A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain

TL;DR

This paper establishes the global existence of small energy solutions for a two-dimensional semilinear strongly damped wave equation with mixed nonlinearities in an exterior domain, using weighted energy methods under certain conditions.

Contribution

It provides the first global existence result for this class of equations with mixed nonlinearities in exterior domains, employing weighted energy techniques and small initial data assumptions.

Findings

Global existence of small energy solutions is proven.

Weighted energy methods are effectively applied.

Conditions on nonlinear powers ensure solution existence.

Abstract

We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $|u|^p+|u_t|^q$ in an exterior domain, where $p,q>1$. Assuming the smallness of initial data in exponentially weighted spaces and some conditions on powers of nonlinearity, we prove global (in time) existence of small data energy solution with suitable higher regularity by using a weighted energy method.