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arXiv · 1707.08540

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

Yun-guang Lu·Yuusuke Sugiyama
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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.

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Yun-guang Lu, Yuusuke Sugiyama. 2017-07-26. Existence and nonexistence theorems for global weak solutions to quasilinear wave equations for the elasticity. https://arxiv.org/abs/1707.08540

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