Lifespan estimates for wave equations with damping and potential posed on asymptotically Euclidean manifolds

TL;DR

This paper studies the lifespan and blow-up behavior of solutions to semilinear wave equations with damping and potential on asymptotically Euclidean manifolds, addressing conjectures related to wave equation nonlinearities.

Contribution

It provides new lifespan estimates for wave equations with mixed nonlinearities on asymptotically Euclidean manifolds, connecting to the Strauss and Glassey conjectures.

Findings

Derived upper bounds for solution lifespan

Identified conditions for finite time blow-up

Extended results to manifolds with asymptotic Euclidean geometry

Abstract

In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with time dependent damping and potential, and mixed nonlinearities $c_1 |u_t|^p+c_2 |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and the Glassey conjecture.