Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation del . (u(n)vertical bar del Delta u vertical bar(q-2)del Delta u) - delta u(m)Delta u = f(x, u) with Navier boundary conditions in multidimensional space. By the truncation method, a fixed point argument and some energy estimates, the existence and asymptotic limit delta -> 0 for the positive weak solutions are given. Second, the parabolic equation u(t) + (u(n)vertical bar u(xxx)vertical bar(q-2)u(xxx))(x) - delta u(m)u(xx) = 0 with a Navier boundary in one-dimensional space is researched. The existence is obtained by applying a semi-discrete method for the time variable and solving the corresponding elliptic problem. The uniqueness is shown for q = 2 depending on an energy estimate. In addition, the iteration relation of the semi-discrete problem gives an exponential decay result for the time t -> infinity. The thin film equation, which is usually used to describe the motion of a very thin layer of viscous in compressible fluids along an inclined plane, is a class of nonlinear fourth-order parabolic equations and the maximum principle does not hold directly. For applying the classic theory of partial differential equation, the paper transforms the fourth-order problem into a second-order elliptic-elliptic system or a second-order parabolic-elliptic system. (C) 2011 Elsevier Ltd. All rights reserved.