Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 48, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.48 EXISTENCE AND STABILITY OF FORCED WAVES FOR p-LAPLACE EQUATIONS IN A SHIFTING HABITAT TIANYUAN XU, GEGE LIU, JINGXUE YIN Abstract. This article concerns the existence and stability of forced waves for p-Laplace equa- tions with a shifting habitat given by a non-decreasing function with a sign change. The existence of forced waves is studied by the monotone iteration method combined with a pair of delicate super- and sub-solutions. Finally, we develop an approximating weighted energy method to prove the Lp stability and exponential convergence of forced waves. 1. Introduction The threats associated with the global warming and the worsening of the environment resulting from industrialization cause shifting of habitat ranges of the species, for instance, the migration of New Guinean birds[8, 19]. In this article, we are interested in the following p-Laplace diffusion equation with a shifting habitat ut = D(|ux|p−2ux)x + u(t, x)[r(x− ct)− u(t, x)], t > 0, x ∈ R. (1.1) Here u(t, x) stands for specific population density, c > 0 represents the shifting speed of the edge of the habitat and D(|ux|p−2ux)x with p > 2 is the p-Laplace type diffusion, i.e., the gradient- dependent diffusion. In particular, equation (1.1) is degenerate at the points where ux = 0. As usual, we assume throughout this paper that the resource function r(ξ) is continuous and nondecreasing, r(±∞) are finite, negative at −∞ and positive at +∞. Depending on the sign- changing property of r(·), the shifting habitats can be divided into the viable habitat {x ∈ R : r(x − ct) > 0} and the hostile habitat {x ∈ R : r(x − ct) ≤ 0}. These assumptions imply the scenario that the favourable habitat of species is rightwards shrinking with a speed c > 0. In recent years, significant efforts have been made in the study of dispersal phenomenon with a shifting habitat, see [3, 4, 5, 6, 7, 16, 20, 21, 23, 24] and the references therein. Berestycki et al. [3] proposed a mathematical model that involves a reaction diffusion equation ut = Duxx + f(x− ct, u), t > 0, x ∈ R and showed that the existence and uniqueness of the forced waves and the large time behavior of solutions for the corresponding initial value problem. For the related results in different type domains, we refer the readers to the extend works of Berestycki et al. [4, 5]. When f satisfies a sublinearity condition, Berestycki and Fang [6] considered existence and multiplicity of the forced waves and the attractivity of these waves except for some critical cases. Later then, Fang et al.[7] showed that the existence, uniqueness and stability of forced time periodic waves. Recently, the authors [16] studied the model with density-dependent diffusion and obtained the existence of forced waves and the stability of these waves in the weighted L1-space. Very recently, Wang et al.[23] established the existence, uniqueness and stability of forced pulsating waves for the competition system in time-periodic media. Shen and Xue [20, 21] investigated the existence, persistence as well as spreading speed properties of forced waves for Keller-Segel models in shifting 2020 Mathematics Subject Classification. 35K65, 35K57, 35C07, 92D25. Key words and phrases. Forced waves; p-Laplacian diffusion; shifting habitats; stability. ©2025. This work is licensed under a CC BY 4.0 license. Submitted December 1, 2024. Published May 8, 2025. 1 2 T. XU, G. LIU, J. YIN EJDE-2025/48 environments. For an overview of other related models with a shifting habitat, we refer to [22] and the references therein. Gradient-dependent diffusion plays a crucial role in dynamics of biological groups, for example the formation of spatial aggregates of animals [18]. The reaction diffusion equations with gradient- dependent dispersal have been investigated in a series of works [1, 2, 9, 10, 11, 14, 15]. Jin and Yin [15] proved the existence and asymptotic behavior of traveling wave solutions for the evolutionary p-Laplacian equation with time delay. In [2], the authors studied the asymptotic behaviour of solutions to the p-Laplacian diffusion equation subjecting to homogeneous Dirichlet boundary conditions in a tubular domain. More recently, Huang et al. [14] established the existence and stability of traveling wave solutions for the Nicholson’s blowflies equation with gradient-dependent diffusion and time delay. Recently there has been a considerable progress in the study of stability of traveling waves for the degenerate diffusion equations (see for instance [12, 13, 14, 16, 17]). By the weighted energy method, Huang and his coauthors investigated the global stability and exponential convergence rate of traveling wave solutions for a series of degenerate diffusion equations [12, 13, 14]. More recently, Liu et al. [17] obtained the stability of traveling waves in the weighted L1-space for the nonlocal degenerate diffusion equation with time delay. The purpose of this paper is to prove the existence and stability of forced waves for the p- Laplace diffusion equation with a shifting habitat. We find that p-Laplace type diffusion and a shifting habitat cause many difficulties in the study of (1.1). For the existence results, we need to determine a pair of super- and sub-solutions to obtain the forced waves. Because the peculiar structure of the p-Laplacian type diffusion term, we construct an approximation of the weighted function to obtain the weighted Lp w-regularity of the perturbed solution. We define an approximating weighted function wk(ξ) by wk(ξ) =  e−λk, ξ < −k, eλξ, |ξ| ≤ k, eλk, ξ > k, where λ > 0 and k > 0. Finally, by the approximating weighted energy method, we prove the Lp stability and exponential convergence of forced waves. The rest of this article is organized as follows. In Section 2, we present the main results. In Section 3, we state the proof of the existence of forced traveling waves with the wave speed c > 0 and the global stability of these waves by the approximating weighted energy method. 2. Main results In this article, we consider the monotone increasing positive traveling wave solution u(x, t) = ϕ(ξ) with ξ = x− ct of (1.1), where c is the same as the habitat shifting speed, i.e., ϕ(ξ) satisfies D(|ϕ′(ξ)|p−2ϕ′(ξ))′ + cϕ′(ξ) + ϕ(ξ)(r(ξ)− ϕ(ξ)) = 0. (2.1) A continuous function ϕ(ξ) is called a forced wave if ϕ(ξ) ∈ W 1,p loc (R) satisfies (2.1) in the sense of distributions, ϕ(−∞) = 0 and ϕ(+∞) = r(+∞). Here, we give the definition of the spaces of Lp w(R) and L∞([0, T ];Lp w(R)). Lp w(R) denotes the weighted Lp-space with the weight function w(ξ) and norm ∥v∥Lp w(R) = (∫ R |v(ξ)|pw(ξ)dξ )1/p . Then L∞([0, T ];Lp w(R)) is equipped with the norm ∥v∥L∞([0,T ];Lp w(R)) = sup t∈[0,T ] ∥v(t)∥Lp w(R). For the next results, we will take p > 2, corresponding to the case of degenerate diffusion. Our main results are stated as follows. Theorem 2.1. The forced waves of (2.1) connecting 0 and r(+∞) exist for any given c > 0. EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 3 Theorem 2.2. For each p > 2, D > 0, let ξ := x − ct and u(t, ξ) := u(t, ξ + ct) be a solution of the Cauchy problem of (1.1). Assume that ϕ(ξ) is a forced wave with a speed c > 2 √ r(+∞)Dp2ρ where ρ = max{∥(uξ) p−2∥L∞(R), ∥(ϕξ) p−2∥L∞(R)}, ∥(uξ) p−2∥L∞(R) is derived in Lemma 3.6. Let v(t, ξ) := u(t, ξ)−ϕ(ξ) and w(ξ) = eλξ be the weight function with λ ∈ (λ1, λ2), where 0 < λ1 < λ2 are the roots of −pr(+∞) + cλ−Dpρλ2 = 0. Suppose that the initial perturbation around the forced wave satisfies v0(ξ) ∈ Lp(R)∩Lp w(R), then v(t, ξ) ∈ L∞([0, T ];Lp(R) ∩ Lp w(R)) for any T > 0 and for some sufficiently small ζ > 0 ∥v(t, ·)∥Lp(R) ≤ C(∥v0∥Lp(R) + ∥v0∥Lp w(R))e −ζt. 3. Proof of the main results The existence of travelling wave solutions for the problem (2.1) is proved by the monotone iteration method combined with a pair of super- and sub-solutions. For simplicity, we denote r(+∞) = r̄ and κ = r̄ − r(−∞) > 0. Here we firstly give the definition of the super- and sub-solutions of (2.1). Definition 3.1. A continuous function ϕ : R → [0, r̄] with ϕ(ξ) ∈ W 1,p loc (R) is said to be a bounded positive sub- (or super-, respectively) solution of (2.1) if ϕ(−∞) = 0, ϕ(+∞) ≤ (≥)r̄ and ϕ(ξ) satisfies F [ϕ] := −cϕ′(ξ)−D(|ϕ′(ξ)|p−2ϕ′(ξ))′ − r(ξ)ϕ(ξ) + ϕ2(ξ) ≤ (≥)0 in the sense of distributions. Lemma 3.2 (Comparison principle). For i = 1, 2, we assume that ϕi ∈ C(R) ∩ W 1,p loc (R), 0 ≤ ϕi ≤ r̄, lim infξ→±∞(ϕ1(ξ)− ϕ2(ξ)) ≥ 0 and ϕi holds the differential inequality − cϕ′ 1(ξ)−D(|ϕ′ 1(ξ)|p−2ϕ′ 1(ξ)) ′ + ϕ2 1(ξ) + κϕ1(ξ) ≥ −cϕ′ 2(ξ)−D(|ϕ′ 2(ξ)|p−2ϕ′ 2(ξ)) ′ + ϕ2 2(ξ) + κϕ2(ξ) in the sense of distributions. Then ϕ1(ξ) ≥ ϕ2(ξ) for all ξ ∈ R. Proof. Taking φ = (ϕ2(ξ)− ϕ1(ξ))+ ∈ W 1,p 0 (R) as a test function, we obtain 0 ≥ ∫ +∞ −∞ D(|ϕ′ 2(ξ)|p−2ϕ′ 2(ξ)− |ϕ′ 1(ξ)|p−2ϕ′ 1(ξ))(ϕ2(ξ)− ϕ1(ξ)) ′ +dξ − c ∫ +∞ −∞ (ϕ2(ξ)− ϕ1(ξ)) ′(ϕ2(ξ)− ϕ1(ξ))+dξ + ∫ +∞ −∞ (ϕ2 2(ξ)− ϕ2 1(ξ))(ϕ2(ξ)− ϕ1(ξ))+dξ + ∫ +∞ −∞ κ(ϕ2(ξ)− ϕ1(ξ))(ϕ2(ξ)− ϕ1(ξ))+dξ ≥ D ∫ {ϕ2>ϕ1} (|ϕ′ 2(ξ)|p−2ϕ′ 2(ξ)− |ϕ′ 1(ξ)|p−2ϕ′ 1(ξ))(ϕ ′ 2(ξ)− ϕ′ 1(ξ))dξ − c 2 ∫ +∞ −∞ ((ϕ2 − ϕ1) 2 +) ′dξ + ∫ {ϕ2>ϕ1} (ϕ2(ξ) + ϕ1(ξ) + κ)(ϕ2(ξ)− ϕ1(ξ))dξ = I1 + I2 + I3. The estimate I1 ≥ 0 follows from a basic inequality. We note that lim infξ→±∞(ϕ1(ξ)−ϕ2(ξ)) ≥ 0, which implies that I2 = 0. Therefore ϕ1(ξ) ≥ ϕ2(ξ) for all ξ ∈ R. □ Next we consider the solvability of the degenerate elliptic problem. 4 T. XU, G. LIU, J. YIN EJDE-2025/48 Lemma 3.3. Let 0 < ϕ ≤ r̄ be a nondecreasing super-solution of (2.1), ϕ(−∞) = 0 and ϕ(+∞) = r̄. Then the degenerate elliptic problem −cϕ′(ξ)−D(|ϕ′(ξ)|p−2ϕ′(ξ))′ + ϕ2(ξ) + κϕ(ξ) = (r(ξ) + κ)ϕ(ξ), lim ξ→−∞ ϕ(ξ) = 0, lim ξ→+∞ ϕ(ξ) = r̄, (3.1) has a nondecreasing solution ϕ(ξ) ∈ W 1,p loc (R) satisfying 0 < ϕ(ξ) ≤ ϕ(ξ) for all ξ ∈ R. Moreover, ϕ(ξ) is also a super-solution of (2.1) and ϕ(ξ) ∈ C1+β(R) for some β ∈ (0, p−1 p ). Proof. For each K > 1, we focus on the solutions of the regularized problem −cϕ′(ξ) = D ( (|ϕ′(ξ)|2 + 1/K)(p−2)/2ϕ′(ξ) )′ − ϕ2(ξ)− κϕ(ξ) + f(ξ), ξ ∈ (−K,K), ϕ(−K) = λ(−K), ϕ(K) = λ(K), (3.2) where f(ξ) := (r(ξ) + κ)ϕ(ξ), λ(−K) = −κ+ √ κ2+4f(−K) 2 and λ(K) = −κ+ √ κ2+4f(K) 2 . In view of the standard quasi-linear parabolic theory, we immediately obtain that (3.2) admits a unique solution ϕK . Let us estimate ϕK and the gradient of ϕK respectively. We assert that 0 < λ(−K) ≤ ϕK(ξ) ≤ λ(K) < r̄. Otherwise, there exists ξ̃ ∈ (−K,K) such that ϕ(ξ̃) > λ(K). Suppose that the maximum of ϕK is attained at ξ0 which is a interior point of (−K,K). It should be note that ϕK(ξ0) > ϕK(K), ϕ′ K(ξ0) = 0 and ϕ′′ K(ξ0) ≤ 0. Then a direct computation yields f(ξ0) ≥ ϕ2 K(ξ0) + κϕK(ξ0) > ϕ2 K(K) + κϕK(K) = f(K). (3.3) Noting that r(ξ) and ϕ(ξ) are nondecreasing, we can deduce f(ξ) is also nondecreasing, which contradicts to (3.3). Hence ϕK(ξ) ≤ λ(K) for any ξ ∈ [−K,K]. Similarly, we can show that ϕK(ξ) ≥ λ(−K). By the above proof, we know that ϕK takes the maximum and minimum value at K and −K, which implies that ϕ′ K(−K) ≥ 0 and ϕ′ K(K) ≥ 0. We claim that ϕ′ K(ξ) ≥ 0 for any ξ ∈ (−K,K). In fact, if this is not true, we can find ξ1 ∈ (−K,K) such that ϕ′ K(ξ1) < 0. Assume that (ξ2, ξ3) is the maximum interval such that ξ1 ∈ (ξ2, ξ3) and each of ϕ′ K(ξ) is negative for ξ ∈ (ξ2, ξ3). It is easy to see that ϕK(ξ2) > ϕK(ξ3), ϕ ′ K(ξ2) = ϕ′ K(ξ3) = 0, and ((|ϕ′ K(ξ)|2 + 1/K)(p−2)/2ϕ′(ξ))′|ξ=ξ2 ≤ 0, ((|ϕ′ K(ξ)|2 + 1/K)(p−2)/2ϕ′(ξ))′|ξ=ξ3 ≥ 0. Therefore − (|ϕ′ K(ξ2)|2 + 1/K)(p−2)/2ϕ′(ξ2)) ′ − cϕ′ K(ξ2) + ϕ2 K(ξ2) + κϕK(ξ2) > −(|ϕ′ K(ξ3)|2 + 1/K)(p−2)/2ϕ′(ξ3)) ′ − cϕ′ K(ξ3) + ϕ2 K(ξ3) + κϕK(ξ3), which implies that f(ξ2) > f(ξ3) for ξ2 < ξ3. This contradicts to the nondecreasing of f . In what follows, for simplicity, we omit the symbol of absolute value in |ϕ′(ξ)|. For 1 < M < K, suppose that α(ξ) ∈ C2 0 (−M,M) is a cut-off function such that α(ξ) ∈ [0, 1], |α′(ξ)| < 2 for any ξ ∈ (−M,M) and α(ξ) = 1 for ξ ∈ (−M + 1,M − 1). Multiplying (3.2) by αp(ξ)ϕK(ξ) and integrating over (−K,K), we obtain∫ K −K Dαp((ϕ′ K(ξ))2 + 1/K)(p−2)/2(ϕ′ K(ξ))2dξ + ∫ K −K αpϕ3 K(ξ)dξ + ∫ K −K καpϕ2 K(ξ)dξ ≤ − ∫ K −K pDαp−1((ϕ′ K(ξ))2 + 1/K)(p−2)/2ϕ′ K(ξ)ϕK(ξ)α′(ξ)dξ + ∫ K −K cαp(ξ)ϕK(ξ)ϕ′ K(ξ)dξ + ∫ K −K αpϕK(ξ)f(ξ)dξ ≤ C ∫ K −K pDαp−1(ϕ′ K(ξ))p−1ϕK(ξ)|α′(ξ)|dξ + ∫ K −K pDαp−1ϕ′ K(ξ)ϕK(ξ)|α′(ξ)|dξ + ∫ K −K cαpϕK(ξ)ϕ′ K(ξ)dξ + ∫ K −K αpϕK(ξ)f(ξ)dξ EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 5 ≤ 1 4 ∫ K −K Dαp(ϕ′ K(ξ))pdξ + C ∫ K −K Dppϕp K(ξ)|α′(ξ)|pdξ + 1 4 ∫ K −K Dαp(ϕ′ K(ξ))pdξ + C ∫ K −K Dαp−2ϕ p p−1 K (ξ)dξ + ∫ K −K αpϕK(ξ)f(ξ)dξ. Since ((ϕ′ K(ξ))2 + 1/K)(p−2)/2(ϕ′ K(ξ))2 ≥ (ϕ′ K(ξ))p for p > 2, we see that D 2 ∫ M−1 −M+1 (ϕ′ K(ξ))pdξ + ∫ M−1 −M+1 ϕ3 K(ξ)dξ + ∫ M−1 −M+1 κϕ2 K(ξ)dξ ≤ C ∫ −M+1 −M + ∫ M M−1 Dppϕp K(ξ)|α′(ξ)|pdξ + CMr̄p/(p−1) + 4Mκr̄ ≤ CDpp2p+1r̄p + 2CMr̄p/(p−1) + 4Mκr̄, which implies that ∥ϕK∥W 1,p(−M+1,M−1) ≤ C, where C is independent of K. By Sobolev embedding theory, we get that for q ∈ (0, p−1 p ), W 1,p(−M + 1,M − 1) is compactly embedded in Cq(−M + 1,M − 1) for any 1 < M < K. Then we can take a subsequence of ϕK which is still denoted by {ϕK(ξ)} such that {ϕK(ξ)} uniformly converges to ϕ(ξ) on any compact interval. Here ϕ(ξ) belongs to Cq(R)∩W 1,p loc (R) and 0 ≤ ϕ ≤ r̄. Note that ϕ(ξ) ∈ Cq(R), then we further obtain ϕ(ξ) ∈ C1+β(R) for some β ∈ (0, q) by the regularity theory of p-Laplacian equations. We observe that ϕ(ξ) is increasing since each ϕK(ξ) is monotonically increasing. Moreover, we can show that ϕ(ξ) is a solution of (3.2) and ϕ(ξ) > 0 for all ξ ∈ R. In fact, assume that there exists ξ̂ ∈ R such that ϕ(ξ̂) = 0. Clearly, ϕ(ξ̂) is the minimum value. We note that ϕ′(ξ̂) = 0, ϕ′′(ξ̂) ≥ 0, and f(ξ̂) = (r(ξ̂)+κ)ϕ(ξ̂) ≤ 0 by (3.2), which contradicts to the assumption that ϕ(ξ) > 0 for all ξ ∈ R. According to Lemma 3.2, one may immediately obtain that ϕ(ξ) ≤ ϕ(ξ). A further computation for ξ ∈ R yields (r(ξ) + κ)ϕ(ξ) ≥ (r(ξ) + κ)ϕ(ξ), which implies that ϕ(ξ) is a super-solution. □ Now, let us formulate a super-solution and a sub-solution. Because of the assumption that r(−∞) < 0 < r̄, we can choose two constants ξ0, ξ1 such that r(ξ0) < 0 and r(ξ1) > r̄ 2 > 0 which will be used to establish the super- and sub-solutions later. Lemma 3.4. For each c > 0, the function ϕ(ξ) := { r̄eµ0(ξ−ξ0), ξ < ξ0, r̄, ξ ≥ ξ0, (3.4) where µ0 is a sufficiently small positive constant, is a super-solution of (2.1) and holds that F [ϕ](ξ) ≥ 0 for all ξ ∈ R. Proof. If ξ < ξ0, ϕ(ξ) = r̄eµ1(ξ−ξ0), then F [ϕ](ξ) = −cr̄µ0e µ0(ξ−ξ0) −D(p− 1)r̄p−1µp 0e (p−1)µ0(ξ−ξ0) − r̄r(ξ)eµ0(ξ−ξ0) + r̄2e2µ0(ξ−ξ1) ≥ r̄eµ0(ξ−ξ0)(−cµ0 −D(p− 1)r̄p−2µp 0 − r(ξ0)). The quantity F [ϕ] will be nonnegative if we choose µ0 > 0 small enough such that cµ0 +D(p− 1)r̄p−2µp 0 + r(ξ0) < 0. (3.5) Note that F [ϕ](ξ) ≥ 0 holds naturally when ξ > ξ0. □ Lemma 3.5. For each c > 0, let µ1 = − r(ξ1) c < 0 with 0 < r̄ 2 < r(ξ1). Defined ϕ(ξ) := r̄max{0, eµ1(ξ−ξ1) −Mep̂µ1(ξ−ξ1)} with M > 1 a constant, then for sufficiently large M and 1 < p̂ < 2, ϕ is a sub-solution of (2.1) and holds that F [ϕ](ξ) ≤ 0 for all ξ ∈ R. 6 T. XU, G. LIU, J. YIN EJDE-2025/48 Proof. Let ξ2 = lnM (1−p̂)µ1 + ξ1. ϕ(ξ) = 0 if ξ ≤ ξ2, and ϕ(ξ) = r̄eµ1(ξ−ξ1) −Mr̄ep̂µ1(ξ−ξ1) if ξ > ξ2. For ξ ∈ (−∞, ξ2), it is easy to check that F [ϕ](ξ) ≤ 0. For ξ ∈ (ξ2 + ∞), ϕ′(ξ) = r̄µ1e µ1(ξ−ξ1) − Mr̄p̂µ1e p̂µ1(ξ−ξ1). Let ξ3 = ln(Mp̂) (1−p̂)µ1 + ξ1 > ξ2 be the unique solution of ϕ′(ξ) = 0. It is not difficult to see that ϕ′(ξ) ≥ 0 for ξ ∈ (ξ2, ξ3] and ϕ′(ξ) ≤ 0 for ξ ∈ (ξ3,+∞). Noting that (|ϕ′(ξ)|p−2ϕ′(ξ))′ = (p− 1)|ϕ′(ξ)|p−2ϕ′′(ξ) := h(ξ). Let us estimate h(ξ). Case (i) ξ ∈ (ξ2, ξ3]. By direct calculations, we obtain h(ξ) = (p− 1)(r̄µ1e µ1(ξ−ξ1) −Mr̄p̂µ1e p̂µ1(ξ−ξ1))p−2(r̄µ2 1e µ1(ξ−ξ1) −Mr̄p̂2µ2 1e p̂µ1(ξ−ξ1)) = (p− 1)r̄p−1e(p−1)µ1(ξ−ξ1)(−µ1) p(Mp̂e(p̂−1)µ1(ξ−ξ1) − 1)p−2(1−Mp̂2e(p̂−1)µ1(ξ−ξ1)). Since (−µ1) p(Mp̂e(p̂−1)µ1(ξ−ξ1) − 1)p−2 > 0, then we see that (−µ1) p(Mp̂e(p̂−1)µ1(ξ−ξ1) − 1)p−2(1−Mp̂2e(p̂−1)µ1(ξ−ξ1)) ≥ −(−µ1) p(Mp̂e(p̂−1)µ1(ξ−ξ1) − 1)p−2Mp̂2e(p̂−1)µ1(ξ−ξ1) ≥ −Mp−1p̂p(−µ1) pe(p−1)(p̂−1)µ1(ξ−ξ1). Using that ϕ(ξ) > 0, we know that 1 M > e(p̂−1)µ1(ξ−ξ1). Therefore, h(ξ) ≥ −(p− 1)r̄p−1Mp−1p̂p(−µ1) pe(p−1)p̂µ1(ξ−ξ1) ≥ −(p− 1)r̄p−1p̂p(−µ1) p( 1 M ) p−2 p̂−1−1ep̂µ1(ξ−ξ1) (3.6) for all ξ ∈ (ξ2, ξ3]. Case (ii) ξ ∈ (ξ3,+∞). We know that ϕ′(ξ) < 0, then h(ξ) = (p− 1)(Mr̄p̂µ1e p̂µ1(ξ−ξ1) − r̄µ1e µ1(ξ−ξ1))p−2(r̄µ2 1e µ1(ξ−ξ1) −Mr̄p̂2µ2 1e p̂µ1(ξ−ξ1)) = (p− 1)r̄p−1e(p−1)µ1(ξ−ξ1)(−µ1) p(1−Mp̂e(p̂−1)µ1(ξ−ξ1))p−2(1−Mp̂2e(p̂−1)µ1(ξ−ξ1)). Let ξ4 = ln(Mp̂2) (1−p̂)µ1 + ξ1 > ξ3 (here p̂ > 1) be the unique solution of ϕ′′(ξ) = 0, then h(ξ) ≥ 0 for all ξ ≥ ξ4. When ξ ∈ (ξ3, ξ4), noting that 1 < Mp̂2e(p̂−1)µ1(ξ−ξ1), we see that h(ξ) ≥ −(p− 1)r̄p−1Mp̂2(−µ1) pe(p−2)µ1(ξ−ξ1)+p̂µ1(ξ−ξ1) ≥ −(p− 1)r̄p−1p̂2(−µ1) p( 1 M ) p−2 p̂−1−1ep̂µ1(ξ−ξ1). (3.7) Based on the previous inequality, we obtain that h(ξ) ≥ −(p− 1)r̄p−1p̂p(−µ1) p( 1 M ) p−2 p̂−1−1ep̂µ1(ξ−ξ1) for all ξ ∈ (ξ2, ξ4). Next we need to evaluate F [ϕ](ξ). When ξ ∈ (ξ2,+∞), we know that F [ϕ](ξ) ≤ −cr̄µ1e µ1(ξ−ξ1)(1−Mp̂e(p̂−1)µ1(ξ−ξ1))−Dh(ξ) − r(ξ1)r̄e µ1(ξ−ξ1)(1−Me(p̂−1)µ1(ξ−ξ1)) + r̄2e2µ1(ξ−ξ1)(1−Me(p̂−1)µ1(ξ−ξ1)). (3.8) Noting that cµ1 = −r(ξ1) and 1 > Me(p̂−1)µ1(ξ−ξ1), then (3.8) implies that F [ϕ](ξ) ≤ c(p̂− 1)µ1Mr̄ep̂µ1(ξ−ξ1) −Dh(ξ) + r̄2e2µ1(ξ−ξ1). (3.9) According to the estimate of h(ξ), when ξ ∈ (ξ2, ξ4), (3.9) indicates that F [ϕ](ξ) ≤ c(p̂− 1)µ1Mr̄ep̂µ1(ξ−ξ1) +D(p− 1)r̄p−1Mp−1p̂p(−µ1) p( 1 M ) (p−2)p̂ p̂−1 ep̂µ1(ξ−ξ1) + r̄e2µ1(ξ−ξ1) EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 7 ≤ c(p̂− 1)µ1Mr̄ep̂µ1(ξ−ξ1)(1 + D(p− 1)r̄p−1p̂p(−µ1) p c(p̂− 1)µ1M p−2 p̂−1 + r̄e(2−p̂)µ1(ξ−ξ1) c(p̂− 1)µ1M ) ≤ c(p̂− 1)µ1Mr̄ep̂µ1(ξ−ξ1)(1 + D(p− 1)r̄p−1p̂p(−µ1) p c(p̂− 1)µ1M p−2 p̂−1 + r̄ c(p̂− 1)µ1M ). Since c(p̂− 1)µ1 < 0, 1 < p̂ < 2 and p−2 p̂−1 > 0, we have F [ϕ](ξ) ≤ 0 as long as we select sufficiently large M . We now consider the case in which ξ ≥ ξ4. The (3.9) implies that F [ϕ](ξ) ≤ c(p̂− 1)µ1Mr̄ep̂µ1(ξ−ξ1) + r̄2e2µ1(ξ−ξ1) ≤ r̄ep̂µ1(ξ−ξ1)(c(p̂− 1)µ1M + r̄e(2−p̂)µ1(ξ−ξ1)) ≤ r̄ep̂µ1(ξ−ξ1)(c(p̂− 1)µ1M + r̄). Similarly, we can choose M large enough in such a way that F [ϕ](ξ) ≤ 0 holds. □ In view of the definition of ϕ in Lemma 3.4 and ϕ in Lemma 3.5, it should be note that ξ0 < ξ1 satisfies r(ξ0) < 0 < r(ξ1). We obtain that 0 ≤ ϕ ≤ ϕ ≤ r̄. Proof of Theorem 2.1. For p > 2, c > 0, assume that ϕ(ξ) is defined as emma 3.4 and ϕ0(ξ) = ϕ(ξ). We consider the iterative problem −cϕ′ i(ξ)−D(|ϕ′ i(ξ)|p−2ϕ′ i(ξ)) ′ + ϕ2 i (ξ) + κϕi(ξ) = (r(ξ) + κ)ϕi−1(ξ), lim ξ→−∞ ϕi(ξ) = 0, lim ξ→+∞ ϕi(ξ) = r̄. (3.10) It follows from Lemma 3.3 that there exists a increasing super-solution ϕ1(ξ) of (2.1) and 0 < ϕ1 ≤ ϕ ≤ r̄. According to Lemma 3.2, we see that ϕ1(ξ) ≥ ϕ. Then we can show by induction that the existence of increasing solutions ϕi > 0 of (3.10) and ϕ ≤ ϕi+1(ξ) ≤ ϕi(ξ) ≤ ϕ for each i ∈ N+. Recalling that the proof of Lemma 3.3, we see that ∥ϕi∥W 1,p are bounded uniformly in any compact interval, which indicates that ϕi ∈ C(R) by the embedding theorem. We further derive that there exists a increasing continuous function ϕ such that limi→∞ ϕi(ξ) = ϕ(ξ). By the same argument in Lemma 3.3, we can obtain ϕ > 0. This conclusion is that ϕ is the travelling wave solution of (2.1). □ Now, we consider the existence, regularity and uniqueness of the solution for the following original problem. For each c > 0, ξ = x − ct is substituted into the equation of (1.1), and one obtains (we still denote the solution as u for simplicity) ∂u ∂t = D(|uξ|p−2uξ)ξ + c ∂u ∂ξ + u(r(ξ)− u), u(t,−∞) = 0, u(t,+∞) = r̄, u(0, ξ) = u0(x). (3.11) Lemma 3.6. The global solution u ∈ C(1+α)/2,1+α(R+ ×R) with α = p−1 p of the Cauchy problem (3.11) uniquely exists and satisfies ∥u∥L∞ ≤ max{∥u0∥L∞ + 1, r̄ + 1}, ux ∈ L∞(R+;L p loc(R)), ut ∈ L2 loc(R+ × R). Proof. Suppose that {ul 0} is a sequence of sufficiently smooth functions which converges to u0 as l → ∞ uniformly with respect to ξ on any compact interval and 0 ≤ ul 0(ξ) ≤ u0(ξ). Let us 8 T. XU, G. LIU, J. YIN EJDE-2025/48 consider the approximate problem ∂u ∂t = D(|uξ|2 + 1 l ) p−2 2 uξ)ξ + c ∂u ∂ξ + u(r(ξ)− u), t > 0, ξ ∈ (−l, l), u(t,−l) = 1 l , u(t, l) = r̄ + 1 l e(r(−∞)− 1 l )t, t > 0, u(0, ξ) = ul 0(ξ) + 1 l ξ ∈ (−l, l). (3.12) The existence of solutions for (3.12) can be obtained by the standard theory of quasi-linear para- bolic equations. We further derive that 1 l e(r(−∞)− 1 l )t ≤ ul(t, ξ) ≤ max{r̄ + 1 l , ∥ũ0∥L∞ + 1}. (3.13) Now we are ready to estimate the gradient of ul(t, ξ). For each a ∈ (−l + 2, l − 2), assume that η(ξ) ∈ C∞ 0 (a − 2, a + 2) and 0 ≤ η(ξ) ≤ 1, |η′(ξ)| ≤ 1, η ≡ 1 for ξ ∈ (a − 1, a + 1). Multiplying Eq.(3.12) by uη2p and integrating over R, we have 1 2 d dt ∫ R u2η2pdξ +D ∫ R (|uξ|2 + 1 l ) p−2 2 |uξ|2η2pdξ + ∫ R u3η2pdξ = −2Dp ∫ R (|uξ|2 + 1 l ) p−2 2 uξuη 2p−1ηξdξ − 2pc ∫ R u2η2p−1ηξdξ + ∫ R r(ξ)u2η2pdξ. (3.14) Note that (|uξ|2 + 1 l ) p−2 2 uξ ≤ |uξ|p−1 + 1, when p ≥ 2. Since u, r(ξ) are bounded and η(ξ) has a compact support, the right-hand side in (3.14) is converted to − 2Dp ∫ R (|uξ|2 + 1 l ) p−2 2 uξuη 2p−1ηξdξ − 2pc ∫ R u2η2p−1ηξdξ + ∫ R r(ξ)u2η2pdξ ≤ 2Dp ∫ R |uξ|p−1uη2p−1|ηξ|dξ + 2Dp ∫ R uη2p−1|ηξ|dξ − 2pc ∫ R u2η2p−1ηξdξ + ∫ R r(ξ)u2η2pdξ ≤ D 2 ∫ R |uξ|pη2pdξ + C ∫ R |uηηξ|pdξ − 2pc ∫ R u2η2p−1ηξdξ + ∫ R r(ξ)u2η2pdξ ≤ D 2 ∫ R |uξ|pη2pdξ + C. (3.15) It is easy to obtain (|uξ|2 + 1 l ) p−2 2 |uξ|2 ≥ |uξ|p, when p > 2. Recalling that (3.14) and (3.15), we have 1 2 d dt ∫ R u2η2pdξ + D 2 ∫ R (|uξ|2 + 1 l ) p 2 η2pdξ ≤ D 2 ∫ R (|uξ|2 + 1 l ) p−2 2 η2pdξ + C. Using Young’s inequality, we see that 1 2 d dt ∫ R u2η2pdξ + D 2 ∫ R (|uξ|2 + 1 l ) p 2 η2pdξ ≤ C. (3.16) For each t > 0 and fix σ > 0, integrating (3.16) from t to t+ σ, we conclude that 1 2 ∫ R u2(t+ σ, ξ)η2pdξ + ∫ t+σ t ∫ R (|uξ|2 + 1 l ) p 2 η2pdξds ≤ C. (3.17) The mean value theorem indicates that there exists t∗ ∈ (t, t+ σ) such that∫ R (|uξ(t ∗, ξ)|2 + 1 l ) p 2 η2pdξ ≤ C. (3.18) EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 9 Multiplying (3.12) by utη 4p and integrating over R, Young’s inequality implies that∫ R u2 tη 4pdξ + D p d dt ∫ R (|uξ|2 + 1 l ) p 2 η4pdξ + 1 3 d dt ∫ R u3η4pdξ = −4Dp ∫ R (|uξ|2 + 1 l ) p−2 2 uξutη 4p−1ηξdξ + c ∫ R uξutη 4pdξ + ∫ R r(ξ)uutη 4pdξ ≤ 1 2 ∫ R u2 tη 4pdξ + C ∫ R (|uξ|2 + 1 l )p−1η2(2p−1)dξ + C ∫ R |uξ|pη2pdξ + C. (3.19) Furthermore, we multiply D((|uξ|2 + 1 l ) p−2 2 uξ)ξ = ∂u ∂t − c∂u∂ξ +u2 − r(ξ)u by (|uξ|2 + 1 l ) p−2 2 uξ)ξη 4p and integrate the resultant equation on R to obtain D ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ = ∫ R ( ∂u ∂t − c ∂u ∂ξ + u2 − r(ξ)u)(|uξ|2 + 1 l ) p−2 2 uξ)ξη 4pdξ ≤ D 2 ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ + 1 2D ∫ R ( ∂u ∂t − c ∂u ∂ξ + u2 − r(ξ)u)2η4pdξ ≤ D 2 ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ + 1 D ∫ R u2 tη 4pdξ + 1 D ∫ R (c ∂u ∂ξ − u2 + r(ξ)u)2η4pdξ ≤ D 2 ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ + 1 D ∫ R u2 tη 4pdξ + 2c2 D ∫ R u2 ξη 4pdξ + 2 D ∫ R (u2 − r(ξ)u)2η4pdξ ≤ D 2 ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ + 1 D ∫ R u2 tη 4pdξ + C ∫ R |uξ|pη2pdξ + C. (3.20) Combining (3.19) and (3.20), we obtain∫ R u2 tη 4pdξ + 4D p d dt ∫ R (|uξ|2 + 1 l ) p 2 η4pdξ + 4 3 d dt ∫ R u3η4pdξ + D2 2 ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ ≤ C ∫ R (|uξ|2 + 1 l )p−1η2(2p−1)dξ + C ∫ R |uξ|pη2pdξ + C. Next, we need to estimate ∫ (|uξ|2 + 1 l ) p−1η2(2p−1)dξ. In fact, by Young’s inequality, we see that∫ R (|uξ|2 + 1 l )p−1η2(2p−1)dξ ≤ ∫ R (|uξ|2 + 1 l )p−2uξuξη 2(2p−1)dξ + ∫ R (|uξ|2 + 1 l )p−2η2(2p−1)dξ ≤ ∫ R u ( (2p− 3)(|uξ|2 + 1 l )p−2|uξξ|η2(2p−1) + 2(2p− 1)(|uξ|2 + 1 l )p−2|uξ|η4p−3ηξ ) dξ + ∫ R (|uξ|2 + 1 l )p−2η2(2p−1)dξ ≤ ε ∫ R (|uξ|2 + 1 l )p−2|uξξ|2η4pdξ + Cε ∫ R (|uξ|2 + 1 l )p−2η4p−4dξ + C ∫ R (|uξ|2 + 1 l )p−2|uξ|η4p−3dξ. Assume that A = (|uξ|2 + 1 l ) p−2η4p−4, B = (|uξ|2 + 1 l ) p−2|uξ|η4p−3. Now, let us consider the exponent p in the different cases. When p > 4, we can simply estimate A = (|uξ|2 + 1 l )p−2η4p−4 ≤ η2p(ε1(|uξ|2 + 1 l )p−1η2(p−1) + Cε1(|uξ|2 + 1 l ) p 2 ηp), B ≤ (|uξ|2 + 1 l )p− 3 2 η4p−3 ≤ η2p(ε2(|uξ|2 + 1 l )p−1η2(p−1) + Cε2(|uξ|2 + 1 l ) p 2 ηp). 10 T. XU, G. LIU, J. YIN EJDE-2025/48 When 3 < p ≤ 4, using Young’s inequality, we have A = (|uξ|2 + 1 l )p−2η4p−4 ≤ η2p(|uξ|2 + 1 l ) p 2 ηp + C), B ≤ (|uξ|2 + 1 l )p− 3 2 η4p−3 ≤ η2p(ε3(|uξ|2 + 1 l )p−1η2(p−1) + Cε3(|uξ|2 + 1 l ) p 2 ηp). Similarly, for the case 2 ≤ p ≤ 3, it holds that A = (|uξ|2 + 1 l )p−2η4p−4 ≤ η2p((|uξ|2 + 1 l ) p 2 ηp + C), B ≤ (|uξ|2 + 1 l )p− 3 2 η4p−3 ≤ η2p((|uξ|2 + 1 l ) p 2 ηp + C). Now, let us estimate (3.19). We have∫ R u2 tη 4pdξ + 4D p d dt ∫ R (|uξ|2 + 1 l ) p 2 η4pdξ + 4 3 d dt ∫ R u3η4pdξ + D2 2 ∫ R ((|uξ|2 + 1 l ) p−2 2 uξ)ξ) 2η4pdξ ≤ C ∫ R (|uξ|2 + 1 l ) p 2 η2pdξ + C, which, together with (3.17) and (3.18), indicates that∫ t+σ t ∫ R u2 tη 4pdξds+ sup t ∫ R |uξ|pη4pdξ ≤ C. It then follows that ∫ t+σ t ∫ a+1 a−1 u2 tdξds+ sup t ∫ a+1 a−1 |uξ|pdξ ≤ C, (3.21) where C is independent of l. That is ul ∈ L∞((0,∞),W 1,p loc (R)) and (ul)t ∈ L2((t, t + σ) × (a − 1, a+1)). Hence we deduce the sequence {ul} weak converges to u. Furthermore, it is not difficult to show that u is the solution for the problem (3.11) and u also holds that (3.17) and (3.21), and 0 ≤ u(t, ξ) ≤ max{∥ũ0∥L∞ + 1, r(+∞) + 1}. From the Sobolev embedding inequality, we obtain u ∈ L∞((0,∞);Cα(R)) with α = p−1 p . Now let us show that u ∈ Cα/2,α(R+ × R). For each t1, t2 ∈ R+, x ∈ R, assume that t2 ≤ t1, r = |t1 − t2|1/p. Suppose that Br is a ball of radius r centered at x, we can deduce that∫ Br |u(t1, y)− u(t2, y)|dy = ∫ Br ∣∣∣∣∫ t2 t1 ∂u(t, y) ∂t dt ∣∣∣∣dy ≤ ∫ Br ∫ t2 t1 ∣∣∣∣∂u(t, y)∂t ∣∣∣∣ dtdy. Noticing that (3.21) and using Hölder inequality, the above inequality yields∫ Br |u(t1, y)− u(t2, y)|dy ≤ (∫ Br ∫ t2 t1 ∣∣∣∣∂u(t, y)∂t ∣∣∣∣2 dtdy )1/2(∫ Br ∫ t2 t1 1dtdy )1/2 ≤ C|t1 − t2|1/2r1/2. Then the mean value theorem implies that there exists x0 ∈ Br such that |u(t1, x0)− u(t2, x0)| ≤ C|t1 − t2|1/2r−1/2 = C|t1 − t2| p−1 2p . We further have |u(t1, x)− u(t2, x)| ≤ |u(t1, x)− u(t1, x0)|+ |u(t1, x0)− u(t2, x0)|+ |u(t2, x0)− u(t2, x)| ≤ C(|x− x0| p−1 p + |t1 − t2| p−1 2p ) ≤ C|t1 − t2| p−1 2p , which indicates that u(t, ξ) ∈ Cα/2,α(R+ ×R). Furthermore, we see from the regularity theory of p-Laplacian equations that u(t, ξ) ∈ C 1+α 2 ,1+α(R+ × R). EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 11 Finally, we prove the uniqueness of the solution. Assume that u1, u2 are two solutions of (3.11) and u = u1−u2. It should be note that u(x, 0) = 0. Let αn(x) ∈ C∞ 0 (R) be a cut-off function such that αn(x) ∈ [0, 1], αn(x) = 1 for |x| ≤ n, αn(x) = 0 for |x| ≥ n+ 1, and |α′(x)| < 2. Multiplying (3.11) by e−βtu(t, x)αn(x) with β ≥ 2(r(+∞) + 1) and integrating over Qτ , we have 1 2 ∫ R e−βτu2(x, τ)αn(x)dx+ ∫∫ Qτ ( β 2 + u1 + u2 − r(ξ))u2αn(x)e −βtdxdt +D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)e −βtαn(x)dxdt = −D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1 − u2)e −βtα′ n(x)dxdt ≤ D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+ ∫∫ Qτ u2|α′ n(x)|e−βtdxdt. (3.22) Noticing that α′ n(x) = 0 for |x| < n and |x| > n+ 1, we see that D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+ ∫∫ Qτ u2|α′ n(x)|e−βtdxdt ≤ 2D ∫∫ Qτ (|u1x|2(p−1) + |u2x|2(p−1))|α′ n(x)|dxdt+ ∫∫ Qτ u2|α′ n(x)|dxdt ≤ C, where C is independent of n. Letting n → ∞ in (3.22), we obtain∫∫ Qτ u2e−βtdxdt+D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)e −βtdxdt ≤ C. (3.23) Recalling (3.22) and noticing that uix(i = 1, 2) are bounded, we infer that D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt ≤ D(p− 1)(∥u1x∥L∞ + ∥u2x∥L∞)p−2 ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt +D ∫∫ Qτ u2|α′ n(x)|e−βtdtdx ≤ C ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt. From (3.23), the above inequality reduces to C ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt → 0, as n → ∞. Now, (3.22) shows that 1 2 ∫ R e−βtu2(x, τ)dx+ ∫∫ Qτ e−βt(u2 + (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x))dxdt ≤ 0, which implies that u1 = u2 in Qτ . □ To prove the stability of the forced waves, let us consider the solutions of (1.1) of the form v(t, ξ) := u(t, ξ + ct) − ϕ(ξ) with the initial value v0(ξ) := u0(ξ) − ϕ(ξ). Substituting we obtain the degenerate perturbed equation ∂v ∂t = D(|uξ|p−2uξ − |ϕξ|p−2ϕξ)ξ + c ∂v ∂ξ + (r(ξ)− (u+ ϕ))v, v(0, ξ) = v0(ξ). (3.24) Next we study the regularity of the perturbed solutions for (3.24). Lemma 3.7 (Lp-regularity of perturbed solution). If v is a solution of the perturbed equation (3.24), then v ∈ L∞([0, T ];Lp(R)) for any T > 0. 12 T. XU, G. LIU, J. YIN EJDE-2025/48 Proof. Let αn(ξ) ∈ C∞ 0 (R) be the cut-off function defined as in Lemma 3.6. It is easy to check that α′ n is supported in [−n− 1,−n] ∪ [n, n+ 1]. We multiply (3.24) by α2 n|v|p−2v and integrate over R to obtain 1 p d dt ∥α2 nv p∥L1(R) +D(p− 1) ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2α2 ndξ + ∫ R α2 n(u+ ϕ)|v|pdξ ≤ ( 4c p + 4D) ∫ [−n−1,−n]∪[n,n+1] |v|p−2(v2 + |(|uξ|p−2uξ − |ϕξ|p−2ϕξ)v|)αndξ + ∫ R r(ξ)α2 n|v|pdξ. (3.25) From Lemma 3.6, we know that v, uξ, ϕξ are bounded. It then follows that∫ [−n−1,−n]∪[n,n+1] |v|p−2(v2 + |(|uξ|p−2uξ − |ϕξ|p−2ϕξ)v|)αndξ ≤ C, where C is independent of n. We ignore two positive terms in left-hand side in (3.25), which implies that 1 p d dt ∥α2 nv p∥L1(R) ≤ r(+∞)∥α2 nv p∥L1(R) + C. Gronwall’s inequality indicates that d dt ( e−pr(+∞)t∥α2 nv p∥L1(R) ) ≤ Ce−pr(+∞)t. Then integrating over [0, t] for any 0 < t < T , one has sup t∈[0,T ] ∥α2 nv p∥L1(R) ≤ C. By letting n → ∞, we further have sup t∈[0,T ] ∥vp∥L1(R) ≤ C. □ Lemma 3.8 (Weighted Lp w-regularity of the perturbed solution). ( If v is a solution of the per- turbed equation (3.24), then v ∈ L∞([0, T ];Lp w(R)) for any T > 0. Proof. Here we firstly choose an approximation of the weight function. For λ > 0, let wk(ξ) =  e−λk, ξ < −k, eλξ, |ξ| ≤ k, eλk, ξ > k. By Lemma 3.7, we see that v ∈ L∞([0, T ];Lp(R)). Then we can multiply (3.24) by |v|p−2vwk and integrate over R to obtain 1 p d dt ∥vpwk∥L1(R) +D(p− 1) ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2wkdξ + cλ p ∫ k −k |v|pwkdξ + ∫ R (u+ ϕ)|v|pwkdξ = −λD ∫ k −k (|uξ|p−2uξ − |ϕξ|p−2ϕξ)|v|p−2vwkdξ + ∫ R r(ξ)|v|pwkdξ. (3.26) According to Lemma 3.6, we see that uξ, ϕξ are bounded. From this, we can take ρ = max{∥(uξ) p−2∥L∞(R), ∥(ϕξ) p−2∥L∞(R)}. EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 13 By Young’s inequality and the mean value theorem, we obtain λD ∫ k −k ||uξ|p−2uξ − |ϕξ|p−2ϕξ||v|p−1wkdξ ≤ D 2ρ ∫ k −k (|uξ|p−2uξ − |ϕξ|p−2ϕξ) 2|v|p−2wkdξ + λ2Dρ ∫ k −k |v|pwkdξ ≤ D(p− 1) 2 ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2wkdξ + λ2Dρ ∫ R |v|pwkdξ. (3.27) Combining this with (3.26), we see that d dt ∥vpwk∥L1(R) < (λ2Dpρ+ pr(+∞))∥vpwk∥L1(R). We further have sup t∈(0,T ) ∥vpwk∥L1(R) ≤ C, where C is independent of k. As k → ∞, we obtain sup t∈(0,T ) ∥v∥Lp w(R) ≤ C. □ Proof of Theorem 2.1. According to Lemma 3.8, we know that for any t > 0, |v|pw ∈ L1(R). We can multiply the first equation of (3.24) by |v|p−2vw, where w(ξ) = eλξ and λ > 0 will be specified later, and integrate it over R to obtain 1 p d dt ∥vpw∥L1(R) +D(p− 1) ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2wdξ + cλ p ∥vpw∥L1(R) + ∫ R (u+ ϕ)|v|pwdξ − ∫ R r(ξ)|v|pwdξ = −Dλ ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)|v|p−2vwdξ. (3.28) Let us estimate the right-hand side of (3.28). Noting that uξ, ϕξ are bounded and ρ is defined as in Lemma 3.8, we have −Dλ ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)|v|p−2vwdξ ≤ (p− 1)ρDλ ∣∣ ∫ R vξ|v|p−2vwdξ ∣∣ ≤ ρDλ2 ∫ R |v|pwdξ. (3.29) Combining this with (3.28), we see that d dt ∥vpw∥L1(R) + cλ∥vpw∥L1(R) − p(ρDλ2 + r(+∞))∥vpw∥L1(R) ≤ 0. Multiplying the above inequality by eζ1s and integrating it from 0 to t, we have eζ1t∥vp(t, ξ)w∥L1(R)+(cλ−pρDλ2−pr(+∞)−ζ1)∥eζ1svpw∥L1((0,t)×R) ≤ ∥vp(0, ξ)w∥L1(R). (3.30) Noting that the equation −pr(+∞) + cλ−Dpρλ2 = 0 (3.31) has a double root λ∗ when c∗ = 2 √ Dp2ρr(+∞), and for any c > c∗, (3.31) admits two different positive roots 0 < λ1 < λ2. As λ ∈ (λ1, λ2), −pr(+∞) + cλ − Dpρλ2 > 2ϵ for some constants ϵ > 0. Taking arbitrarily small ζ1 < ϵ, we see that −pr(+∞) + cλ−Dpρλ2 − ζ1 > ϵ. We further have ∥vp(t, ξ)w∥L1(R) + ϵ∥eζ1(s−t)vpw∥L1((0,t)×R) ≤ e−ζ1t∥vp0w∥L1(R), that is ∥v∥Lp w(R) ≤ e−ζ1t∥v0∥Lp w(R). (3.32) 14 T. XU, G. LIU, J. YIN EJDE-2025/48 Using that r(−∞) < 0 and r(ξ) is a nondecreasing continuous function, there exists ξ∗ such that r(ξ∗ + 1) = r(−∞) 2 < 0, which further leads to r(ξ) ≤ r(−∞) 2 < 0 for any ξ ∈ (−∞, ξ∗ + 1). Since w(ξ) = eλξ with λ > 0, then (3.32) can be deduced that∫ +∞ ξ∗ |v|pdξ ≤ e−λξ∗ ∫ +∞ ξ∗ w|v|pdξ ≤ C∥v0∥Lp w(R)e −ζ1t. (3.33) To proceed, we introduce a new cut-off function α̃n(ξ) ∈ C∞ 0 (R) satisfying α̃n(ξ) =  0, ξ ≥ ξ∗ + 1, 1, −n ≤ ξ < ξ∗, 0, ξ ≤ −n− 1, (3.34) where n >> 1 is an integer and 0 ≤ α̃n(ξ) ≤ 1, |α̃′ n(ξ)| ≤ 2. According to Lemma 3.7, we can know that |v|p ∈ L1(R). Multiplying (3.24) by |v|p−2vα̃n and integrating over R, we see that 1 p d dt ∫ R |v|pα̃ndξ +D(p− 1) ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2α̃ndξ + ∫ R (u+ ϕ)|v|pα̃ndξ − ∫ R r(ξ)|v|pα̃ndξ = −D ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)|v|p−2vα̃′ ndξ − c p ∫ R |v|pα̃′ ndξ. (3.35) The first term of the right hand in (3.35) implies that −D ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)|v|p−2vα̃′ ndξ ≤ ρD ∫ R |v|p|α̃′′ n|dξ. (3.36) Combining this with (3.35), we have 1 p d dt ∫ R |v|pα̃ndξ +D(p− 1) ∫ R (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2α̃ndξ + ∫ R (u+ ϕ)|v|pα̃ndξ − ∫ R r(ξ)|v|pα̃ndξ ≤ ρD ∫ R |v|p|α̃′′ n|dξ − c p ∫ R |v|pα̃′ ndξ. (3.37) Since |v|p ∈ L1(R) and u, ϕ, uξ, ϕξ, r(ξ) are all bounded, letting n → ∞ in (3.37), one has 1 p d dt ∫ ξ∗+1 −∞ |v|pα̃dξ +D(p− 1) ∫ ξ∗+1 −∞ (|uξ|p−2uξ − |ϕξ|p−2ϕξ)vξ|v|p−2α̃dξ + ∫ ξ∗+1 −∞ (u+ ϕ)|v|pα̃dξ − ∫ ξ∗+1 −∞ r(ξ)|v|pα̃dξ ≤ ρD ∫ ξ∗+1 −∞ |v|p|α̃′′|dξ + c p ∫ ξ∗+1 −∞ |v|p|α̃′|dξ, (3.38) where α̃ is the limiting function of α̃n as n → +∞. According to the definition of α̃n, we deduce that 0 ≤ α̃(ξ) ≤ 1 for ξ ∈ [ξ∗, ξ∗ +1], α̃(ξ) = 1 for ξ ∈ (−∞, ξ∗) and α̃′(ξ), α̃′′(ξ) are bounded and supported in [ξ∗, ξ∗ + 1]. Hence (3.38) indicates that 1 p d dt ∫ ξ∗+1 ξ∗ |v|pα̃dξ + 1 p d dt ∫ ξ∗ −∞ |v|pdξ − ∫ ξ∗+1 −∞ r(ξ)|v|pdξ ≤ ρD ∫ ξ∗+1 ξ∗ |v|p|α̃′′|dξ + c p ∫ ξ∗+1 ξ∗ |v|p|α̃′|dξ. (3.39) From (3.32), we clearly have that∫ ξ∗+1 ξ∗ |v|pdξ ≤ C∥v0∥Lp w(R)e −ζ1t, EJDE-2025/48 FORCED WAVES FOR p-LAPLACE EQUATIONS 15 which, together with (3.39), implies that 1 p d dt ∫ ξ∗+1 ξ∗ |v|pα̃dξ + 1 p d dt ∫ ξ∗ −∞ |v|pdξ − ∫ ξ∗+1 −∞ r(ξ)|v|pdξ ≤ C∥v0∥Lp w(R)e −ζ1t. (3.40) Multiplying (3.40) by eζ2s(ζ2 > 0 will be determined later) and integrating it from 0 to t, one has eζ2t (∫ ξ∗+1 ξ∗ |v|pα̃dξ + ∫ ξ∗ −∞ |v|pdξ ) − p ∫ t 0 ∫ ξ∗+1 −∞ eζ2sr(ξ)|v|pdξds − ζ2 ∫ t 0 eζ2s (∫ ξ∗+1 ξ∗ |v|pα̃dξ + ∫ ξ∗ −∞ |v|pdξ ) ds ≤ ∫ ξ∗+1 −∞ |v(0, ξ)|pdξ + Ce−ζ1t ∫ t 0 eζ2sds. 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Tianyuan Xu School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, Guangdong 510006, China Email address: tyxu93@163.com Gege Liu (corresponding author) School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China Email address: liugege27@163.com Jingxue Yin School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China Email address: yjx@scnu.edu.cn 1. Introduction 2. Main results 3. Proof of the main results Acknowledgement References