zbMATH — the first resource for mathematics

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.
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