SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SUPERCRITICAL SOURCES

Authors

Yanqiu Guo, University of Nebraska-LincolnFollow

Date of this Version

Spring 2012

Document Type

Article

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Mohammad A. Rammaha. Lincoln, Nebraska: March, 2012

Copyright (c) 2012 Yanqiu Guo

Abstract

We consider the local and global well-posedness of the coupled nonlinear wave equations

u_tt – Δu + g₁(u_t) = f₁(u, v)

v_tt – Δv + g₂(v_t) = f₂(u, v);

in a bounded domain Ω subset of the real numbers (Rⁿ) with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. The nonlinearities f₁(u, v) and f₂(u, v) are with supercritical exponents representing strong sources, while g₁(u_t) and g₂(v_t) act as damping. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H¹(Ω) × L²(∂Ω) with boundary data from L²(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this dissertation are non-dissipative and are not locally Lipschitz from H¹(Ω) into L²(Ω) or L²(∂Ω). By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system.

Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good" part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. Moreover, we prove a blow up result for weak solutions with nonnegative initial energy. Finally, we establish important generalization of classical results by H. Brézis in 1972 on convex integrals on Sobolev spaces. These results allowed us to overcome a major technical difficulty that faced us in the proof of the local existence of weak solutions.