An application of Lp−LqLp−Lq decay estimates to the semi-linear wave equation with parabolic-like structural damping

In this paper we study the global existence of small data solutions to utt−△u+2a(−△)σut=|u|p,u(0,x)=u0(x),ut(0,x)=u1(x), where  a>0a>0, σ∈(0,1/2]σ∈(0,1/2] and  p>1p>1. Assuming small data in some Sobolev spaces, we obtain the global existence for  p>1+2/(n−2σ)p>1+2/(n−2σ), in space dimension  n≤n¯, where  n¯=n¯(σ)↗∞, as  σ→1/2σ→1/2. In particular, our result holds in any space dimension  n≥2n≥2 if  σ=1/2σ=1/2.