Global Well-Posedness for a Nonlinear Wave Equation with p-Laplacian Damping

This dissertation deals with the global well-posedness of the nonlinear wave equation utt − Δu − Δput = f (u) in Ω × (0,T), {u(0), ut(0)} = {u0,u1} ∈ H10 (Ω) × L 2 (Ω), u = 0 on Γ × (0, T ), in a bounded domain Ω ⊂ ℜ n with Dirichlét boundary conditions. The nonlinearities f (u) acts as a strong source, which is allowed to have, in some cases, a super-supercritical exponent. Under suitable restrictions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of local solutions. We also provide two types of restrictions on either the power of the source or the initial energy that give global existence of solutions. Finally, we give decay rates for the energy of the system for suitable initial data, with the proof of the decay and decay rates the focus of the talk. Adviser: Mohammad A. Rammah