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

作者：D'Abbicco M; Ebert M R^*

来源：Journal of Differential Equations, 2014, 256(7): 2307-2336.

DOI：10.1016/j.jde.2014.01.002

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)(-).

• 出版日期2014-4-1