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On some quasilinear hyperbolic-parabolic initial boundary value problems. (English) Zbl 0706.35098
The paper concerns the initial boundary value problem for the following coupled hyperbolic-parabolic system \[ (1)\quad \partial^ 2_ tu-Div S(\nabla u,\theta)=f, \] \[ (\theta +T_ 0)\{a(\nabla u,\theta)\partial_ t\theta -tr[(\partial_{\theta}S(\nabla u,\theta))^ T(\partial_ t\nabla u)]\}+Div q(\nabla u,\theta,\nabla \theta)=g. \] Here \(u=(u_ 1,u_ 2,u_ 3)^ T\) is a vector function, \(\theta\) is a scalar function depending on \(t\in {\mathbb{R}}^+_ 0\), \(x\in \Omega \subset {\mathbb{R}}^ 3\) with \(\Omega\) being a bounded domain or an unbounded domain with bounded complement and smooth boundary \(\partial \Omega\). S, a, q are matrix-valued, scalar-valued and vector valued functions, respectively, \(T_ 0>0\) is a constant. The initial data for (1) are given by \[ (2)\quad u(0,x)=u^ 0(x),\quad \partial_ tu(0,x)=u^ 1(x),\quad \theta (0,x)=\theta^ 0(x) \] with Dirichlet boundary conditions \[ (3)\quad u(t,)|_{\partial \Omega}=0,\quad \theta (t,)|_{\partial \Omega}=0\text{ on } \partial \Omega. \] The system (1)-(3) arises in thermoelasticity, where u is the displacement vector, wile \(\theta\) is the temperature difference.
The main goal of the work is to prove the existence of a local (in time) classical solution for (1)-(3).
The proof is based on a suitable modification to the Kato approach. References to the recent works of Kawashima and Matsumura [Mixed problems for symmetric hyperbolic-parabolic systems, manuscript (1989)] and Y. Shibata [On a local existence theorem for some quasilinear hyperbolic-parabolic coupled systems with Neumann type boundary condition, manuscript (1989)] treating a similar nonlinear hyperbolic-parabolic problem are given too.
Reviewer: V.Georgiev

MSC:
35M20 PDE of composite type (MSC2000)
74B20 Nonlinear elasticity
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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