s∗ > 0, C > 0 such that for each s ≥ s∗ and each solution (v, w) to (3.3), we have M(s) ≤M(s∗) + Ce(3−p1)sM(s)p1 + Ce 3−p1 2 sM(s) p1+1 2 . (5.1) Proof. As in the proof of Proposition 4.3, by Lemmas 4.10 and 4.11 we have d ds E5(s) + L4(s) ≤ η ( L4(s) + e−2λsα2(s) ) + C(η)e−λsE5(s) + C(η)e−2λ1sE5(s)p1 + C(η)e−λ1sE5(s) p1+1 2 , letting η = 1/2 and using Lemma 4.8, we deduce that d ds E5(s) ≤ Ce−λsE5(s) + Ce−2λ1sE5(s)p1 + Ce−2λ1sE5(s)p1 + Ce−λ1sE5(s) p1+1 2 . (5.2) In the following, it suffices to estimate the upper bound for L4(s). Indeed, note that L4(s) = ( 1 2 − 2λ)(C0E0(s) + C1E1(s) + E2(s)) + C0L0(s) + C1L1(s) + L2(s) + α̇2(s). From (4.28), (4.29), and (4.31), it follows that C0E0(s) + C1E1(s) + C2E2(s) ≤ CE5(s). Lemma 4.5 gives C0L0(s) + C1L1(s) + L2(s) + α̇2(s) = C0 ∫ R (1 2 F 2 y + 1 a(es/2y) G2 ) dy + C1 ∫ R ( a(es/2y)f2y + g2 − f2 ) dy + ∫ R (1 2 y2a(es/2y)f2y + y2g2 ) dy + 2 ∫ R ya(es/2y)fy(f + g)dy + α̇2(s) ≤ C ( ‖f‖2L2 + ‖G‖2L2 + ‖fy‖2L2 + ‖g‖2L2 + ‖yfy‖2L2 + ‖yg‖2L2 + α̇2(s) ) ≤ C ( ‖g‖2H0,1 + α̇2(s) ) + CE5(s), EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 29 while from Lemma 3.2 it follows that ‖g‖2H0,1 + α̇2(s) = esb2(t(s)) ( e−s b2(t(s)) ( ‖g‖2H0,1 + α̇2(s) )) ≤ Cesb2(t(s))E5(s) ≤ Ce 1−β 1+β sE5(s); therefore, L4(s) ≤ Ce 1−β 1+β sE5(s). (5.3) Substituting (5.3) into (5.2) gives d ds E5(s) ≤ Ce−λsE5(s) + Ce−2λ1sE5(s)p1 + Ce−2λ1sE5(s)p1 + Ce−λ1sE5(s) p1+1 2 . By (2.10), it is easy to see that −2λ1 = 3− p1 > 0, so d ds E5(s) ≤ Ce−λsE5(s) + Ce−2λ1sE5(s)p1 + Ce(3−p1)sE5(s)p1 + Ce−λ1sE5(s) p1+1 2 . Then d ds ( E5(s)Λ(s) ) ≤ Ce−2λ1sE5(s)p1Λ(s) + Ce(3−p1)sE5(s)p1Λ(s) + Ce−λ1sE5(s) p1+1 2 Λ(s) ≤ Ce(3−p1)sE5(s)p1Λ(s) + Ce 3−p1 2 sE5(s) p1+1 2 Λ(s). (5.4) As the indexes satisfy 3−p1 > 0, after integrating, there exists a s∗ > 0 sufficiently large such that (5.1) holds for each s ≥ s∗. � Proof of Theorem 2.6. By Corollary 4.2, the lifespan S(ε) of the solution (v, w) to (3.3) satisfies S(ε) > s∗ provided ε is sufficiently small, where s∗ is given by Lemma 5.1. Moreover, in view of (4.37), M(s∗) ≤ Cε2I20 . If M(s) cannot reach 2Cε2I20 at any time, then lims→S(ε)M(s) ≤ 2Cε2I20 , which contradicts (2.8). So let S ≥ s∗ be the first time such that M(S) = 2Cε2I20 . By Lemma 5.1, we have 2Cε2I20 ≤ Cε2I20 + Ce(3−p1)S ( Cε2I20 )p1 + Ce 3−p1 2 S ( Cε2I20 ) p1+1 2 , so Cε2 ≤ Ce 3−p1 2 Sεp1+1. Then ε 2(1−p1) 3−p1 ≤ CeS ≤ CeS(ε) ≤ C ( B(T (ε)) + 1 ) , namely, B(T (ε)) + 1 ≥ Cε 2(1−p1) 3−p1 . � Remark 5.2. We aim to establish an upper bound for (1.1), analogous to the lower bound found in (2.11). For simplicity, let us consider the equation ∂2t u− ∂x(a(x)∂xu) + b(t)∂tu = |u|p, t > 0, x ∈ R, u(0, x) = εu0(x), ∂tu(0, x) = εu1(x), x ∈ R. (5.5) 30 Y. LI, H. LIU, F. GUO EJDE-2024/04 If a(x) = 1, we note that the lower bound (2.11) for (5.5) coincides with [8, Theorem 1.2], which suggests that (2.11) is sharp for (5.5). By employing a test function argument, we can establish the following upper bound for (5.5) B(T (ε)) . { ε− 2(p−1) 3−2p , 1 < p < 3/2, eε −(p−1) , p = 3/2. (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). Acknowledgments. This work was supported by the NSF of China (11731007), by the Priority Academic Program Development of Jiangsu Higher Education In- stitutions, and by the NSF of Jiangsu Province (BK20221320). References [1] Th. Cazenave, A. Haraux, Y. Martel; An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998. [2] S. R. Dunbar, H. G. Othmer; On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, Nonlinear Oscillations in Biology and Chemistry, Springer-Verlag, Berlin, 66, 1986. [3] H. Fujita; On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. [4] Th. Gallay, G. Raugel; Scaling variables and asymptotic expansions in damped wave equa- tions, J. Differential Equations., 150 (1998), 42-97. [5] L. Hörmander,; The Analysis of Linear Partial Differential Operators, III, Springer, Berlin, 2007. [6] K. P. Hadeler; Reaction telegraph equations and random walk systems, Stochastic and Spatial Structures of Dynamical Systems, 45 (1996), 133-161. [7] M. Ikeda, T. Ogawa; Lifespan of solutions to the damped wave equation with a critical nonlinearity, J. Differential Equations., 261 (2016), 1880-1903. [8] M. Ikeda, M. Sobajima, Y. Wakasugi; Sharp lifespan estimates of blow-up solutions to semi- linear wave equations with time-dependent effective damping, J. Hyperbolic Differ. Equ., 16 (2019), 495-517. [9] M. Ikeda, Y. Wakasugi; A note on the lifespan of solutions to the semilinear damped wave equation, Proc. Amer. Math. Soc., 143 (2015), 163-171. [10] M. Kac; A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math., 4 (1974), 497-509. [11] N. A. Lai, Y. Zhou; The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions, J. Math. Pures Appl., 123 (2019), 229-243. [12] Y. C. Li; Classical solutions for fully nonlinear wave equations with dissipation (in Chinese), Chinese Ann. Math. Ser A., 17 (1996), 451-466. [13] T. T. Li, Y. Zhou; Breakdown of solutions to �u+ut = |u|1+α, Discrete Contin. Dyn. Syst., 1 (1995), 503-520. [14] G. I. Taylor; Diffusion by continuous movements, Proc. London Math. Soc., 20 (1920), 196- 212. [15] G. Todorova, B. Yordanov; Critical exponent for a nonlinear wave equation with damping, J. Differential Equations., 174 (2001), 464-489. [16] Y. Wakasugi; Scaling variables and asymptotic profiles for the semilinear damped wave equa- tion with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452-487. [17] J. Wirth; Solution representations for a wave equation with weak dissipation, Math. Meth. Appl. Sci., 27 (2004), 101-124. [18] J. Wirth; Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations., 222 (2006), 487-514. [19] J. Wirth; Wave equations with time-dependent dissipation II. Effective dissipation, J. Dif- ferential Equations., 232 (2007), 74-103. EJDE-2024/04 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 31 [20] E. Zauderer; Partial Differential Equations of Applied Mathematics, Wiley, New York, (1983). [21] Qi S. Zhang; A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris., 333 (2001), 109-114. Yuequn Li School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Email address: liyuequn1997@163.com Hui Liu School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Email address: 1412797891@qq.com Fei Guo School of Mathematical Sciences and Key Laboratory for NSLSCS, Ministry of Edu- cation, Nanjing Normal University, Nanjing 210023, China Email address: guof@njnu.edu.cn 1. Introduction 2. Main results 3. Proof of Theorem ?? 3.1. Preliminaries 3.2. Proof of Theorem ?? 4. Proof of Theorem ?? 4.1. Proof of Proposition ?? Proof of Theorem ?? Proof of Lemma ?? 5. Proof of Theorem ?? Acknowledgments References