Local existence and uniqueness for a semi-linear accretive wave equation

TL;DR

This paper establishes local existence and uniqueness of solutions for a semi-linear wave equation with a specific nonlinear term, within certain functional spaces, contributing to the mathematical understanding of such equations.

Contribution

It provides the first proof of local well-posedness for a semi-linear wave equation with an accretive nonlinear term involving the time derivative.

Findings

Proves local existence of solutions in specified function spaces.

Establishes uniqueness of solutions under given conditions.

Analyzes the behavior of solutions for the semi-linear wave equation.

Abstract

We prove local existence and uniqueness of the solution $(u,u_t)\in C^0([0,T];H^1\times L^2(\mathbb{R}^N))$ of the semilinear wave equation $u_{tt}-\Delta u=u_t|u_t|^{p-1}$.