STABILITY OF INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR VISCOELASTIC EQUATIONS

We investigate the stability of the initial-boundary value problem for the quasilinear viscoelastic equation vertical bar u(t)vertical bar(rho)u(tt) - Delta u(tt) - Delta u + integral(t)(0) g(t - s)Delta u(s)ds = 0, in Omega x (0, +infinity), u = 0, in partial derivative Omega x (0, +infinity), u(., 0) = u(0)(x), u(t)(., 0) = u(1)(x), in Omega, where Omega is a bounded domain of R-n (n >= 1) with smooth boundary partial derivative Omega, rho 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) <= -xi(t)g(p)(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.