Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damp
- PDF / 282,005 Bytes
- 19 Pages / 595.28 x 793.7 pts Page_size
- 65 Downloads / 196 Views
RESEARCH
Open Access
Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping Fei Liang1,2 and Hongjun Gao1* * Correspondence: gaohj@hotmail. com 1 Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, PR China Full list of author information is available at the end of the article
Abstract In this paper, we consider the system of nonlinear viscoelastic equations
⎧ t ⎪ ⎪ ⎨ utt − u + g1 (t − τ )u(τ )dτ − ut = f1 (u, v), (x, t) ∈ × (0, T), 0
t ⎪ ⎪ ⎩ vtt − v + g2 (t − τ )v(τ )dτ − vt = f2 (u, v),
(x, t) ∈ × (0, T)
0
with initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions gi, fi (i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time. 2000 Mathematics Subject Classifications: 35L05; 35L55; 35L70. Keywords: decay, blow-up, positive initial energy, viscoelastic wave equations
1. Introduction In this article, we study the following system of viscoelastic equations: t ⎧ utt − u + 0 g1 (t − τ )u(τ )dτ − ut = f1 (u, v), (x, t) ∈ × (0, T), ⎪ ⎪ ⎪ t ⎪ ⎨ vtt − v + 0 g2 (t − τ )v(τ )dτ − vt = f2 (u, v), (x, t) ∈ × (0, T), u(x, t) = v(x, t) = 0, x ∈ ∂ × (0, T), ⎪ ⎪ ⎪ u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ , ⎪ ⎩ v(x, 0) = v0 (x), vt (x, 0) = v1 (x), x ∈ ,
(1:1)
where Ω is a bounded domain in ℝn with a smooth boundary ∂Ω, and gi(·) : ℝ+ ® ℝ 2 + , f i (·, ·): ℝ ® ℝ (i = 1, 2) are given functions to be specified later. Here, u and v denote the transverse displacements of waves. This problem arises in the theory of viscoelastic and describes the interaction of two scalar fields, we can refer to Cavalcanti et al. [1], Messaoudi and Tatar [2], Renardy et al. [3]. To motivate this study, let us recall some results regarding single viscoelastic wave equation. Cavalcanti et al. [4] studied the following equation: t utt − u + g(t − τ )u(τ )dτ + a(x)ut + |u|γ u = 0, in × (0, ∞) 0
© 2011 fei and Hongjun; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liang and Gao Boundary Value Problems 2011, 2011:22 http://www.boundaryvalueproblems.com/content/2011/1/22
Page 2 of 19
for a : Ω ® ℝ+, a function, which may be null on a part of the domain Ω. Under the conditions that a(x) ≥ a0 > 0 on Ω1 ⊂ Ω, with Ω1 satisfying some geometry restrictions and −ξ1 g(t) ≤ g (t) ≤ −ξ2 g(t),
t ≥ 0,
the authors established an exponential rate of decay. This latter result has been improved by Cavalcanti and Oquendo [5] and Berrimi and Messaoudi [6]. In their study,
Data Loading...
