The Cauchy Problem For A Class Of Shallow Water Wave Equations

In this thesis, we discuss the Cauchy problem for a class of Shallow Water Wave equations arising from modern mechanics and physics. More precisely, we study the existence and uniqueness of global solutions and blow-up phenomena to the Cauchy problem of one dimensional singularly perturbed Boussinesq-type equation, the existence and uniqueness of local classical solution and blow-up phenomena to the Cauchy problem of higher dimensional singularly perturbed Boussinesq-type equation, and the existence and uniform decays to the Cauchy problem of damping term of singularly perturbed Boussinesq-type equation.The main contents of the work are organized as follows:In Chapter 1, we introduce the physical background and research status to the equations discussed in this thesis. Then we give a list of main results of the paper.In Chapter 2, we mainly discuss the Cauchy problem of one dimensional singularly perturbed Boussinesq-type equation. First, making use of the Galerkin method and energy estimates,we obtain the existence and uniqueness of solutions to the corresponding auxiliary problem. Second, taking advantage of variable substitution, we get the existence and uniqueness of the generalized solution and global classical solution to the Cauchy problem of one dimensional singularly perturbed Boussinesq-type equation. Finally, by using method of the concavity method, we give some sufficient conditions for blow-up of the solution in a finite time.In Chapter 3, we mainly study the Cauchy problem of higher dimensional singularly perturbed Boussinesq-type equation. First, making use of the Galerkin method and energy estimates,we show the existence and uniqueness of local classical solution to the corresponding periodic boundary problem. Second, by the sequence of the periodic boundary problem and the corollary of resonance theorem, we prove that the existence and uniqueness of local classical solution to the Cauchy problem of higher dimensional singularly perturbed Boussinesq-type equation. Finally, by using method of the concavity method, we give some sufficient conditions for blow-up of the solution in a finite time.In Chapter 4, we mainly concern the Cauchy problem of damping term of singularly perturbed Boussinesq equation. First, applying the Fourier transformation and the Duhamel principle,the Cauchy problem for the corresponding linear Boussinesq equation is written an equivalence integral equation. Second, by symbolic analysis methods, we establish existence and the decay estimate of the solution to the corresponding linear Boussinesq equation. Finally, using the contraction mapping principle and integral estimates we give the existence and uniform decays of the Cauchy problem of damping term of singularly perturbed Boussinesq-type equation in the case that the initial data are small.