Existence and asymptotic behavior of solutions to a thin film equation with Dirichlet boundary

This paper is devoted to studying a fourth-order parabolic equation ut+ε(unuxxx)x−δuxx=0ut+ε(unuxxx)x−δuxx=0 with Dirichlet boundary. By transforming the fourth order problem into an elliptic-parabolic system and applying a semidiscretization method, the existence, uniqueness and regularization of solutions are obtained. By introducing an energy functional through the semi-discrete problem, we use the iteration method to show that the solutions of the developing equation exponentially converge to a constant steady state solution as the time variable t→∞t→∞. Finally, we get the result of the asymptotic limit δ→0δ→0.