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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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