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On a nonlinear generalized thermoelastic system with obstacle. (English) Zbl 1254.35153
The authors study the $$n$$-dimensional semilinear thermoelastic system with unilateral boundary conditions (Signorini’s conditions), which describes the motion of an thermoelastic body in contact with a rigid obstacle without attrition. The authors first reformulate the original contact problem as a variational inequality problem and give an approximate variational problem by introducing a penalty term. Then, by careful energy estimates and the compactness argument, the authors prove the global existence of weak solutions. In the one-dimensional case, the authors are able to exploit the one-dimensional features, such as the better regularity under Signorini’s boundary conditions, to show that the solutions decay exponentially as time goes to infinity.
##### MSC:
 35L86 Unilateral problems for nonlinear hyperbolic equations and variational inequalities with nonlinear hyperbolic operators 35B40 Asymptotic behavior of solutions to PDEs 35L53 Initial-boundary value problems for second-order hyperbolic systems 35L71 Second-order semilinear hyperbolic equations 74F05 Thermal effects in solid mechanics 74M15 Contact in solid mechanics