zbMATH — the first resource for mathematics

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

35M20 PDE of composite type (MSC2000)
74B20 Nonlinear elasticity
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
[1] Sobolev Spaces, Academic Press, New Yourk, 1975. · Zbl 0314.46030
[2] Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965.
[3] Dafermos, Arch. Rat. Mech. Anal. 87 pp 267– (1985)
[4] Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity, Bonner Mathematische Schriften 192. Bonn. 1989.
[5] and , On some quasilinear hyperbolic-parabolic initial boundary value problems, SFB 256 Preprint, No. 42, University of Bonn, 1988.
[6] ’Abstract differential equations and nonlinear mixed problems’, Center for Pure and Appl. Math. Report, University of California, Barkeley. Published in Fermi Lectures, Scuola Normale Sup., Pisa, 1985.
[7] ’Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics’, Thesis, Kyoto University, 1983.
[8] and , ’Mixed problems for symmetric hyperbolic-parabolic systems’, manuscript’, 1989.
[9] Kawashima, Proc. Japan Acad. Ser. A 63 pp 243– (1987)
[10] Li, Chin. Ann. Math. 8B pp 252– (1987)
[11] Milani, Boll. U.M.I. 2-B pp 641– (1983)
[12] Racke, Math Meth. in the Appl. Sci. 10 pp 517– (1988)
[13] Racke, Math. Zeit schrift (1990)
[14] ’On a local existence theorem for some quasilinear hyperbolic-parabolic coupled system with Neumann type boundary condition’, manuscript, 1989.
[15] Shibata, Tsukuba J. Math. 13 pp 223– (1989)
[16] Shibata, Nonlinear Anal., T.M.A. 11 pp 335– (1987)
[17] Slemrod, Arch. Rat. Mech. Anal. 76 pp 97– (1981)
[18] Slobodezkii, Dokl 123 pp 616– (1958)
[19] Vol’pert, Math. USSR Sbornik 16 pp 517– (1972)
[20] Partielle Differentialgleichungen, B. G. Teubner, Stuttgert, 1982. · doi:10.1007/978-3-322-96662-9
[21] Zheng, Chin. Ann. Math. 4B pp 443– (1983)
[22] Zheng, Sci. Sinica, Ser A. 30 pp 1133– (1987)
[23] Chrzgszczyk, Arch Mech. 39 pp 605– (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.