Existence Of Solutions Of Two Classes Of Nonlinear Evolution Equations

In this paper we study the initial boundary value problem of a class of semilinear pseudoparabolic equations and the Cauchy problem of the strongly damped wave equations.Firstly, the existence of the W^1,2 and W^1,p solution of a class of semilinear pseudoparabolic equations are proved. It is proved that if f∈C',f' is bounded above and satisfies the growth condition, then for any T > 0, the problem admits a unique solution. Secondly, using the potential well method we studied the Cauchy problem of a class of nonlinear wave equations, where the semilinear term f(u) satisfies certain growth conditions. It isproved that if the initial energy is less than the depth of potential well, the problem has a global weak solution which belongs to the potential wells. Finally, using the family of potential wells method we study the invariant sets and vacuum islolating of solutions for a class of nonlinear wave equations. It is proved that the problem has invariant sets and vacuum islolating of solutions when the initial energy is less than the depth of potential well.