Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 69, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.69 LONG-TIME BEHAVIOR OF SOLUTIONS FOR TIME-PERIODIC REACTION-DIFFUSION EQUATIONS AND APPLICATIONS LINLIN LI, ZHUO CHEN Abstract. This article concerns the asymptotic behavior of solutions for time-periodic reaction-diffusion equations with a drift term in one dimensional space. Assuming that drift term is decaying, we analyze the effect of the de- caying rate of the drift term on the propagation speed of solutions. Also we show some applications of our results in high-dimensional domains. 1. Introduction In this article, we consider the initial value problem of the time-periodic reaction- diffusion equation, ut = uxx + k(x)ux + f(t, u), x ∈ R, t > 0, u(0, x) = u0(x), x ∈ R, (1.1) where 0 ≤ u0(x) ≤ 1. Throughout this article, we assume that f(t, u) is periodic in t, that is, there is T ∈ R such that f(t+T, u) = f(t, u) for all u ∈ R and t ∈ R. We also assume that f(t, ·) is bistable for u ∈ [0, 1], that is, there is a periodic function θt ∈ (0, 1) such that f(t, 0) = f(t, 1) = 0, f(t, θt) = 0, f(t, ·) < 0 on (0, θt), f(t, ·) > 0 on (θt, 1), fu(t, 0), fu(t, 1) < 0 for all t ∈ R. which means that 0 and 1 are stable zeros of f . This implies that there exist ρ ∈ (0, 1/2) and τ > 0 such that −fu(t, u) ≥ τ for t ∈ R and u ∈ [0, ρ] ∪ [1− ρ, 1]. (1.2) The drift term k(x) is assumed to be decaying to 0 as x → +∞. More precise assumptions on k(x) and the initial value u0(x) will be given later. We aim to analyze the effect of the decaying rate of k(x) on the propagation speed of the solution. Before showing our results, we recall some well-known results of reaction-diffusion equations. In the pioneering work [5], Fife and McLeod studied the one-dimensional 2020 Mathematics Subject Classification. 35B09, 35B40, 35E15, 35K15, 35K57. Key words and phrases. Reaction-diffusion equation; asymptotic behavior; propagation speed. ©2024. This work is licensed under a CC BY 4.0 license. Submitted January 6, 2024. Published November 11, 2024. 1 2 L. LI, Z. CHEN EJDE-2024/69 homogeneous reaction-diffusion equation ut = uxx + f(u), t ∈ R, x ∈ R, (1.3) where f(u) is of bistable type, that is, there is θ ∈ (0, 1) such that f(0) = f(1) = f(θ) = 0, f < 0 on (0, θ) and f > 0 on (θ, 1). (1.4) They proved that (1.3) admits a unique traveling front which is an entire solution having the form ϕ(x− ct) and satisfying ϕ′′ + cϕ′ + f(ϕ) = 0, in R, ϕ(−∞) = 1, ϕ(+∞) = 0. The function ϕ and the constant c are called profile and propagation speed of the traveling front respectively. By [5], the profile ϕ and the speed c are uniquely deter- mined by the reaction term f , and the sign of c is the same as the sign of ∫ 1 0 f(u)du. For the initial value problem of (1.3) with the initial value u(0, x) = u0(x), they proved that if u0(x) satisfies lim infx→−∞ u0(x) > θ and lim supx→+∞ u0(x) < θ, then sup x∈R |u(t, x)− ϕ(x− ct+ x0)| → 0, as t → +∞, for some constant x0. This implies that the traveling front ϕ(x−ct) is asymptotically stable. Moreover, if c > 0 and the initial value u0(x) satisfies lim sup |x|→+∞ u0(x) < θ and u0(x) > θ for |x| < L, where L is a sufficiently large constant, then sup x∈R |u(t, x)− (ϕ(x− ct+ ξ1) + ϕ(−x− ct+ ξ2)− 1)| → 0 as t → +∞, for some constants ξ1 and ξ2. For the high dimensional version of (1.3), ut = ∆u+ f(u), t > 0, x ∈ RN (with N ≥ 2), (1.5) Jones [7] proved that if c > 0 and the solution of the compactly supported initial value grows in the sense that u(t, x) → 1 locally uniformly in RN as t → +∞, its level sets will go around and around. Roughly speaking, the solution is approxi- mately radially symmetric solution as t → +∞. From the results of [7], we also know that the solution u(t, x) satisfies ∀0 < c1 < c, inf |x|≤c1t u(t, x) → 1 as t → +∞, ∀c2 > c, sup |x|≥c2t u(t, x) → 0 as t → +∞. The constant c is also called propagation speed of the solution u by the fact that the area where u close to 1 extends almost at the speed c as t → +∞. The propagation speeds of solutions of initial value problems for other types of reaction terms and environment have been extensively investigated, see [2, 4, 9] and references therein. If we look at the radially symmetric solutions u(t, r) = u(t, |x|) of (1.5), it is not hard to show that u(t, r) satisfies the one-dimensional equation ut = urr + N − 1 r ur + f(u). (1.6) EJDE-2024/69 TIME-PERIODIC REACTION-DIFFUSION EQUATIONS 3 By comparing with (1.3), the above equation contains an additional drift term. The influence of more general drifts to the long-time behavior of solutions has been investigated by Uchiyama [10], through considering the following equation ut = uxx + k(x)ux + f(u), t > 0, x ∈ R, with the initial value u(0, x) = u0(x). Here, f is still bistable, that is, satisfying (1.4), and the drift k(x) satisfies lim x→+∞ k(x) = 0 and k′(x) = O( 1 x ln3 x ) as x → +∞. Then, if lim supx→+∞ u0(x) < θ and u(t, x) grows in the sense that u(t, x) → 1 locally uniformly in R as t → +∞, it holds sup x>0 |u(t, x)− ϕ(x− ct+m(t) + x0)| → 0 as t → +∞, (1.7) where x0 is a constant and m(t) is the solution of m′(t) = k(ct − m(t)) for t ≥ 0 and m(0) = −L for some L > 0. As mentioned in [10], if k′(x) = O(x− 3 2 \ lnx), then the asymptotic form of m(t) can be given. Especially if k(x) = (N − 1)/x as in (1.6), then m(t) = (N − 1)/c ln t + O(1). By applying this result to the high- dimensional equation (1.5), it implies that the high-dimensional diffusion will cause a logarithmic delay to the propagation speed of the solution. For the time-periodic reaction-diffusion equation (1.1) without drifts, Alikakos, Bates and Chen [1] proved that there exists a unique time-periodic traveling front ϕ(t, x− ct) satisfying ϕt − cϕξ − ϕξξ − f(t, ϕ) = 0, t ∈ R, ξ ∈ R, ϕ(t+ T, ·) = ϕ(t, ·), ϕ(t,−∞) = 1, ϕ(t,+∞) = 0, t ∈ R. (1.8) The profile ϕ and the speed c are uniquely determined by the reaction term f , and the sign of c has the same sign of ∫ T 0 ∫ 1 0 f(t, s)dsdt. They also studied the stability of the time-periodic traveling front ϕ(t, x−ct). Precisely, they proved that for (1.3) with k(x) ≡ 0, if the initial value u0(x) satisfies lim infx→−∞ u0(x) > 1 − ρ and lim supx→+∞ u0(x) < ρ where ρ is given by (1.2), then the solution u(t, x) satisfies sup x∈R |u(t, x)− ϕ(t, x− ct+ x0)| → 0, as t → +∞, (1.9) for a constant x0. In this article, we first investigate the influence of the drift term on the long- time behavior of the solution of (1.1), especially to the propagation speed. Some a priori assumptions will be needed throughout this paper. We always assume that∫ T 0 ∫ 1 0 f(t, s)dsdt > 0 which means c > 0 and lim x→+∞ k(x) = 0 and k′(x) = O( 1 x ln3 x ), as x → +∞. (1.10) Moreover, we assume that the solution u(t, x) of (1.1) grows in the sense that u(t, x) → 1 locally uniformly in R as t → +∞. (1.11) This assumption is not empty. For example, according to the results of [1], (1.11) will hold if u0(x) = 1 for x ∈ [−L,L] and supx∈R k(x) is small enough for large enough L. We aim to generalize the results of Uchiyama [10] to the time-periodic case. However, by the effect of the time periodicity, such generalization is not trivial. For 4 L. LI, Z. CHEN EJDE-2024/69 example, the Lyapunov functional in [10] can not be used to prove the convergence of the solution to the traveling front. Now, we present our main results. Theorem 1.1. Assume that (1.10) holds. If lim supx→+∞ u0(x) < ρ and the solu- tion u(t, x) of (1.1) grows in the sense of (1.11), then sup x>0 |u(t, x)− ϕ(t, x− ct+m(t) + x0)| → 0, as t → +∞, (1.12) for some constant x0, where m(t) is the solution of m′(t) = k(ct−m(t)) for t ≥ 0 with m(0) = −L for some large L. By changing u(t, x) to u(t,−x), one can easily show that similar results hold for x < 0. Corollary 1.2. Assume that k(x) → 0 and k′(x) = O(1/(|x| ln3 |x|)) as x → −∞. If lim supx→−∞ u0(x) < ρ and the solution u(t, x) of (1.1) grows in the sense of (1.11), then sup x<0 |u(t, x)− ϕ(t,−x− ct+m(t) + x0)| → 0, as t → +∞ (1.13) for a constant x0, where m(t) is the solution of m′(t) = −k(−ct+m(t)) for t ≥ 0 with m(0) = −L for some large L. From Theorem 1.1, we have that if k(x) = 1/x for large x, then m(t) = 1 c ln t+ O(1) which means that there is a logarithmic delay for the propagation speed of u. If k(x) = 1/xr for large x with r > 1, then m(t) = o(1/tr−1) + O(1) which means that if the drift delays very fast, then the influence to the propagation speed of u is small. On the other hand, when k(x) ≡ 0, Theorem 1.1 also leads to the asymptotic stability of the time-periodic traveling front proved in [1]. The above results can be applied to initial value problems in high-dimensional domains. Precisely, we consider the following initial value problem vt = ∆v + f(t, v), x ∈ Ω, t > 0, v(0, x) = v0(x), x ∈ Ω, ∂νv = 0, x ∈ ∂Ω. (1.14) where Ω is an unbounded connected set of RN with smooth boundary and 0 ≤ v0(x) ≤ 1 is compactly supported. The first application is to the whole space RN . Theorem 1.3. Assume that Ω = RN and the solution v(t, x) of (1.14) grows in the sense that v(t, x) → 1 locally uniformly in RN as t → +∞. Then, for any ε > 0, there exist positive constants L(ε) and T such that v(t, x) ≥ 1− ε, for |x| ≤ ct− N − 1 c ln t− L(ε) and t ≥ T , (1.15) v(t, x) ≤ ε, for |x| ≥ ct− N − 1 c ln t+ L(ε) and t ≥ T . (1.16) The second application is to exterior domains. Here, an exterior domain is defined by Ω = RN \K, where K is a compact set. Theorem 1.4. Assume that Ω = RN \K and the solution v(t, x) of (1.14) grows in the sense that v(t, x) → 1 locally uniformly in Ω as t → +∞. Then, for any EJDE-2024/69 TIME-PERIODIC REACTION-DIFFUSION EQUATIONS 5 ε > 0, there exist positive constants L(ε) and T such that v(t, x) ≥ 1− ε, for x ∈ Ω such that |x| ≤ ct− N − 1 c ln t− L1(ε) and t ≥ T , (1.17) v(t, x) ≤ ε, for x ∈ Ω such that |x| ≥ ct− N − 1 c ln t+ L2(ε) and t ≥ T . (1.18) Remark 1.5. From the results in [3], one knows that if v0(x) is close to 1 in a sufficiently large ball and K is star-shaped or directionally convex with respect to a hyperplane, the solution v(t, x) grows. This paper is organized as follows. In Section 2, we prove Theorem 1.1. In Section 3, we show the applications, that is, we prove Theorems 1.3 and 1.4. 2. Asymptotic behavior of solutions This section is devoted to the proof of Theorem 1.1. We first need some param- eters and some auxiliary functions. Let L > 0 be a sufficiently large constant and mL(t) be the solution of m′ L(t) = k(ct−mL(t)), for t ≥ 0 and mL(0) = −L. (2.1) Then, by taking L sufficiently large and since k(x) → 0 as x → +∞, one has that mL(t) < ct for all t > 0 and ct−m(t) = O(t) → +∞ as t → +∞. Take a function γ(t) = 2 3 ln 3 2 (t+ 1), for t > 0. Then, it is increasing and has the following properties γ(0) = 0, sup t≥0 γ′(t) ≤ 1, ∫ +∞ 0 e−bγ(t)dt < +∞ for all b > 0. For each M ≥ 0, we let λL,M (t) = sup x>1, |x−ct|≤M+γ(t) |k(x−mL(t))− k(ct−mL(t))|. (2.2) Since k′(x) = O(1/(x ln3 x)) as x → +∞, one knows that λL,M (t) is integrable on (0,+∞) and lim L→+∞ ∫ +∞ 0 λL,M (t)dt = 0. Remember that there exist ρ ∈ (0, 1 2 ) and τ > 0 such that −fu(t, u) ≥ τ for t ∈ R and u ∈ [0, ρ] ∪ [1− ρ, 1]. (2.3) Since ϕ(t,−∞) = 1 and ϕ(t,+∞) = 0, there is R > 0 such that 0 < ϕ(t, ξ) ≤ ρ 2 for t ∈ R and ξ ≥ R, 1− ρ 2 ≤ ϕ(t, ξ) < 1 for t ∈ R and ξ ≤ −R. (2.4) Since ϕξ(t, ξ) < 0 by [1], there is a > 0 such that −ϕξ(t, ξ) ≥ a for t ∈ R and −R ≤ ξ ≤ R. (2.5) By [1], one also knows that there exist C1 > 0 and η > 0 such that |ϕξ(t, ξ)| ≤ C1e −η|ξ|, for t ∈ R and ξ ∈ R, (2.6) 6 L. LI, Z. CHEN EJDE-2024/69 and 1− ϕ(t, ξ) ≤ C1e ηξ, for t ∈ R and ξ < 0, (2.7) Let q = min{τ 2 , η}. We show some upper and lower bounds for the solution u(t, x) under some initial conditions and boundary conditions. Lemma 2.1. For each 0 < r < ρ 2 , there are constants M and L0 > 0 such that (i) if L ≥ L0, u0(x) ≥ ϕ(0, x− L)− r for x > 0, and u(t, 1) ≥ 1− re−qγ(t) for t > 0, then u(t, x) ≥ ϕ(t, x− ct+mL(t) + α(t))− re−qγ(t), for t > 0 and x ≥ 1. (2.8) (ii) if L ≥ L0 and u0(x) ≤ ϕ(0, x− L) + r for x > 0, then u(t, x) ≤ ϕ(t, x− ct+mL(t)− α(t)) + re−qγ(t), for t > 0 and x ≥ 1, (2.9) where α(t) = ∫ t 0 (λM,L(s) + rCe−qγ(s))ds and C is a large constant. Proof. (i) For t > 0 and x ∈ R, we define u−(t, x) = max{ϕ(t, x− ct+mL(t) + α(t))− re−qγ(t), 0} We are going to show that u−(t, x) is a subsolution of (1.1) for t > 0 and x ≥ 1. Firstly, we check the initial and boundary conditions. For t = 0, one has that u−(0, x) = max{ϕ(0, x− L)− r, 0} ≤ u0(x) for x > 0. For x = 1, one has that u−(t, 1) = max{ϕ(t, 1− ct+mL(t) + α(t))− re−qγ(t), 0} ≤ 1− re−qγ(t) ≤ u(t, 1) for t > 0. Then, we have to show only that Q[u−] := u− t − u− xx − k(x)u− x − f(t, u−) ≤ 0, (2.10) for t > 0 and x ≥ 1 such that u−(t, x) > 0. By (1.8), it follows that Q[u−] = ϕξ(m ′ L(t) + α′(t)− k(x)) + rqγ′(t)e−qγ(t) + f(t, ϕ)− f(t, u−) = ϕξ(k(ct−mL(t))− k(x) + λL,M (t) + rCe−qγ(t)) + rqγ′(t)e−qγ(t) + f(t, ϕ)− f(t, u−) (2.11) where ϕ and ϕξ take values at (t, x− ct+mL(t) + α(t)). For t > 0 and x ≥ 1 such that x− ct+mL(t) + α(t) ≤ −R, one has that 1− ρ 2 ≤ ϕ(t, x− ct+mL(t) + α(t)) < 1 and 1− ρ ≤ u−(t, x) < 1, since r < ρ/2. Then, by (2.3) and the mean value theorem, it follows that f(t, ϕ)− f(t, u−) ≤ −τre−qγ(t). EJDE-2024/69 TIME-PERIODIC REACTION-DIFFUSION EQUATIONS 7 By ϕξ < 0, q ≤ τ/2, γ′(t) ≤ 1 and (2.11), one has that Q[u−] ≤ ϕξ(k(ct−mL(t))− k(x) + λM,L(t))− τ 2 re−qγ(t). (2.12) Let K = sup x>0 |k(x)|. Notice that α(t) is bounded for all t ≥ 0. Take M sufficiently large such that M − α(t) > 0 and 2KC1e −η(M−α(t)) ≤ τr 2 (2.13) where C1 and η are defined by (2.6). Then, if 1 ≤ x ≤ ct−mL(t)− γ(t)−M , one has that x− ct+mL(t) + α(t) ≤ −γ(t)−M + α(t) < 0 and |ϕξ(t, x− ct+mL(t) + α(t))| ≤ C1e −η(γ(t)+M−α(t)) ≤ τr 4K e−ηγ(t). In this case, it follows from (2.12) that Q[u−] ≤ −2Kϕξ − τ 2 re−qγ(t) ≤ 0, by λM,L(t) ≥ 0 and q ≤ η. If x ≥ 1 and x ≥ ct−mL(t)− γ(t)−M , it follows from x− ct+mL(t) + α(t) ≤ −R and (2.2) that k(ct−mL(t))− k(x) + λM,L(t) ≥ 0. Thus, in this case, Q[u−] ≤ 0 by (2.12). For t > 0 and x ≥ 1 such that −R ≤ x − ct +mL(t) + α(t) ≤ R. One has that −ϕξ ≥ a. Moreover, by (2.2), k(ct−mL(t))− k(x) + λM,L(t) ≥ 0. Then, it follows from (2.11) that Q[u−] ≤ −arCe−qγ(t) + rqe−qγ(t) + ∥fu(t, u)∥L∞re−qγ(t) ≤ 0, by taking C sufficiently large. For t > 0 and x ≥ 1 such that x− ct+mL(t) + α(t) ≥ R, one has that 0 < ϕ(t, x− ct+mL(t) + α(t)) ≤ ρ 2 and 0 ≤ u−(t, x) ≤ ρ. Then, by (2.3), it follows that f(t, ϕ)− f(t, u−) ≤ −τre−qγ(t). By ϕξ < 0, q ≤ τ 2 , γ ′(t) ≤ 1 and (2.11), one has that Q[u−] ≤ ϕξ(k(ct−mL(t))− k(x) + λM,L(t))− τ 2 re−qγ(t). If x ≥ ct−mL(t) + γ(t) +M , one has that x− ct+mL(t) + α(t) ≥ γ(t) +M + α(t) > 0, and |ϕξ(t, x− ct+mL(t) + α(t))| ≤ C1e −η(γ(t)+M+α(t)) ≤ τr 4K e−ηγ(t), by (2.13). Then, Q[u−] ≤ 0 since λM,L(t) ≥ 0 and q ≤ η. If x ≤ ct−mL(t)+γ(t)+ M , it follows from x− ct+mL(t) + α(t) ≥ R and (2.2) that k(ct−mL(t))− k(x) + λM,L(t) ≥ 0. Then Q[u−] ≤ 0. 8 L. LI, Z. CHEN EJDE-2024/69 This completes the proof of (2.10). Then, (2.8) follows from the comparison principle. (ii) The proof is almost parallel to the proof of (i). For t > 0 and x ∈ R, we define u+(t, x) = min{ϕ(t, x− ct+mL(t)− α(t)) + re−qγ(t), 1} Let us check that u+(t, x) is a supersolution of (1.1) for t > 0 and x ≥ 1. We first verify the initial condition. For t = 0, one has that u+(0, x) = min{ϕ(0, x− L) + r, 1} ≥ u0(x), for x > 0. Since ct − mL(t) = O(t) → +∞ and γ = o(t) as t → +∞, it follows from (2.12) and q ≤ η that ϕ(t, 1− ct+mL(t)− α(t)) + re−qγ(t) ≥ 1, for t > 0, even if it means increasing L and hence, u+(t, x) = 1 ≥ u0(1) for t > 0. Then, we have to check only that Q[u+] := u+ t − u+ xx − k(x)u+ x − f(t, u+) ≥ 0, (2.14) for t > 0 and x ≥ 1 such that u+(t, x) < 1. By (1.8), it follows that Q[u+] = (m′ L(t)− α′(t)− k(x))ϕξ − rqγ′(t)e−qγ(t) + f(t, ϕ)− f(t, u+) = ϕξ(k(ct−mL(t))− k(x)− λL,M (t)− rCe−qγ(t))− rqγ′(t)e−qγ(t) + f(t, ϕ)− f(t, u+) where ϕ and ϕξ take values at (t, x− ct+mL(t)− α(t)). For t > 0, such that x− ct+mL(t)− α(t) ≤ −R, one has that 1− ρ 2 ≤ ϕ(t, x− ct+mL(t)− α(t)) < 1 and 1− ρ ≤ u+(t, x) < 1, since r < ρ/2. Then, by (2.3), it follows that f(t, ϕ)− f(t, u+) ≥ τre−qγ(t). By ϕξ < 0, q ≤ τ/2 and γ′(t) ≤ 1, one has that Q[u+] ≥ ϕξ(k(ct−mL(t))− k(x)− λL,M (t)) + τ 2 re−qγ(t). If x ≤ ct−mL(t)−γ(t)−M , one has that x−ct+mL(t)−α(t) ≤ −γ(t)−M −α(t) and |ϕξ(t, x− ct+mL(t)− α(t))| ≤ C1e −η(γ(t)+M+α(t)) ≤ τr 4K e−ηγ(t), by (2.13). In this case, Q[u+] ≥ −2Kϕξ − τ 2 re−qγ(t) ≥ 0, by λM,L(t) ≥ 0 and q ≤ η. If x ≥ ct −mL(t) − γ(t) −M , it follows from x − ct + mL(t)− α(t) ≤ −R and (2.2) that k(ct−mL(t))− k(x)− λM,L(t) ≤ 0. Thus, Q[u+] ≥ 0. For t > 0 such that −R ≤ x − ct +mL(t) − α(t) ≤ R. One has that −ϕξ ≥ a. Moreover, by (2.2), k(ct−mL(t))− k(x)− λM,L(t) ≤ 0. EJDE-2024/69 TIME-PERIODIC REACTION-DIFFUSION EQUATIONS 9 Then, Q[u+] ≥ arCe−qγ(t) − rqe−qγ(t) − ∥fu(t, u)∥L∞re−qγ(t) ≥ 0, by taking C sufficiently large. For t > 0 such that x− ct+mL(t)− α(t) ≥ R, one has that 0 < ϕ(t, x− ct+mL(t)− α(t)) ≤ ρ and 0 ≤ u+(t, x) ≤ ρ. Then, by (2.3), it follows that f(t, ϕ)− f(t, u−) ≤ −τre−qγ(t). By ϕξ < 0, q ≤ τ/2 and γ′(t) ≤ 1, one has that Q[u+] ≤ ϕξ(k(ct−mL(t))− k(x)− λM,L(t)) + τ 2 re−qγ(t). If x ≥ ct−mL(t) + γ(t) +M , one has that x− ct+mL(t)− α(t) ≥ γ(t) +M − α(t) > 0, and |ϕξ(t, x− ct+mL(t)− α(t))| ≤ C1e −η(γ(t)+M−α(t)) ≤ τr 4K e−ηγ(t). by (2.13). Then, Q[u+] ≥ 0 since λM,L(t) ≥ 0 and q ≤ η. If x ≤ ct−mL(t)+γ(t)+ M , it follows from x− ct+mL(t)− α(t) ≥ R and (2.2) that k(ct−mL(t))− k(x)− λM,L(t) ≤ 0. Then, Q[u+] ≥ 0. This completes the proof of (2.14). Then, (2.9) follows from the comparison principle. □ We then show that the solution u(t, x) of (1.1) satisfies the initial and boundary conditions in Lemma 2.1 after some time T . Lemma 2.2. Let u(t, x) be the solution of (1.1) and assume that u(t, x) grows in the sense of (1.11). For any 0 < c1 < c, there is a positive constant δ such that sup 0 0. Since k(x) → 0 as x → +∞, one can pick L > 0 large enough such that α := c− c′ − sup x>L k(x) > 0. Take r ∈ (0, ρ) such that r ≤ min { aα ∥fu(t, u)∥L∞ + τ , C1 } , (2.16) where ρ, τ and a are defined by (2.3) and (2.5) respectively. Take β ∈ (0, η) where η is defined by (2.6) such that β2 + β sup x>L |k(x)| ≤ τ. For t > 0 and x ∈ R, define u−(t, x) := max{ϕ(t, x− c′t)− re−βx, 0}. 10 L. LI, Z. CHEN EJDE-2024/69 We are going to show that u−(t, x) is a subsolution of (1.1). Since ϕ(0, x) ≤ C1e −ηx for x > 0 and by (1.11) and u grows, there is T1 > 0 such that u(T1, x) ≥ 0 ≥ ϕ(0, x)− re−βx, for x ≥ 0, and u(t, 0) ≥ 1− r, for t ≥ T1. Thus, u(T, x) ≥ u−(0, x) for x ≥ 0 and u(t, 0) ≥ u−(t, 0) for t ≥ T1. Then, we only have to show that Q[u−] := (u−)t − (u−)xx − k(x)(u−)x − f(t, u−) ≤ 0, for t ≥ T1 and x ≥ 0 such that u− > 0. By (1.8), it follows that Q[u−] = (c− c′ − k(x))ϕξ + (rβk(x)− rβ2)e−βx + f(t, ϕ)− f(t, u−) ≤ αϕξ + τre−βx + f(t, ϕ)− f(t, u−), where ϕ and ϕξ take values at (t, x − c′t). Let R be defined by (2.4). Then, for t ≥ T1 and x ≥ 0 such that x− c′t ≤ −R and x− c′t ≥ R respectively, one has that 1 − ρ ≤ u−(t, x) ≤ 1 and 0 ≤ u−(t, x) ≤ ρ respectively. Thus, by the mean value theorem and (2.3), f(t, ϕ)− f(t, u−) ≤ −τre−βx. Since ϕξ < 0, it follows that Q[u−] ≤ 0. For t ≥ T1 and x ≥ 0 such that −R ≤ x− c′t ≤ R, one has that −ϕξ ≥ a. Then, by (2.16), Q[u−] ≤ −aα+ (τ + ∥fu(t, u)∥L∞)re−βx ≤ 0. Consequently, from the comparison principle it follows that u(t+ T1, x) ≥ u−(t, x) ≥ ϕ(t, x− c′t)− re−βx, for t ≥ 0 and x ≥ 0. Therefore, (2.15) holds. This completes the proof. □ Lemma 2.3. Let u(t, x) be the solution of (1.1). If lim supx→+∞ u0(x) < ρ, then lim supx→+∞ u(t, x) converges to 0, as t → +∞. Proof. Let r := lim supx→+∞ u0(x) < ρ. Let v(t, x) be the solution of vt = vxx + k(x)vx + f(t, v), for t > 0 and x ∈ R, and v(0, x) = { 1, if x ≤ 0, r, if x > 0. Let Φ(t) be the solution of Φ′(t) = f(t,Φ) with Φ(0) = r. Then Φ(t) → 0 as t → +∞ by (2.3). It follows from the maximum principle and standard parabolic estimates that v(t,+∞) = Φ(t). Since lim supx→+∞ u0(x) = r and 0 ≤ u0(x) ≤ 1, there is L > 0 such that u0(x) ≤ v(0, x−L) and hence, u(t, x) ≤ v(t, x−L). Then the conclusion follows. □ Proof of Theorem 1.1. Let u(t, x) be the solution of (1.1). Fix r ∈ (0, ρ). Then, by Lemma 2.3, there is T1 = k1T, k1 ∈ Z, where T is the periodic such that lim supx→+∞ u(T1, x) ≤ r 2 . Since ϕ(t,−∞) = 1 and ϕ(t,+∞) = 0, there is L > 0 such that u(T1, x) ≤ ϕ(0, x− L) + r for x ≥ 0. EJDE-2024/69 TIME-PERIODIC REACTION-DIFFUSION EQUATIONS 11 By Lemma 2.1, one has that u(T1 + t, x) ≤ ϕ(t, x− ct+mL(t)− α(t)) + re−qγ(t), for t > 0 and x > 0 Since u(t, x) grows and γ(t) = o(t) as t → +∞, by Lemma 2.2 there is T2 = k2T > 0, k2 ∈ Z such that u(T2, x) ≥ ϕ(0, x− L)− r, for x ≥ 0, and u(t, 1) ≥ 1− re−qγ(t), for t ≥ T2. Then, by Lemma 2.1, one has that u(T2 + t, x) ≥ ϕ(t, x− ct+mL(t) + α(t))− re−qγ(t), for t > 0 and x ≥ 1. Since limt→+∞ α(t) < +∞ and mL(t) satisfies (2.1), there exist constants α1 and α2 such that ϕ(t, x− ct+mL(t) + α1)− re−qγ(t−T2) ≤ u(t, x) ≤ ϕ(t, x− ct+mL(t) + α2) + re−qγ(t−T1), (2.17) for t ≥ max{T1, T2} and x ≥ 1. Now, take a sequence {tn := nT}n∈N such that tn → +∞ as n → +∞. Let un(t, x) = u(t+ tn, x+ ctn −mL(tn)). Then by 2.17, one has that ϕ(t, x− ct+mL(t+ tn) + α1 −mL(tn))− re−qγ(t+tn−T2) ≤ un(t, x) ≤ ϕ(t, x− ct+mL(t+ tn)−mL(tn) + α2) + re−qγ(t+tn−T1), (2.18) for t ≥ −tn+max{T1+T2} and x ≥ 1−ctn+mL(tn). Since mL(t) satisfies (2.1), it follows that mL(t+ tn)−mL(tn) → 0 as tn → +∞ locally uniformly for t ∈ R. By parabolic estimates, the sequence un(t, x) converges to an entire solution u∞(t, x) locally uniformly for (t, x) ∈ R× R of the equation (u∞)t = (u∞)xx + f(t, u∞), t ∈ R, x ∈ R. By (2.18), one also has that ϕ(t, x− ct+ α1) ≤ u∞(t, x) ≤ ϕ(t, x− ct+ α2), for t ∈ R, x ∈ R. By the stability result of ϕ(t, x− ct) in [1], there is x0 ∈ R such that u∞(t, x) ≡ ϕ(t, x− ct+ x0). Thus, for any r > 0, there is N > 0 such that |u(tN , x+ ctN −mL(tN ))− ϕ(0, x+ x0)| < r, that is, ϕ(tN , x− ctN +mL(tN ) + x0)− r ≤ u(tN , x) ≤ ϕ(tN , x− ctN +mL(tN ) + x0) + r Again by lemma 2.1, one has that ϕ(t, x− ct+mL(t) + x0 + αN (t))− re−qγ(t−tN ) ≤ u(t, x) ≤ ϕ(t, x− ct+mL(t) + x0 − αN (t)) + re−qγ(t−tN ) (2.19) 12 L. LI, Z. CHEN EJDE-2024/69 for t > tN and x ≥ 1, where αN (t) = ∫ t tN (λM,L(s) + rCe−qγ(s))ds. For tN large enough, αN (t) can be arbitrary small. Since r is arbitrary small and |ϕξ| is bounded. One has that sup x≥1 |u(t, x)− ϕ(t, x− ct+mL(t) + x0)| is arbitrary small for t > tN by taking tN large enough. Therefore, sup x≥1 |u(t, x)− ϕ(t, x− ct+mL(t) + x0)| → 0, as t → +∞. This completes the proof. □ 3. Applications In this section, we apply our results to high-dimensional domains. More precisely, we show asymptotic speeds for solutions of initial value problems in two kinds of high-dimensional domains: the whole space RN and exterior domains. For convenience, we modify (1.1) a little bit. Let u(t, r) be the solution of ut = urr + N − 1 r ur + f(t, u), t > 0, r > 0 ur(t, 0) = 0. (3.1) with compactly supported initial value u0(x). Then, by the same proof as of The- orem 1.1, if u(t, r) grows, then sup r>1 |u(t, r)− ϕ(t, r − ct+m(t) + x0)| → 0, as t → +∞, (3.2) for some constants x0, where m(t) is the solution of m′(t) = (N − 1) \ (ct−m(t)) and has the form m(t) = N − 1 c ln t+O(1). (3.3) Proof of Theorem 1.3. Since v0(x) is compactly supported, there is R1 > 0 such that v0(x) = 0 for |x| ≥ R1. Let u1(t, r) be the solution of (3.1) with u1(0, r) = 1 for 0 < r ≤ R1. Then, obviously v0(x) ≤ u1(0, |x|) for x ∈ RN and u1(t, |x|) satisfies vt = ∆v + f(t, v) for t > 0 and x ∈ RN . It follows from the comparison principle that v(t, x) ≤ u1(t, |x|), for t > 0 and x ∈ RN . Since (3.2) holds for u1(t, r) and m(t) satisfies (3.3), we have (1.16). On the other hand, since v(t, x) grows, there are T > 0 and R2 > 0 such that v(T, x) ≥ 1 − ϵ for |x| ≤ R2 where ϵ > 0 is a small constant. One can take T sufficiently large such that ϵ is a sufficiently small constant and R2 is a sufficiently large constant. Let u2(t, x) be the solution of (3.1) with u2(0, r) = 1 − ϵ for 0 < r ≤ R2 and u2(0, r) = 0 for r > R2. By v0(x) ≥ u2(0, |x|) and the comparison principle, one has that v(t+ T, x) ≥ u2(t, |x|), for t > 0 and x ∈ RN . Since (3.2) holds for u2(t, x) and m(t) satisfies (3.3), one immediately has (1.15). This completes the proof. □ EJDE-2024/69 TIME-PERIODIC REACTION-DIFFUSION EQUATIONS 13 Proof of Theorem 1.4. By lemma 2.2, there is a δ > 0 such that sup 0 0 such that v0(x) = 0 for x ∈ Ω such that |x| ≥ R1 and K ⊂ B(0, R1). Let u1(t, r) be the solution of (3.1) with u1(0, r) = 1 for 0 < r ≤ R1 and u1(0, r) = 0 for r > R1. We define u1(t, x) = u1(t, |x|) + βξ(x)e−δt, where β > 0 is to be given. One can calculate that ∂νu1(t, x) = (u1)r x |x| · ν + β∇ξ · νe−δt for x ∈ ∂Ω. Since K ⊂ B(0, R1) and (u1)r = o(e−δt) for x ∈ ∂Ω as t → +∞, one can pick β > 0 such that ∂νu1(t, x) > 0 for x ∈ ∂Ω. Then, we show that u1(t, x) is a supersolution of (1.14). Obviously, v0(x) ≤ u1(0, x) for x ∈ Ω. Assume that ξ(x) ≡ 0 in RN \ B(0, L) for some L > 0. Assume without loss of generality that u1(t, |x|) ≥ 1−ρ for x ∈ B(0, L) and t > 0. Otherwise, one can consider u1(t+T, |x|) for large T > 0 by (3.4). Then, one only has to check that (u1)t −∆u1 − f(t, u1) ≥ 0, for t > 0 and x ∈ Ω ∩B(0, L). One can compute that (u1)t −∆u1 − f(t, u1) = −βδξ(x)e−δt − β∆ξe−δt + f(t, u1(t, |x|))− f(t, u1) ≥ −βδξ(x)e−δt − β∆ξe−δt − τβξ(x)e−δt ≥ 0, for x ∈ Ω ∩B(0, L). Then, by the comparison principle, one has that v(t, x) ≤ u1(t, x), for t > 0 and x ∈ Ω. By (3.2), one has (1.18). On the other hand, since v(t, x) grows, there are T > 0 and R2 > 0 such that v(T, x) ≥ 1− ϵ for x ∈ Ω ∩B(0, R2). One can take T sufficiently large such that ϵ is small enough and R2 is large enough such that K ⊂ B(0, R2). Let u2(t, r) be the solution of (3.1) with u2(0, r) = 1− ϵ for 0 < r ≤ R2 and u2(0, r) = 0 for r > R2. We define u2(t, x) = u2(t, x)− βξ(x)e−δt Similar as above arguments, one can show that u2(t, x) is a subsolution of (1.14). By the comparison principle, one has that v(t, x) ≥ u2(t, x), for t > 0 and x ∈ Ω. By (3.1), one has (1.17). This completes the proof. □ Acknowledgments. Linlin Li was supported by the NSF of China under Grant 12301202. 14 L. LI, Z. CHEN EJDE-2024/69 References [1] N. Alikakos, P. W. Bates, X. Chen; Periocid traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805. [2] D. G. Aronson, H. F. Weinberger; Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33–76. [3] H. Berestycki, F. Hamel, H. Matano; Bistable travelling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788. [4] H. Berestycki, G. Nadin; Asymptotic spreading for general heterogeneous Fisher-KPP type equations, Memoirs Amer. Math. Soc., 280 (2022): 1381. [5] P. C. Fife, J. B. McLeod; The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [6] H. Guo, H. Monobe; V -shaped fronts around an obstacle, Math. Ann., 379 (2021), 661–689. [7] C. K. R. T. Jones; Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364. [8] L. Li; Time-periodic planar fronts around an obstacle, J. Nonlinear Sci., 31 (2021), 90. [9] L. Rossi; The Freidlin-Gärtner formula for general reaction terms, Adv. Math., 317 (2017), 267-298. [10] K. Uchiyama; Asymptotic behavior of solutions of reaction-diffusion equations with carying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311. Linlin Li College of Science, University of Shanghai for Science and Technology, Shanghai, China Email address: lilinlin@usst.edu.cn Zhuo Chen College of Science, University of Shanghai for Science and Technology, Shanghai, China Email address: imchenz@163.com 1. Introduction 2. Asymptotic behavior of solutions 3. Applications Acknowledgments References