Existence of Blow-Up Solution to the Cauchy Problem of Inhomogeneous Damped Wave Equation

This paper is concerned with the non-existence of a global solution to the initial value problem of the inhomogeneous damped wave equation with a nonlinear memory term and a nonlinear gradient term. The critical exponent and formation of singularity of solution are closely related to symmetry in the study of blow-up dynamics for nonlinear wave equations, which provides a profound mathematical tool for analyzing the explosion of solutions within finite time. The proofs of blow-up results of solutions are based on the test function method, where the test function is variable separated. The influences of two types of damping terms, two types of nonlinearities, and an inhomogeneous term on exponents of the problem in blow-up cases are explained. It is worth pointing out that the inhomogeneous term in the problem is discussed with respect to the exponent $\sigma$ in three cases (namely, $\sigma =0$, $-1<\sigma <0$, and $\sigma >0$). As far as we know, the results in Theorems 1–4 are new.