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

In this paper, we study diffusion phenomena for the wave equation with structural dampingutt−Δu+2a(−Δ)σut=0,u(0,x)=u0(x),ut(0,x)=u1(x), with a>0a>0 and σ∈(0,1/2)σ∈(0,1/2). We show that the solution u   behaves like the solution v+v+ tovt++12a(−Δ)1−σv+=0,v+(0,x)=v0+(x), for suitable choice of initial data v0+. More precisely, we derive Lp−LqLp−Lq decay estimates for the difference u−v+u−v+ and its time and space derivatives, where 1⩽p⩽q⩽∞1⩽p⩽q⩽∞, possibly not on the conjugate line, satisfying some additional condition related to σ.In particular, we show that, under suitable assumptions on p,q,σp,q,σ, a double diffusion phenomenon   appears, that is, the difference u−v+u−v+ behaves like the solution tovt−+2a(−Δ)σv−=0,v−(0,x)=v0−(x), for a suitable choice of initial data v0−.