Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation ∇⋅(un|∇Δu|q−2∇Δu)−δumΔu=f(x,u)∇⋅(un|∇Δu|q−2∇Δu)−δumΔu=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 δ→0δ→0 for the positive weak solutions are given. Second, the parabolic equation ut+(un|uxxx|q−2uxxx)x−δumuxx=0ut+(un|uxxx|q−2uxxx)x−δumuxx=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=2q=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→∞t→∞. 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.