On the well-posedness of a nonlinear pseudo-parabolic equation

In this paper we consider the Cauchy problem for the pseudo-parabolic equation:partial derivative partial derivative t (u + mu(-Delta)(s1)u) + (-Delta)(s2)u = f(u), x is an element of Omega, t 0.Here, the orders s(1), s(2) satisfy 0 s(1) not equal s(2) 1 (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source f is globally Lipschitz. In the case when the source term f satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given.