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