Weak and Young Measure Solutions for Hyperbolic Initial Boundary Value Problems of Elastodynamics in the Orlicz--Sobolev Space Setting

Abstract

We establish the existence result for a global weak solution (respectively, Young measure solution) in the Orlicz--Sobolev space setting for the nonlinear hyperbolic initial boundary value problem to $u_{tt}=\text{div}(\sigma(Du))+\mu(\triangle u)_t$ (respectively, $u_{tt}=\text{div}(\sigma(Du))$), where the function $\sigma=\partial W/\partial F$ is continuous and the stored-energy function $W\colon \mathbb{M}^{d\times n}\to \mathbb{R}$ may be nonconvex. Our study is motivated by one-dimensional elastodynamics. The present paper gives first solvability results for nonlinear hyperbolic partial differential equations with nonpower-growth-type nonlinearity for $Du$ in the monotonicity case and in the case with lack of monotonicity. The results are new even for the one-dimensional case with $\sigma(\tau)=\ln^q(1+\vert{\tau}\vert)\vert{\tau}\vert^{p-2}\tau+a\tau$ for $q>0$ and $p\geq2$ and $a\in \mathbb{R}$; here $a>0$ corresponds to the strong ellipticity of $W$, $a=0$ --- the convexity of $W$, and $a<0$ ---...