On a dynamic boundary condition for a class of nonlinear parabolic equations

The main objective of this paper is to study a weak solution for a certain parabolic problem, defined on an open subset of R^n with smooth boundary. By using the Galerkin approximation and a family of potential wells, we obtain the existence of a global solution and finite time blow-up under some suitable conditions. On the other hand, the results for asymptotic behavior for certain solution with positive initial energy are also given.