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Weak solutions for a nonlinear system in viscoelasticity. (English) Zbl 0635.73047
This paper contains an extension of work by R. J. DiPerna [Arch. Ration. Mech. Anal. 82, 27-70 (1983; Zbl 0519.35054)] on the global existence of \(L^{\infty}\)-weak solutions to the Cauchy problem \(w_ t=v_ x\), \(v_ t=\sigma_ x\), \(x\in {\mathbb{R}}\), \(t>0\). The constitutive equation \(\sigma =\phi (w)\) used previously by DiPerna to model a one- dimensional elastic medium is here replaced by the assumption \(\sigma =\phi (w(x,t))+\int^{t}_{0}k(t-\tau)\phi (w(x,\tau)d\tau\), where the integral term is used to represent a type of fading memory. The solutions (w,v) to the Cauchy problem belong, for arbitary finite \(T>0\), to the class \(L^{\infty}((0,T)\); L 2(\({\mathbb{R}}))\cap L^{\infty}((0,T)\times {\mathbb{R}})\).
Reviewer: R.Saxton

MSC:
74Hxx Dynamical problems in solid mechanics
35A15 Variational methods applied to PDEs
45K05 Integro-partial differential equations
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74B20 Nonlinear elasticity
35L99 Hyperbolic equations and hyperbolic systems
35L65 Hyperbolic conservation laws
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