The Existence And Uniform Decays To The Cauchy Problem Of The Dissipative Boussinesq Equation

We study the existence and uniform decays to the Cauchy problem of the .issipative Boussinesq equation in the case that the initial data are small, where u0, u1 are initial data functions,the nonlinear terms are f(u),and β>0, η> 0.First, we apply Fourier transform and Duhamel principle to its equivalent in-tegral equation, second we get the the decay estimate and existence of the solution to the linearized equation, then using the contraction mapping principle and inte-gral estimates we give the existence and uniform decays to the cauchy problem of the nonlinear equation in the case that the initial data are small.