Blow-up solutions to the semilinear wave equation with overdamping term

Abstract

This article deals with the Cauchy problem to the following damped wave equation \begin{equation*} {\left\lbrace \begin{array}{ll} u_{tt}-\Delta u+b(t) u_t=M(u),~&(t,x)\in R^{+}\times R^{N},\\ u(0,x)=u_0(x),~u_t(0,x)=u_1(x),~&x\in R^{N}, \end{array}\right.}CP \end{equation*} with the focusing nonlinearity $M(u)=|u|^{p-1}u,~p>1.$ For the focusing nonlinearity $M(u)=\pm |u|^{p},~p>1,$ Ikeda and Wakasugi in [8] have showed that the solution to Problem (CP) exists globally for small data and fails to exist globally for large data. Meanwhile, they also proposed an open problem [8, Remark 1.3]. In this note, we give a positive answer to this open problem by using a method different from the test-function method. In addition, an inverse Hölder inequality associated with the solution and a differential inequality argument are used to establish a lower bound for the blow-up time.