Global existence of solutions for semilinear damped wave equation with nonlinearities of derivative type

TL;DR

This paper proves that semilinear damped wave equations with derivative nonlinearities have unique global solutions for small initial data in low dimensions, expanding understanding of their long-term behavior.

Contribution

It establishes the global existence of solutions for all p > 1 in low dimensions, a new result for derivative-type nonlinearities in damped wave equations.

Findings

Global solutions exist for all p > 1 in dimensions 1 and 2.

Unique solutions are obtained for small initial data.

The results extend previous knowledge on semilinear damped wave equations.

Abstract

In this paper, we consider the semilinear damped wave equation with nonlinearities of derivative type $|u_t|^p$. We observe that this problem admits a unique global (in time) solution with small initial data for all $p > 1$ in low spatial dimensions $n=1,2$. This result provides new insights into semilinear damped wave equations and complements the existing literature.