Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t, x)u(t) and nonlinear absorption vertical bar u vertical bar(p-1)u (Ikawa (2000) [17]). We consider 1 < p < (n+2)/(n-2) and separable a(t, x) = lambda(x)eta(t) with lambda(x) similar to (1 + vertical bar x vertical bar)(-alpha) and eta(t) similar to (1 + t)(-beta) satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L-2 and Lp+1 norms of solutions. We also observe the following behavior: if alpha is an element of [0, 1), beta is an element of (-1, 1) and 0 < alpha + beta < 1, there are three different regions for the decay of solutions depending on p; if alpha is an element of (-infinity, 0) and beta is an element of (-1, 1), there are only two different regions for the decay of the solutions depending on p. (C) 2011 Elsevier Ltd. All rights reserved.