Blow-up of solutions to semilinear wave equations with a time-dependent strong damping

Abstract

The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^{\beta }}\Delta u_t$) and a nonlinearity $|u{|}^p$. We will show the influence of the the parameter $\beta$ in the blow-up results under some hypothesis on the initial data and the exponent $p$ by using the test function method. We also study the local existence in time of mild solution in the energy space $H^1({\mathbb{R}}^n)\times L^2({\mathbb{R}}^n)$.