Blow up for small-amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds

TL;DR

This paper studies finite-time blow-up and lifespan estimates for small-amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds, connecting to the Strauss and Glassey conjectures.

Contribution

It provides new lifespan bounds and existence results for solutions, extending to damped wave equations with integrable, space-independent dissipation.

Findings

Lifespan estimates match upper and lower bounds in certain cases

Existence results for solutions with specific nonlinearities

Applicability to damped wave equations with integrable dissipation

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 mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and Glassey conjecture. In some cases, we obtain existence results, where the lower bound of the lifespan agrees with the upper bound in order. In addition, our results apply for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.