Blow-up of solutions to semilinear strongly damped wave equations with different nonlinear terms in an exterior domain

TL;DR

This paper investigates the blow-up behavior of solutions to semilinear strongly damped wave equations in exterior domains, establishing local existence and conditions leading to finite-time blow-up for certain nonlinearities.

Contribution

It provides new blow-up results for semilinear damped wave equations with derivative and mixed nonlinearities, using fixed-point and test function methods.

Findings

Local existence of mild solutions proven using Banach fixed-point theorem.

Blow-up results established under specific initial data and exponent conditions.

Analysis extends understanding of nonlinear wave equations in exterior domains.

Abstract

In this paper, we consider the initial boundary value problem in an exterior domain for semilinear strongly damped wave equations with power nonlinear term of the derivative-type $|u_t|^q$ or the mixed-type $|u|^p+|u_t|^q$, where $p,q>1$. On one hand, employing the Banach fixed-point theorem we prove local (in time) existence of mild solutions. On the other hand, under some conditions for initial data and the exponents of power nonlinear terms, the blow-up results are derived by applying the test function method.