Stability of initial-boundary value problem for quasilinear viscoelastic equations

Abstract

We investigate the stability of the initial-boundary value problem for the quasilinear viscoelastic equation |ut|ρutt - ∆utt - ∆u + ∫t0 g(t - s)∆u(s)ds = 0, in Ω x (0, +∞), u = 0, in ∂Ω x (0, +∞), u(‧, 0) = u0(x), ut(‧, 0) = u1(x), in Ω, where Ω is a bounded domain in 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.