Blow-up of solution to semilinear wave equations with strong damping and scattering damping

Abstract

This paper is devoted to investigating the initial boundary value problem for a semilinear wave equation with strong damping and scattering damping on an exterior domain. By introducing suitable multipliers and applying the test-function technique together with an iteration method, we derive the blow-up dynamics and an upper-bound lifespan estimate of the solution to the problem with power-type nonlinearity $|u|^{p}$ | u | p , derivative-type nonlinearity $|u_{t}|^{p}$ | u t | p , and combined type nonlinearities $|u_{t}|^{p}+|u|^{q}$ | u t | p + | u | q in the scattering case, respectively. The novelty of the present paper is that we establish the upper-bound lifespan estimate of the solution to the problem with strong damping and scattering damping, which are associated with the well-known Strauss exponent and Glassey exponent.