Global Solution and Regularizing Properties on a Class of Nonlinear Evolution Equation

In this paper we will consider the equation[formula]where[formula]The initial value problem is proved to be locally well posed for initial data taken inD(A2)×D(A3/2) and globally well posed for small data, in this case we also show the exponential decay of the solution as time goes to infinity. The main result of this paper is to prove that the solution has the smooting effect property on the initial data. This means that, if the initial data belongs toD(A2)×D(A3/2) then the solutionubelongs toC∞(]0, +∞[; D(Ak)) k , providedM,N, andRareC∞-function.