Existence Analysis and Discretization of Nonlinear Partial Differential equations

Abstract

This thesis investigates three representative wave equations, including one lin- ear and two nonlinear waves equations, with viscoelastic effects under various bound- ary conditions such as dynamic, Neumann, and Dirichlet conditions. The global ex- istence and uniqueness of weak solutions are established using the Faedo-Galerkin method. Energy techniques and the multiplier method are employed to analyze the long-time behavior, leading to exponential energy decay under suitable assump- tions on the nonlinear and viscoelastic terms. For the numerical approximation, the finite element method is used for spa- tial discretization, resulting in semi-discrete schemes, while implicit time-stepping methods construct fully discrete approximations. Optimal a priori error estimates are derived for both semi-discrete and fully discrete schemes, ensuring stability and convergence. Numerical experiments validate the theoretical results and demonstrate the ef-ficiency and accuracy of the proposed methods. Overall, this thesis provides a uni-fiedtheoretical and numerical framework for nonlinear partial differential equa- tions with strong damping and various boundary conditions.