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Weak and Young measure solutions for hyperbolic initial boundary value problems of elastodynamics in the Orlicz-Sobolev space setting. (English) Zbl 1338.35302

Summary: 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 \(u_{tt}=\operatorname{div}(\sigma(Du))+\mu(\triangle u)_t\) (respectively, \(u_{tt}=\operatorname{div}(\sigma(Du))\)), 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 \(\sigma(\tau)=\ln^q(1+|\tau|)|\tau|^{p-2}\tau+a\tau\) for \(q>0\) and \(p\geq 2\) and \(a\in \mathbb R\); here \(a>0\) corresponds to the strong ellipticity of \(W\), \(a=0\) – the convexity of \(W\), and \(a<0\) – the nonconvexity of \(W\) under the Andrews-Ball inequality.

MSC:

35L72 Second-order quasilinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
74D10 Nonlinear constitutive equations for materials with memory
74B20 Nonlinear elasticity
28A33 Spaces of measures, convergence of measures
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35D30 Weak solutions to PDEs
74H20 Existence of solutions of dynamical problems in solid mechanics
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