Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

TL;DR

This paper investigates the well-posedness of a (1+1)-dimensional wave equation with a nonlinear boundary condition, establishing conditions for existence, uniqueness, and conservation laws, and analyzing special solutions called breathers.

Contribution

It provides a rigorous analysis of well-posedness for a wave equation with a quasilinear boundary condition, including cases with nonlinear Neumann conditions linked to Maxwell problems.

Findings

Global existence and uniqueness when f is an increasing homeomorphism

Conservation of energy and momentum in the system

Non-wellposedness when f is a decreasing homeomorphism

Abstract

We consider the linear wave equation $V(x) u_{tt}(x, t) - u_{xx}(x, t) = 0$ on $[0, \infty)\times[0, \infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x(0, t) = (f(u_t(0,t)))_t$ at $x=0$. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where $f\colon\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where $f$ is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at $x=0$ have enough regularity to solve the initial value problem with their own initial data.