Diffusion phenomena for the wave equation with structural damping in the L-p - L-q framework

Author(s)

D'Abbicco, M. ^[1] ; Ebert, M. R. ^[2]

Affiliation

^[1] Univ Bari, Dipartimento Matemat, I-70125 Bari - Italy

^[2] Univ Sao Paulo, FFCLRP, Dept Computacao & Matemat, BR-14040901 Ribeirao Preto, SP - Brazil

• Document type: Journal article
• Source: Journal of Differential Equations; v. 256, n. 7, p. 2307-2336, APR 1 2014.
• Web of Science Citations: 20

Abstract

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. (AU)

• FAPESP's process: 12/19085-3 - Global existence for a class of semi-linear wave equations with variable coefficients
• Grantee: Marcelo Rempel Ebert
• Support Opportunities: Research Grants - Visiting Researcher Grant - International