On a nonlinear generalized thermoelastic system with obstacle. (English) Zbl 1254.35153

Oliveira, M. L.; Clark, M. R.; Marinho, A. O.

Adv. Differ. Equ. 17, No. 1-2, 75-103 (2012).

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.

Reviewer: Song Jiang (Beijing)

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

Keywords:

contact problem; global existence; exponential decay; unilateral boundary conditions; Signorini’s conditions; penalty term