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Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity. (English) Zbl 0723.35044
In this paper existence global in time and uniqueness of solutions of the equations of nonlinear thermoelasticity is proved. The solution is defined for all (x,t)\(\in [0,\infty)\times [0,\infty)\), and at the boundary \(x=0\) the displacement gradient and the temperature satisfy homogeneous Dirichlet boundary conditions. To prove global existence the local solution is continued using decay estimates for the linearized equations obtained by means of Fourier sine and cosine transformations. This procedure also yields the decay estimates \[ \| u(t)\|_{L^{\infty}}=O((1+t)^{-1/2}),\quad \| D^ 1u(t)\|_{L^{\infty}}=O((1+t)^{-1}), \] \[ \| u(t)\|_{L^ 2}=O((1+t)^{-1/4}),\quad \| D^ 1u(t)\|_{L^ 1}=O((1+t)^{-1/2}) \] for the solution u(t).

MSC:
35L50 Initial-boundary value problems for first-order hyperbolic systems
74B20 Nonlinear elasticity
35L60 First-order nonlinear hyperbolic equations
35Q72 Other PDE from mechanics (MSC2000)
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