On a hyperbolic wave equation in unbounded domains

Abstract

The aim of this thesis is the study of the quasilinear damped wave equation of Kirchhoff’s type with variable diffusion coefficient in all of R^N. For the functional analysis of the time dependent problem, we make use of the homogeneous Sobolev spaces and of the generalized Sobolev embeddings, followed by the preceding studies of the Kirchhoff’s type problem. In strong connection with the corresponding natural phenomena, we obtain results concerning the local (unique) existence of the solutions using the Faedo-Galerkin approximation and the Banach Fixed-Point Theorem. We also prove the global existence and energy estimates of the solutions using the method of the modified potential well. We complete our study with the blow-up analysis of the solutions for initial data of negative energy using the concavity method, where for the discrete case (a=2) we prove the modification of the upper-bound of the time T.