Mathematical analysis to a nonlinear fourth-order partial differential equation

✍ Scribed by Bo Liang

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 275 KB
• Volume: 74
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2011.03.035

✦ Synopsis

The paper first study the steady-state thin film type equation

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 → 0 for the positive weak solutions are given. Second, the parabolic equation

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 → ∞. 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.