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Global existence of small solutions to the initial value problem for nonlinear thermoelasticity. (English) Zbl 0725.35065
The authors consider the initial value problem in three-dimensional nonlinear thermoelasticity (the coupling of a hyperbolic system to a parabolic system) described by $(*)\quad U_{tt}-D'SDU+\gamma \nabla \theta =f^ 1(\nabla U,\nabla^ 2U,\theta,\nabla \theta);\quad \theta_ t-\kappa \Delta \theta +\gamma \nabla 'U_ t=f^ 2(\nabla U,\nabla U_ t,\nabla^ 2U,\theta,\nabla \theta,\nabla^ 2\theta)\text{ in } {\mathbb{R}}^ 3\times (0,\infty)$ $U(x,0)=U^ 0(x),\quad \theta (x,0)=\theta^ 0(x)\text{ on } {\mathbb{R}}^ 3$ where $$\kappa$$ $$(>0)$$, $$\gamma$$ ($$\neq 0)$$ are constants, $$U=(U_ 1,U_ 2,U_ 3)$$, $$\theta$$ are the displacement and the temperature respectively, and where D is a differential (6$$\times 3)$$ matrix operator and S is a positive definite (6$$\times 6)$$ marix (consisting of Lamé constants). Further $$f^ 1=(f^ 1_ 1,f^ 1_ 2,f^ 1_ 3)$$ and $$f^ 2$$ are the difference from the general state to the initially isotropic one expressed in terms of the nonlinearity. They assume that $$f^ 1,f^ 2$$ are smooth and have the nonlinearity of order 2 satisfying $$f^ 1(\nabla U,\nabla^ 2U,0,0)=0$$ and $$f^ 2(\nabla U,\nabla U_ t,\nabla^ 2U,0,0,0)=0.$$
Then they prove the existence and uniqueness of global smooth solutions to the problem (*), transforming it to an integral equation, using the $$L^ p$$-L$${}^ q$$ estimate of the local solutions in some Sobolev spaces and applying the usual continuation arguments, in case that the initial data are smooth and small. The asymptotic behaviour of solution as $$t\to \infty$$ is also described.
Reviewer: T.Kakita (Tokyo)

##### MSC:
 35M20 PDE of composite type (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 35K45 Initial value problems for second-order parabolic systems 35L55 Higher-order hyperbolic systems 74B99 Elastic materials
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