Well-posedness Research On Generally Boussinesq Equations

This work carries out qualitative studies on several generalized Boussinesq equations by using potential wells method. The threshold results (sharp conditions) for global existence of solutions to the problems are given. Consequently, this work further enriches and develops potential well theory.Firstly, this paper studies the Cauchy problem for a class of generalized Boussinesq equa-tion. As we know, the generalized Boussinesq-type equation were introduced to describe the motion of water wave with small-amplitude long waves, which are frequently used in computer models for the simulation of water waves in shallow seas and harbors. By applying Fourier transformation the energy conservation is obtained. And for both positive energy and non-positive energy the thesis gives some properties for potential wells and derives the vacuum isolation of solutions. Based on above derived properties, this thesis proves the sharp condition of global-in-existence and nonexistence of solutions to above problems. At the critical energy level, the similar sharp condition of global well-poseness of above problems is also obtained. Further, this thesis considers the Cauchy problems for a class of the multi-dimensional general-ized Boussinesq equations and a class of the damped multidimensional generalized Boussinesq equations. Under some proper assumptions and by using potential wells method, we prove the existence and nonexistence of global weak solution without establishing the local existence theory.