Existence and Stability of Some Nonlinear Boundary Value Problems

Abstract

In this thesis, we investigate dynamic contact problems, with or without friction, between a deformable body and a rigid foundation. The material behavior is modeled by a nonlinear viscoelastic or thermo-viscoelastic law with long memory effects. Unlike quasi-static models, the system’s acceleration is retained to capture the fully dynamic nature of the pro cess. The contact is governed by Tresca’s friction law, simplified Coulomb’s friction Law and a rigid-plastic foundation. We establish both local and global existence and uniqueness of weak solutions. The analytical approach combines operator theory, Galerkin schemes, fixed-point arguments, and nonlinear semigroup techniques. Under suitable assumptions on the problem data, we also prove the exponential stability of the solutions using energy methods.