The upper estimates of the functions that satisfy some differential integral inequality are established in this paper. We obtain the uniform estimates of maximum of the solutions for a general class of parabolic equations and extend some known results.

We investigate the asymptotic behaviour of solution of the initial-boundary value problem for the equation, which describes the evolution of graphs with speed depending on curvature function of current graphs. In contrast with [1-5] and others, in which the speed of flow of graphs is directly proportional to the curvature of graphs, here we discuss is that the speed is inversely proportional to the curvature of graphs, and our methods is different from theirs.

The existence of the attractors of the periodic initial value problem for the Landau-Lifshitz-Maxwell equations in one and two space dimensions is proved. We also get accurate estimates of the upper bounds of Hausdorff and fractal dimensions for the attractors by means of uniform a priori estimates for time and Lyapunov functional method.

This paper is a continuation of the previous paper in the Journal of Partial Differential Equations [1]. We derive in this paper the renormalized energy to further determine the locations of vortices in some case for the variational problem related to the superconducting thin films having variable thickness.

Some abstract results about convergences for a sequence of parameterized nonempty closed convex sets, in the Mosco sense as well as in the sense of the local gap have been proved. By using these results, the convergences of some sequences of closed convex sets by certain general concrete structures arc discussed.