Critical exponent for semi-linear structurally damped wave equation of derivative type

Abstract

Main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: $$u_{tt}-\Delta u+\mu (-\Delta {)}^{\sigma /2}{u}_t=|u_t{|}^p,u(0,x)=u_0(x),u_t(0,x)=u_1(x),$$ with $\mu >0$, $n\ge 1$, $\sigma \in (0,2]$ and $p>1$. In particular, we are going to prove the non-existence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.

9 pages