Global Nonlinear Stability of Traveling Wave Solution to Time-Like Extremal Hypersurface in Minkowski Space

Abstract

Abstract There are a few results about the global nonlinear stability of nontrivial large solution to quasilinear wave equations. Time-like extremal surface in Minkowski space is an important model of quasilinear wave equation. There are two folds in this paper. Firstly, we get the existence of traveling wave solution to the time-like extremal hypersurface in $\mathbb {R}^{1+(n+1)}$, which can be considered as the generalized Bernstein theorem. For $n=2$, we are also concerned with global stability of traveling wave solutions with speed of light to time-like extremal hypersurface equation in $1+(2+1)$ dimensional Minkowski space, which is corresponding with quasilinear wave equation in two space dimensions.