Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 85, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STABILITY OF INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR VISCOELASTIC EQUATIONS KUN-PENG JIN, JIN LIANG, TI-JUN XIAO Abstract. We investigate the stability of the initial-boundary value problem for the quasilinear viscoelastic equation |ut|ρutt −∆utt −∆u+ ∫ t 0 g(t− s)∆u(s)ds = 0, in Ω× (0,+∞), u = 0, in ∂Ω× (0,+∞), u(·, 0) = u0(x), ut(·, 0) = u1(x), in Ω, where Ω is a bounded domain of Rn (n ≥ 1) with smooth boundary ∂Ω, ρ is a positive real number, and g(t) is the relaxation function. We present a general polynomial decay result under some weak conditions on g, which generalizes and improves the existing related results. Moreover, under the condition g′(t) ≤ −ξ(t)gp(t), we obtain uniform exponential and polynomial decay rates for 1 ≤ p < 2, while in the previous literature only the case 1 ≤ p < 3/2 was studied. Finally, under a general condition g′(t) ≤ −H(g(t)), we establish a fine decay estimate, which is stronger than the previous results. 1. Introduction In this article, we consider the stability of the initial-boundary value problem for quasilinear viscoelastic equations, |ut|ρutt −∆utt −∆u+ ∫ t 0 g(t− s)∆u(s)ds = 0, in Ω× (0,+∞), u = 0, in ∂Ω× (0,+∞), u(·, 0) = u0(x), ut(·, 0) = u1(x), in Ω, (1.1) where Ω is a bounded domain of Rn(n ≥ 1) with smooth boundary ∂Ω, ρ is a positive real number, and g(t) the relaxation function. In [16], under the assumption that the bounded C1-function g : R+ → R+ satisfies 1− ∫ +∞ 0 g(t)ds > 0, g′(t) ≤ −ξgp(t), 1 ≤ p < 3 2 , (1.2) where ξ > 0 is a constant, Messaoudi and Tatar obtained decay rates in [16, Theo- rem 3.1]. 2010 Mathematics Subject Classification. 35Q74, 35B35, 74H55, 74H40, 93D15. Key words and phrases. Quasilinear viscoelastic equation; polynomial and exponential decay; relaxation function; uniform decay. c©2020 Texas State University. Submitted November 11, 2019. Published July 30, 2020. 1 2 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 More recently, Messaoudi and Al-Khulaifi [13] improved this result [16, Theorems 3.1] by using the assumption that the non-increasing differentiable function g : R+ → R+ satisfies 1− ∫ +∞ 0 g(t)ds > 0, g′(t) ≤ −ξ(t)gp(t), 1 ≤ p < 3 2 , (1.3) here ξ(t) : R+ → R+ is a non-increasing differentiable function with ξ(0) > 0. Messaoudi and Mustafa [14] also studied problem (1.1) and the corresponding decay results were obtained for the following condition on g(t), g′(t) ≤ −H(g(t)), t ≥ 0, (1.4) where H is a positive function and satisfies some conditions (see details in [14, hypotheses (A2) and (A3)]). For more related information on the stability of problem (1.1) and some related equations or systems, we refer the reader to [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23, 24] and references therein. In this article, we investigate the stability for problem (1.1) by using more general (weaker) assumptions on the relaxation functions g(t). We establish ideal stability theorems with exact uniform polynomial decay rates t−1 for the solutions to this problem, under some basic conditions (see Theorem 3.2). Furthermore, in Theorems 3.4 and 3.6, our results hold for all 1 ≤ p < 2, while in the previous literature only the case: 1 ≤ p < 3 2 was studied. Therefore, all of our results, with much weaker conditions on the relaxation function g(t), are optimal so far. In the next section, we prove some estimates (lemmas) which will be used in Section 3. Finally, we will state and prove our main results in Section 3. 2. Basic estimates In this article we use the following assumptions: (A1) 0 < ρ, if n = 1, 2; and 0 < ρ ≤ 2 n− 2 , if n ≥ 3; (A2) g(t) : [0,+∞)→ [0,+∞) is a non-increasing differentiable function with meas(J0) = 0, g(0) > 0, g′(t) ≤ 0, µ0 > 0, where J0 := {s ≥ 0; g(s) > 0, g′(s) = 0} = 0, µ0 := 1− ∫ +∞ 0 g(t) dt. In the sequel, C,Ci > 0, i = 1, 2, . . . represent positive constants which are possibly different in different places. We denote G(t) := ∫ +∞ t g(s)ds, for t ≥ 0; M(δ) := ∫ +∞ 0 g(s) Kδ(s) ds, Kδ(s) := −g′(s) g(s) + δ, where δ ∈ (0, 1) is a constant. We define I1(t) := ∫ Ω ∫ t 0 G(t− s)|∇u(s)|2 ds dx, EJDE-2020/85 STABILITY OF VISCOELASTIC EQUATIONS 3 I2(t) := M(δ) ( δ ∫ Ω ∫ t 0 G(t− s)|∇u(s)|2 ds dx+ E(t) ) . Lemma 2.1. For t ≥ 0, d dt I1(t) ≤ −1 2 ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx+ 2G(0) ∫ Ω |∇u(t)|2dx, (2.1) and d dt I2(t) ≤ −1 2 M(δ) ∫ Ω ∫ t 0 Kδ(t− s)g(t− s)|∇u(t)−∇u(s)|2 ds dx + 2δM(δ)G(0) ∫ Ω |∇u(t)|2dx. (2.2) Moreover, δM(δ)→ 0, as δ → 0. (2.3) Proof. Noting that −(a± b)2 ≤ −1 2 a2 + b2, we see by a direct calculation that, for t ≥ 0, d dt I1(t) = − ∫ Ω ∫ t 0 g(t− s)|∇u(s)|2 ds dx+G(0) ∫ Ω |∇u(t)|2dx = − ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)−∇u(t)|2 ds dx +G(0) ∫ Ω |∇u(t)|2dx ≤ −1 2 ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx + ∫ Ω ∫ t 0 g(t− s)|∇u(t)|2 ds dx+G(0) ∫ Ω |∇u(t)|2dx ≤ −1 2 ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx+ 2G(0) ∫ Ω |∇u(t)|2dx. This means that (2.1) holds. From the definition of Kδ(s), (2.1) and (3.2), it follows that d dt I2(t) ≤ −1 2 M(δ) ∫ Ω ∫ t 0 (δg(t− s) + g′(t− s)) |∇u(t)−∇u(s)|2 ds dx + 2δM(δ)G(0) ∫ Ω |∇u(t)|2dx ≤ −1 2 M(δ) ∫ Ω ∫ t 0 Kδ(t− s)g(t− s)|∇u(t)−∇u(s)|2 ds dx + 2δM(δ)G(0) ∫ Ω |∇u(t)|2dx. According to [8, P. 1525, lines 8-10], we know that (2.3) is true. Thus, we completed the proof. � 4 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 We define F1(t) := 1 ρ+ 1 ∫ Ω |ut|ρutu dx+ ∫ Ω ∇u · ∇ut dx, Lemma 2.2. For t ≥ 0, d dt F1(t) ≤ −µ0 2 ∫ Ω |∇u|2dx+ 1 2µ0 ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx + ∫ Ω |∇ut|2dx+ 1 ρ+ 1 ∫ Ω |ut|ρ+2dx. (2.4) Proof. Clearly, we can rewrite the first equation in (1.1) as |ut|ρutt −∆utt − ( 1− ∫ t 0 g(s)ds ) ∆u− ∫ t 0 g(t− s) (∆u(t)−∆u(s)) ds = 0. (2.5) It follows from (2.5) that d dt F1(t) = ( 1− ∫ t 0 g(s)ds )∫ Ω u∆udx+ ∫ Ω u(t) ∫ t 0 g(t− s) (∆u(t)−∆u(s)) ds dx + ∫ Ω |∇ut|2dx+ 1 ρ+ 1 ∫ Ω |ut|ρ+2dx = − ( 1− ∫ t 0 g(s)ds )∫ Ω |∇u|2dx − ∫ Ω ∇u(t) · ∫ t 0 g(t− s) (∇u(t)−∇u(s)) ds dx + ∫ Ω |∇ut|2dx+ 1 ρ+ 1 ∫ Ω |ut|ρ+2dx ≤ −µ0 ∫ Ω |∇u|2dx+ µ0 2 ∫ Ω |∇u|2dx + 1 2µ0 ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx + ∫ Ω |∇ut|2dx+ 1 ρ+ 1 ∫ Ω |ut|ρ+2dx ≤ −µ0 2 ∫ Ω |∇u|2dx+ 1 2µ0 ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx + ∫ Ω |∇ut|2dx+ 1 ρ+ 1 ∫ Ω |ut|ρ+2dx. This completes the proof. � Now, we define F2(t) := ∫ Ω ( ∆ut − 1 ρ+ 1 |ut|ρut )∫ t 0 g(t− s)(u(t)− u(s)) ds dx. EJDE-2020/85 STABILITY OF VISCOELASTIC EQUATIONS 5 Lemma 2.3. There is a constant C1 > 0 such that, for t ≥ t0, d dt F2(t) ≤ − G(0) 2(ρ+ 1) ∫ Ω |ut(t)|ρ+2dx− G(0) 2 ∫ Ω |∇ut(t)|2dx + µ0G(0) 16 ∫ Ω |∇u(t)|2dx+ C1 ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx − C1 ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx, (2.6) where t0 a positive large number so that∫ t0 0 g(s)ds = 3G(0) 4 . Proof. By (2.5), we obtain d dt F2(t) = − 1 ρ+ 1 ∫ t 0 g(s)ds ∫ Ω |ut(t)|ρ+2dx+ ∫ t 0 g(s)ds ∫ Ω ut(t)∆ut(t)dx + ∫ Ω ut ∫ t 0 g′(t− s)(∆u(t)−∆u(s)) ds dx − 1 ρ+ 1 ∫ Ω |ut|ρut ∫ t 0 g′(t− s)(u(t)− u(s)) ds dx − ( 1− ∫ t 0 g(s)ds )∫ Ω ∆u(t) · ∫ t 0 g(t− s)(u(t)− u(s)) ds dx − ∫ Ω ∫ t 0 g(t− s)(∆u(t)−∆u(s))ds ∫ t 0 g(t− s)(u(t)− u(s)) ds dx = − 1 ρ+ 1 ∫ t 0 g(s)ds ∫ Ω |ut(t)|ρ+2dx− ∫ t 0 g(s)ds ∫ Ω |∇ut(t)|2dx − ∫ Ω ∇ut · ∫ t 0 g′(t− s)(∇u(t)−∇u(s)) ds dx − 1 ρ+ 1 ∫ Ω |ut|ρut ∫ t 0 g′(t− s)(u(t)− u(s)) ds dx + ( 1− ∫ t 0 g(s)ds )∫ Ω ∇u(t) · ∫ t 0 g(t− s)(∇u(t)−∇u(s)) ds dx + ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx. (2.7) Next, let us to estimate the third, fourth and fifth terms on the right of (2.7). First we estimate the fourth term. By Young’s and Holder’s inequality, for any ζ1 > 0, we have − 1 ρ+ 1 ∫ Ω |ut|ρut ∫ t 0 g′(t− s)(u(t)− u(s)) ds dx 6 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 ≤ 1 ρ+ 1 ζ1 ∫ Ω |ut|2ρ+2dx− g(0) 4ζ1(ρ+ 1) ∫ Ω ∫ t 0 g′(t− s)|u(t)− u(s)|2 ds dx. By (A1), (A2) and the Sobolev embedding inequality, we obtain∫ Ω |ut|2ρ+2dx ≤ Cs(2E(0))ρ ∫ Ω |∇ut|2dx . By Poincaré’s inequality, we have − ∫ Ω ∫ t 0 g′(t− s)|u(t)− u(s)|2 ds dx ≤ −Cp ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx, where Cp is the Poincaré’s constant and Cs the Sobolev embedding constant. There- fore, − 1 ρ+ 1 ∫ Ω |ut|ρut ∫ t 0 g′(t− s)(u(t)− u(s)) ds dx ≤ Cs ρ+ 1 (2E(0))ρζ1 ∫ Ω |∇ut|2dx − g(0)Cp 4ζ1(ρ+ 1) ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx. (2.8) Now, we estimate the third and fifth terms. It is not hard to see that, for any ζ2, ζ3 > 0, − ∫ Ω ∇ut · ∫ t 0 g′(t− s)(∇u(t)−∇u(s)) ds dx ≤ ζ2 ∫ Ω |∇ut|2dx− g(0) 4ζ2 ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx, (2.9) and ( 1− ∫ t 0 g(s)ds )∫ Ω ∇u(t) · ∫ t 0 g(t− s)(∇u(t)−∇u(s)) ds dx ≤ ζ3 ∫ Ω |∇u(t)|2dx+ 1 4ζ3 ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx. (2.10) Thus, combining (2.8), (2.9), (2.10) with (2.7), we know that d dt F2(t) ≤ − 1 ρ+ 1 ∫ t 0 g(s)ds ∫ Ω |ut(t)|ρ+2dx − (∫ t 0 g(s)ds− ζ2 − Cs ρ+ 1 (2E(0))ρζ1 )∫ Ω |∇ut(t)|2dx − (g(0) 4ζ2 + g(0)Cp 4ζ1(ρ+ 1) )∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx + ζ3 ∫ Ω |∇u(t)|2dx+ ( 1 + 1 4ζ3 )∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx. Setting ζ1 = (ρ+ 1)G(0) 8Cs(2E(0))ρ , ζ2 = G(0) 8 , ζ3 = µ0G(0) 16 , we obtain the estimate (2.6). This completes the proof. � EJDE-2020/85 STABILITY OF VISCOELASTIC EQUATIONS 7 3. Main results and their proofs We firstly state an existence and uniqueness result for problem (1.1), which can be proved by using similar arguments as in [4, 15] so we omit it here. Theorem 3.1. Let (A1) and (A2) hold. Then for any u0 ∈ H1 0 (Ω), u1 ∈ H1 0 (Ω), the problem (1.1) has a unique global solution on [0,∞) with the regularity u ∈ C1 ( R+;H1 0 (Ω) ) . We introduce the energy functional E(t) := 1 ρ+ 2 ∫ Ω |ut|ρ+2dx+ 1 2 ∫ Ω |∇ut|2dx+ 1 2 ( 1− ∫ t 0 g(s)ds )∫ Ω |∇u|2dx + 1 2 ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx. (3.1) Then, for t ≥ 0, d dt E(t) = −1 2 g(t) ∫ Ω |∇u|2dx+ 1 2 ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ 1 2 ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx, (3.2) and E(t) ∼ ∫ Ω ( |ut|ρ+2 + |∇ut|2 + |∇u|2 ) dx + ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx. (3.3) The following is our general uniform decay theorem for the solution energy of prob- lem (1.1). Theorem 3.2. Let (A1) and (A2) hold. Then, for u0, u1 ∈ H1 0 (Ω), the solution energy E(t) of the problem (1.1) satisfies∫ +∞ 0 E(t) ≤ CE(0), t ≥ 0, E(t) ≤ CE(0)(t+ 1)−1, t ≥ 0, where C > 0 is a constant. Proof. The proof is mainly based on the construction of an auxiliary function L(t) satisfying L(t0) ≤ CE(0), L(t) ≥ 0, t ≥ 0, and d dt L(t) ≤ −ε0E(t), t ≥ t0. (3.4) Clearly, integrating (3.4) we obtain the desired estimate. Now, we apply the lemmas obtained in the previous section to construct this auxiliary function L(t). We define J(t) := NE(t) + F1(t) + 4 G(0) F2(t). By the definitions of F1(t) and F2(t) and a simple calculation, we see that, there is a constant c0 > 0 such that, for t ≥ 0, |F1(t)|, |F1(t)| ≤ c0E(t). 8 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 Taking N > 8C1/G(0) large enough, we obtain c1E(t) ≤ J(t) ≤ c2E(t), t ≥ 0, where c1, c2 > 0 are constants. Thus, by (2.4), (2.6) and (3.2), for t ≥ t0, we have d dt J(t) ≤ −µ0 4 ∫ Ω |∇u(t)|2dx− 1 ρ+ 1 ∫ Ω |ut(t)|ρ+2dx− ∫ Ω |∇ut|2dx + ( 4C1 G(0) + 1 2µ0 )∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx. (3.5) Moreover, ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx ≤ ∫ Ω ∫ t 0 g(s) Kδ(s) ds ∫ t 0 Kδ(t− s)g(t− s)|∇u(t)−∇u(s)|2 ds dx ≤M(δ) ∫ Ω ∫ t 0 Kδ(t− s)g(t− s)|∇u(t)−∇u(s)|2 ds dx. Hence, by (3.5), for t ≥ t0, we see that d dt J(t) ≤ −µ0 4 ∫ Ω |∇u(t)|2dx− 1 ρ+ 1 ∫ Ω |ut(t)|ρ+2dx− ∫ Ω |∇ut|2dx + ( 4C1 G(0) + 1 2µ0 ) M(δ) ∫ Ω ∫ t 0 Kδ(t− s)g(t− s)|∇u(t)−∇u(s)|2 ds dx. (3.6) Now we define L(t) := J(t) + µ0 32G(0) I1(t) + 2 ( 4C1 G(0) + 1 2µ0 ) I2(t). Then, by (2.1), (2.2) and (3.6), for t ≥ t0, we obtain d dt L(t) ≤ − (3µ0 16 − 4G(0) ( 4C1 G(0) + 1 2µ0 ) δM(δ) )∫ Ω |∇u(t)|2dx − 1 ρ+ 1 ∫ Ω |ut(t)|ρ+2dx− ∫ Ω |∇ut|2dx − µ0 64G(0) ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx. (3.7) Convergence (2.3) shows that there exists δ0 > 0 such that, for any 0 < δ < δ0, δM(δ) ≤ µ0 64G(0) ( 4C1 G(0) + 1 2µ0 ) . Thus, by (3.7) and (3.3), we deduce that, for 0 < δ < δ0, there exists a constant ε0 > 0 such that, for t ≥ t0, d dt L(t) ≤ −ε0E(t). (3.8) EJDE-2020/85 STABILITY OF VISCOELASTIC EQUATIONS 9 Since L(t) ≥ 0 for t ≥ 0, and L(t0) ≤ CE(0), it follows by integrating (3.8) over [t0, τ) that for any τ > t0, ∫ τ t0 E(t)dt ≤ CE(0). So, ∫ +∞ 0 E(t)dt ≤ CE(0). (3.9) Noting that E′(t) ≤ 0, by(3.9), we obtain E(t) ≤ CE(0)(t+ 1)−1, t ≥ 0. This completes the proof. � Remark 3.3. (1) As showed in Theorem 3.2, the polynomial decay rates can be obtained without the control conditions on g′(t) used previously. There are many functions g(t) satisfying the assumptions (A2) without satisfying the previous restriction that g(t) controls g′(t) as in (1.2), (1.3) and (1.4). For example, if g(t) = (√ 2 + sin t ) e−t, t ≥ 0, then g′(t) = − (√ 2− cos t+ sin t ) e−t = − √ 2 ( 1− cos(t+ π 4 ) ) e−t, t ≥ 0. Clearly, g′(t) ≤ 0, for t ≥ 0; g′(t) = 0, for t = 2kπ − π 4 , k = 1, 2, . . . . Hence, g(t) satisfies (A2), while g(t) does not satisfy (1.2), (1.3) or (1.4). That is, g′(t) is not controlled by g(t). Functions g(t) as above have not been studied in the literature. However, we can treat the problem (1.1) with these general relaxation functions, and according to Theorem 3.2 here, we know the energy E(t) of problem (1.1) decays at least at the rate (t+ 1)−1. (2) The decay rates given in Theorem 3.2 are optimal in a sense according to [13, Example 3.1, Remark 3.2] and [8, Remark 3.3(ii)]. When the derivative g′(s) is controlled by the relaxation function g(t), we can prove the following results. Theorem 3.4. Let (A1) and (A2) hold, and g′(t) ≤ −ξ(t)gp(t), t ≥ 0, (3.10) where ξ(t) : R+ → R+ is a non-increasing differentiable function with ξ(0) > 0 and 1 ≤ p < 2 is a constant. Then there are constants C, η > 0 such that for t ≥ 0, E(t) ≤ CE(0)e−η ∫ t 0 ξ(s)ds, p = 1, CE(0) ( 1 1+ ∫ t 0 ξ(s)ds ) 1 p−1 1 < p < 2. (3.11) 10 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 Proof. A key idea in the proof is to construct a Lyapunov function satisfying R(t) ∼ E(t) and d dt R(t) ≤ −ε2ξ(t)Rp(t). To find this function, we will use the results of Theorem 3.2 and J(t) defined above. Clearly, ∫ Ω (∫ t 0 g(t− s)|∇u(t)−∇u(s)|ds )2 dx ≤ G(0) ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx. Thus, by (3.5) and (3.3), for t ≥ t0, we have d dt J(t) ≤ −ε1E(t) + C2 ∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx, (3.12) where ε1 > 0 is a constant. On the other hand, by Theorem 3.2, we know that∫ +∞ 0 E(t)dt ≤ CE(0), and E(t) ≤ CE(0)(t+ 1)−1. Since∫ Ω ∫ t 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ (∫ t 0 ∫ Ω |∇u(t)−∇u(s)|2dxds )1− 1 p (∫ Ω ∫ t 0 gp(t− s)|∇u(t)−∇u(s)|2 ds dx )1/p ≤ C (∫ t 0 (E(t) + E(s))ds )1− 1 p (∫ Ω ∫ t 0 gp(t− s)|∇u(t)−∇u(s)|2 ds dx )1/p ≤ CE1− 1 p (0) (∫ Ω ∫ t 0 gp(t− s)|∇u(t)−∇u(s)|2 ds dx )1/p , by (3.12) it follows that for t ≥ t0, d dt J(t) ≤ −ε1E(t)+C3E 1− 1 p (0) (∫ Ω ∫ t 0 gp(t−s)|∇u(t)−∇u(s)|2 ds dx )1/p . (3.13) Multiplying (3.13) by ξ(t)Ep−1(t), for t ≥ t0, we obtain ξ(t)Ep−1(t) d dt J(t) ≤ −ε1ξ(t)Ep(t) + C3E 1− 1 p (0)ξ(t)Ep−1(t) (∫ Ω ∫ t 0 gp(t− s)|∇u(t)−∇u(s)|2 ds dx )1/p ≤ −ε1 2 ξ(t)Ep(t) + C4ξ(t) ∫ Ω ∫ t 0 gp(t− s)|∇u(t)−∇u(s)|2 ds dx. (3.14) Since ξ(t), E(t) are non-increasing functions, from (3.10) it follows that for t ≥ 0, d dt ( ξ(t)Ep−1(t)J(t) ) = ξ(t)Ep−1(t) d dt J(t) + J(t) d dt ( ξ(t)Ep−1(t) ) ≤ ξ(t)Ep−1(t) d dt J(t), EJDE-2020/85 STABILITY OF VISCOELASTIC EQUATIONS 11 and ξ(t) ∫ Ω ∫ t 0 gp(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ ∫ Ω ∫ t 0 ξ(t− s)gp(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ − ∫ Ω ∫ t 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ −2 d dt E(t). Hence, by (3.14), for t ≥ t0, we have d dt ( ξ(t)Ep−1(t)J(t) + 2C4E(t) ) ≤ −ε1 2 ξ(t)Ep(t). (3.15) Now, we define R(t) := ξ(t)Ep−1(t)J(t) + 2C4E(t). Then, R(t) ∼ E(t). By (3.15), for t ≥ t0, we obtain d dt R(t) ≤ −ε2ξ(t)Rp(t), where ε2 > 0 is a constant. This completes the proof. � Remark 3.5. (1) Theorem 3.4 extends the results in [13, 14, 16], where g′(t) was assumed to satisfy (3.10) with p ∈ [1, 3/2), since Theorem 3.4 holds for all p ∈ [1, 2). Moreover, the decay rates obtained in [13] are E(t) ≤ Ke−λ ∫ t t0 ξ(s)ds , p = 1, E(t) ≤ K ( 1 1 + ∫ t t0 ξ2p−1(s)ds ) 1 2p−2 , 1 < p < 3 2 . In addition, if ∫ +∞ 0 ( 1 tξ2p−1(t) + 1 ) dt < +∞, 1 < p < 3 2 , (3.16) reference [13] shows the improved estimate E(t) ≤ K ( 1 1 + ∫ t t0 ξp(s)ds ) 1 p−1 , 1 < p < 3 2 . Since ξ(t) is nonnegative and non-increasing, it is clear that ξp(s) . ξ(s), and then( 1 1 + ∫ t 0 ξ(s)ds ) 1 p−1 . ( 1 1 + ∫ t t0 ξp(s)ds ) 1 p−1 . Therefore, the decay rates given in Theorem 3.4 is stronger than the previous conclusion in the [13, Theorem 3.1] for all p ∈ [1, 2). On the other hand, we obtain the stronger estimate without the other restrictions on ξ(t) (as (3.16) in [13, Theorem 3.1]). As can be seen, Theorem 3.4 here give stronger conclusions essentially under weaker conditions on g(t). (2) The decay rates given in Theorem 3.4 are optimal in according to [13, Ex- ample 3.1, Remark 3.2] and [8, Remark 3.3(ii)]. 12 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 Theorem 3.6. Let the assumptions of Theorem 3.2 hold, and g′(t) ≤ −H(g(t)), t ≥ 0, (3.17) where H ∈ C1 (R+) is a positive function with H(0) = 0, and it is also a linear or strictly increasing and strictly convex C2 function on (0, r], for some r < 1. Then there are constants k1, k2, k3, ε0 > 0 such that E(t) ≤ k3G −1(k1t+ k2), t ≥ 0, (3.18) where G(t) = ∫ 1 t 1 sH ′(ε0s) ds. Proof. By Theorem 3.2, we obtain∫ +∞ 0 E(t)dt ≤ CE(0) and E(t) ≤ CE(0)(t+ 1)−1. So, ∫ Ω ∫ t 0 |∇u(t)−∇u(s)|2 ds dx ≤ CE(0) < +∞. (3.19) According to (3.17) and (3.19), we can and do take t1 > t0 large enough such that for any t ≥ t1, ∫ Ω ∫ t t1 |∇u(t)−∇u(s)|2 ds dx < min{r,H(r)}, (3.20) − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx < min{r,H(r)}, (3.21)∫ Ω ∫ t−t1 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx < min{r,H(r)}, (3.22) max{g(t),−g′(t)} < min{r,H(r)}. (3.23) Using (3.17), (3.20)-(3.23) and Jensen’s inequality, for t ≥ t1, we obtain − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ≥ ∫ Ω ∫ t−t1 0 H(g(t− s))|∇u(t)−∇u(s)|2 ds dx ≥ H (∫ Ω ∫ t−t1 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx ) . (3.24) Then for t ≥ t1,∫ Ω ∫ t−t1 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ H−1 ( − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ) . (3.25) Moreover, by [14, P. 1860, equation (3.24)], for t ≥ t1, we obtain d dt W1(t) ≤ −ε3E(t) + C5 ∫ Ω ∫ t−t1 0 g(t− s)|∇u(t)−∇u(s)|2 ds dx, (3.26) where W1(t) ∼ E(t) and ε3 > 0 is a constant. EJDE-2020/85 STABILITY OF VISCOELASTIC EQUATIONS 13 By (3.25) and (3.26), for t ≥ t1, we have d dt W1(t) ≤ −ε3E(t) + C5H −1 ( − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ) . (3.27) Now, we define W2(t) := H ′ ( ε0 E(t) E(0) ) W1(t) +ME(t), where 0 < ε0 < r, M > 0 are constants, which will be specific later. Clearly, W2(t) ∼ E(t) because of the assumption on H. Therefore, for t ≥ t1, d dt W2(t) = H ′ ( ε0 E(t) E(0) ) d dt W1(t) + ε0 E′(t) E(0) H ′′ ( ε0 E(t) E(0) ) W1(t) +ME′(t) ≤ C5H ′ ( ε0 E(t) E(0) ) H−1 ( − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ) − ε3E(t)H ′ ( ε0 E(t) E(0) ) +ME′(t), (3.28) where we have used E′(t) ≤ 0, H ′′ ≥ 0, and (3.27). Next, we estimate the first term on the right of (3.28). Let H? be the convex conjugate of H in the sense of Young (see [2, P. 61-64] and [14, P. 1863]). Then H?(s) = s(H ′)−1(s)−H[(H ′)−1(s)], s ∈ (0, H ′(r)), (3.29) and it satisfies ab ≤ H?(a) +H(b), for a ∈ (0, H ′(r)], b ∈ (0, r]. (3.30) Setting a = H ′ ( ε0 E(t) E(0) ) , b = H−1 ( − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ) , and using (3.29), (3.30) and (3.21), we obtain H ′ ( ε0 E(t) E(0) ) H−1 ( − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ) ≤ H? ( H ′ ( ε0 E(t) E(0) )) − ∫ Ω ∫ t−t1 0 g′(t− s)|∇u(t)−∇u(s)|2 ds dx ≤ ε0 E(t) E(0) H ′ ( ε0 E(t) E(0) ) − 2E′(t). (3.31) From (3.28) and (3.31), it follows that for t ≥ t1, d dt W2(t) ≤ − (ε3E(0)− C5ε0) E(t) E(0) H ′ ( ε0 E(t) E(0) ) + (M − 2C5)E′(t). (3.32) Therefore, if we take M > 0 large enough and ε0 > 0 small sufficiently, then we obtain, for t ≥ t1, d dt W2(t) ≤ −ε4H̃ (E(t) E(0) ) , (3.33) 14 K. P. JIN, J. LIANG, T.-J. XIAO EJDE-2020/85 where ε4 > 0 is a constant and H̃(t) = tH ′(ε0t). We define W (t) := γ W2(t) E(0) , where γ > 0 small enough such that W (t) < E(t) E(0) . Clearly, W (t) ∼ E(t) ∼ W2(t), and H̃(t), H̃ ′(t) ≥ 0. So, by (3.33), we know that there exists ε5 > 0 such that for t ≥ t1 d dt W (t) ≤ −ε5H̃ (W (t)) . 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Zhang; Stability of Riemann solutions to pressureless Euler equations with Coulomb- type friction by flux approximation, Electron. J. Differential Equations, 2019 (65) (2019), 1–22. Kun-Peng Jin School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Email address: kjin11@fudan.edu.cn Jin Liang (corresponding author) School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China Email address: jinliang@sjtu.edu.cn Ti-Jun Xiao Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathe- matical Sciences, Fudan University, Shanghai 200433, China Email address: tjxiao@fudan.edu.cn 1. Introduction 2. Basic estimates 3. Main results and their proofs Acknowledgments References