Global well -posedness for systems of nonlinear wave equations

Abstract

This dissertation deals with the global well-posedness of the system of nonlinear wave equations [special characters omitted]in a bounded domain Ω ⊂ [special characters omitted], n = 1, 2, 3; with Dirichlét boundary conditions. The nonlinearities f₁(u, v) and f₂(u, v) act as a strong source in the system and the exponents of velocities are restricted to the range 0 < m, r ≤ 1. The non-critical case 0 < m, r < 1 and the critical case m = r = 1 are analyzed in Chapter 2 and Chapter 3, respectively. Under some restrictions on the parameters in the system we obtain several results on the existence of local solutions, global solutions, and uniqueness. In addition, we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent the source term is more dominant than the exponents of both damping terms.