Diffusion phenomena for the wave equation with structural damping in the Lp - Lq framework

In this paper, we study diffusion phenomena for the wave equation with structural damping u(tt) - Delta u + 2a(-Delta)(sigma)u(t)=0, u(0,x)=u(0)(x), u(t)(0,x)=u(1)(x), with a > 0 and sigma epsilon (0, 1/2). We show that the solution a behaves like the solution v(+) to v(t)(+) + 1/2a(-Delta)(1-sigma)v(+)=0, v(+)(0,x)=v(0)(+)(x), for suitable choice of initial data v(0)(+). precisely, we derive L-p - L-q decay estimates for the difference u - v(+) and its time and space derivatives, where 1 <= p <= q <= infinity, possibly not on the conjugate line, satisfying some additional condition related to sigma. In particular, we show that, under suitable assumptions on p, q, sigma, a double diffusion phenomenon appears, that is, the difference u - v(+) behaves like the solution to v(t)(-)+2a(-Delta)(sigma)v(-) = 0, v(-)(0,x)=v(0)(-)(x), for a suitable choice of initial data v(0)(-). (C) 2014 Elsevier Inc. All rights reserved.