Test function method for blow-up phenomena of semilinear wave equations and their weakly coupled systems

TL;DR

This paper introduces a simplified test function method to analyze blow-up phenomena and lifespan estimates for semilinear wave equations and their weakly coupled systems, including critical cases with new $(p,q)$-curves.

Contribution

A new, simpler framework for deriving sharp lifespan bounds for nonlinear wave equations and systems, extending previous methods with novel $(p,q)$-curves and solutions.

Findings

Derived sharp upper bounds for solution lifespans.

Identified new $(p,q)$-curves for coupled wave systems.

Extended lifespan estimates to new parameter regions.

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 method and give a simple proof of the derivation of sharp upper bound of lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical case, we use a family of self-similar solution to the standard wave equation including Gauss's hypergeometric functions which are originally introduced by Zhou (1992). However, our framework is much simpler than that. As a consequence, we found new $(p,q)$-curve for the system $\partial_t^2u-\Delta u=|v|^q$, $\partial_t^2v-\Delta v=|\partial_tu|^p$ and lifespan estimate for small solutions for new region.