The blow up of solutions to semilinear wave equations on asymptotically Euclidean manifolds

TL;DR

This paper studies the blow-up behavior and lifespan estimates of solutions to semilinear wave equations on asymptotically Euclidean manifolds with exponential metric perturbations, extending understanding of wave dynamics in curved spaces.

Contribution

It introduces new techniques for constructing positive solutions to related elliptic equations and applies them to analyze blow-up phenomena on curved manifolds.

Findings

Blow-up results for semilinear wave equations on asymptotically Euclidean manifolds.

Sharp upper bounds for the lifespan of solutions.

Extension of methods to damped wave equations with integrable dissipation.

Abstract

In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean metric. One of the main ingredients in our proof is the construction of (unbounded) positive entire solutions for $\Delta_{g}\phi_\lambda=\lambda^{2}\phi_\lambda$, with certain estimates which are uniform for small parameter $\lambda\in (0,\lambda_0)$. In addition, our argument works equally well for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.