Blow-up of solutions to semilinear wave equations with spatial derivatives

TL;DR

This paper establishes sharp upper bounds on the lifespan of solutions to small-amplitude semilinear wave equations with spatial derivatives across all spatial dimensions greater than one, using integral equations and differential inequalities.

Contribution

It provides the first complete sharp lifespan estimates for these equations, including critical and subcritical cases, especially for the spherically symmetric scenario.

Findings

Sharp lifespan bounds are obtained for all spatial dimensions n>1.

The method combines integral equations, ODE inequalities, and iteration techniques.

Results unify and extend previous lifespan estimates for semilinear wave equations.

Abstract

For small-amplitude semilinear wave equations with power type nonlinearity on the first-order spatial derivative, the expected sharp upper bound on the lifespan of solutions is obtained for both critical cases and subcritical cases, for all spatial dimensions $n>1$. It is achieved uniformly by constructing the integral equations, deriving the ordinary differential inequality system, and iteration argument. Combined with the former works, the sharp lifespan estimates for this problem are completely established, at least for the spherical symmetric case.