Existence and nonexistence theorems for global weak solutions to quasilinear wave equations for the elasticity

TL;DR

This paper investigates the conditions under which global weak solutions to certain nonlinear degenerate wave equations exist or do not exist, extending previous classical solution results using advanced mathematical techniques.

Contribution

It establishes new existence and nonexistence theorems for global weak solutions to degenerate wave equations using compensated compactness and kinetic formulation methods.

Findings

Proves existence of global weak solutions under specific conditions.

Establishes nonexistence of solutions when certain initial data thresholds are not met.

Extends previous classical solution results to weak solutions for nonlinear wave equations.

Abstract

In this paper, by using the theory of compensated compactness coupled with the kinetic formulation by Lions, Perthame, Souganidis and Tadmor \cite{LPT,LPS}, we prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations $v_{tt} =c (|v|^{s-1} v)_{xx}$ with the nonnegative initial data $v_{0}(x)$ and $ s > 1$. This result is an extension of the results in the second author's paper \cite{Su}, where the existence and the nonexistence of the unique global classical solution were studied with a threshold on $\int_{-\infty}^{\infty} v_{1}(x) dx$ and the non-degeneracy condition $ v_{0}(x) \geq c_{0} > 0$ on the initial data.