Well-posedness and long-time behavior of Lipschitz solutions to extremal surface equations

TL;DR

This paper establishes the well-posedness and long-time behavior of Lipschitz solutions to extremal surface equations in one dimension, linking them to entropy solutions of a conservation law system, and analyzing their convergence to traveling waves.

Contribution

It provides an explicit representation formula, proves uniqueness, and analyzes the convergence and stability of entropy solutions for extremal surface equations.

Findings

Lipschitz solutions are equivalent to entropy solutions in 1D.

Explicit representation formula for solutions.

Solutions converge to traveling waves as time goes to infinity.

Abstract

We show that in one space dimension Lipschitz solutions of extremal surface equations are equivalent to entropy solutions in $L^\infty(\R)$ of a non-strictly hyperbolic system of conservation laws. We obtain an explicit representation formula and the uniqueness of the entropy solutions to the Cauchy problem of the system. By using this formula, we also obtain the convergence and convergence rates as $t \rightarrow +\infty$ of the entropy solutions to explicit traveling waves in the $L^1(\R)$ norm. Moreover, when initial data are constants outside of a finite space interval, the entropy solutions become the explicit traveling waves after a finite time. Finally, we prove $L^1$ stabilities of the entropy solutions.