Blow-up phenomena of semilinear wave equations and their weakly coupled systems

Abstract

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function methods and give a simple proof of the derivation of sharp upper bounds for lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical cases, we use a family of self-similar solutions to the linear wave equation including Gauss's hypergeometric functions, which are originally introduced by Zhou [59]. We emphasize that our framework does not require the pointwise positivity of the initial data even in the high dimensional case N≥4. Moreover, we find a new (p,q)-curve for the system ∂t2u−Δu=|v|q, ∂t2v−Δv=|∂tu|p with lifespan estimates for small solutions in a new region.