Blow-up of solutions to the semilinear wave equation with scale invariant damping on exterior domain

Abstract

Abstract This paper is concerned with the blow-up of solutions to the initial boundary value problem for the wave equation with scale invariant damping term on the exterior domain, where the nonlinear terms are power nonlinearity | u | p $|u|^{p} $ , derivative nonlinearity | u t | p $|u_{t}|^{p} $ and combined nonlinearities | u t | p + | u | q $|u_{t}|^{p}+ |u|^{q} $ , respectively. Upper bound lifespan estimates of solutions to the problem are obtained by constructing suitable test functions and utilizing the test function technique. The main novelty is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.

Keywords

• Scale invariant damping
• Exterior domain
• Blow-up
• Lifespan estimates
• Test function technique