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

Authors

Zahava Wilstein, University of Nebraska-LincolnFollow

Date of this Version

5-2011

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Mohammad A. Rammaha. Lincoln, Nebraska: May, 2011
Copyright 2011 Zahava Wilstein

Abstract

This dissertation deals with the global well-posedness of the nonlinear wave equation
u_tt − Δu − Δ_pu_t = f (u) in Ω × (0,T),
{u(0), u_t(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. Rammaha