An in-depth Analysis of Certain Boussinesq Issues

Research on Boussinesq equations has grown considerably over the past three decades due to their adaptability in explaining nonlinear phenomena, particularly in the analysis of water wave dynamics. These equations are widely used in various fields, including the study of oceanic waves and coastal engineering. This book explores the existence and finite-time blow-up of solutions in the Cauchy problem associated with a generalized Boussinesq equation. The study of the generalized Boussinesq equation and its solutions, including existence, non-existence, and blow-up, is of great interest. The book investigates solutions in both bounded and unbounded domains, as well as in the presence of logarithmic nonlinearity. Local solutions are proven to exist and be unique in both cases. Under certain restrictions on the initial data of our problems, we establish the existence and uniqueness of global solutions, as well as the possibility of blowing up solutions in finite time.