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Global existence and asymptotic behavior of smooth solutions in one- dimensional nonlinear thermoelasticity. (English) Zbl 0682.73007

Bonner Mathematische Schriften, 192. Bonn: Univ. Bonn, Math. Inst., Thesis 1988. 60 p. DM 7.50 (1989).
Let \(u\), \(v\) and \(\theta\) denote, respectively, the displacement gradient velocity and temperature difference of a one-dimensional nonlinear dynamic coupled thermoelasticity for a semi-space \(B = (0,\infty)\), \(t\geq0\) (\(t\)=time). The following initial boundary value problem is studied. Find a triplet \((u,v,\theta)\) on \(\bar B \times [0,\infty)\) that satisfies the field equations \[ \left\{ \begin{aligned} &u_ t - v_ t = 0,\\ &v_ t - a(u,\theta)u_ x + b(u,\theta)\theta_ x = 0, \quad\text{on }B \times (0,\infty)\\ &c(u,\theta)\theta_ t + b(u,\theta)v_ x - d(\theta,\theta_ x)\theta_{xx} =0, \end{aligned} \right\} \tag{1} \] the initial conditions (2) \(u(\cdot,0) = u_0\), \(v(\cdot,0) = v_0\), \(\theta (\cdot,0) = \theta_0\) on \(B\) and the boundary conditions (3) \(u |_{\partial B} = \theta |_{\partial B} = 0\), \(t\geq 0\).
Here \(a\), \(b\), \(c\) and \(d\) in (1) are prescribed functions characterizing the nonlinear thermoelastic semi-space, and \(u_0\), \(v_0\), \(\theta_0\) are prescribed disturbances; and subscripts in (1) denote partial derivatives.
The author proves that under suitable smoothness hypothesis on the material functions \(a\), \(b\), \(c\) and \(d\), and under suitable compatibility conditions imposed on “small” initial data \(u_0\), \(v_0\), \(\theta_0\), there exists a unique global smooth solution to the nonlinear problem (1)–(3), and this solution reveals a decay behaviour as \(t\to\infty\). The proof is divided into three parts. Firstly, a study of existence and uniqueness of a smooth solution to an associated linear initial boundary value problem is presented. Next, the study is used to prove a local existence and uniqueness theorems for the nonlinear problem (1)–(3). Finally, it is shown how the main theorem follows from the local results. The decay behaviour of the solution to the nonlinear problem is obtained by using, among others, results on time-decay rates of a solution to the associated constant-coefficient linear thermoelastic problem due to R. Leis [Initial boundary value problems in mathematical physics (1986; Zbl 0599.35001)].
Reviewer: J.Ignaczak

MSC:

74F05 Thermal effects in solid mechanics
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74A15 Thermodynamics in solid mechanics
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0599.35001