Long Time Behavior for Some Higher Order Partial Differential Equations

Abstract

The paper mainly concerns some fourth order parabolic equations which have important in thin film theory and phase transformation theory. The long time behavior for a fourth order parabolic equation with Dirichlet boundary conditions is studied including the one-dimensional and higher-dimensional space. By using entropy method and choosing proper multiplier functions, the large time decay rate is gained and the solutions tend to its steady state solution as the time tends to infinity.