Existence and nonexistence theorems for global weak solutions to quasilinear wave equations for the elasticity. (arXiv:1707.08540v2 [math.AP] UPDATED)

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. 查看全文>>