The Cauchy Problem For The Double Damped Boussinesq Equation

This paper considers the Cauchy problem for the sixth-order Boussinesq equation with double damped terms in Sobolev space Hs. The existence and uniqueness of the local solutions and global solutions and asymptotic behavior are established under the assumptions of initial data. Furthermore, the blow-up result for a class of Boussinesq equation with strong damped term will be discussed. The main results include the following three aspects:In Chapter two, we firstly study the Cauchy problem to the linear equation, then we obtain the existence and uniqueness of local solutions for the nonlinear Cauchy problem by means of the contraction mapping theorem. Secondly, some energy inequalities are obtained by utilizing an estimate for the local solution, then the existence and uniqueness of the global solution are proved.In Chapter three, using a multiplier method and some important inequalities, we obtain integral estimates and discuss the asymptotic behavior of the solution, it is proved that the global solution decays to zero exponentially as the time approaches infinity.In Chapter four, we will use the concavity method to discuss blow-up of the solution to the problem and obtain the sufficient conditions of blow-up of the solution.