A Generalized Sixth Order Thin Film Equation

As a very important class of nonlinear diffusion equations, thin film equations have been widely researched in recent twenty years. Many mathematicians did a lot on this field. Thin film equation comes from the diffusion phenomena on the surface of crystal. Similar models were found in other physical phenomena, such as competition, exclusion of biological groups and the migration of the river bed. Thin film equation of higher order was first studied by F. Bernis and A. Friedman [1]. They did research on the higher order equation as followsThey got the existence of generalized solution and the positivity of initial value led to the positivity of solutions.J. C. Flitton, J. R. King [3] examined initial boundary value problems for a sixth order degenerate parabolic equation of the formIt arises when considering the motion of a thin film of viscous fluid driven by an over-lying elastic plate. Analytical and numerical methods were exploited to characterize the solutions, which turn out to be rather sensitive to the value of n. The authors mainly considered the three boundary conditions.J. D. Evans, V. A. Galaktionov and J. R. King [12] studied blow-up behavior of solutions of the sixth-order thin film equation For the first critical exponent p= p0= n+1+(?),n∈(0, (?)) the free-boundary problem with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface allowed a countable set of continuous branches of radially symmet-ric self-similar blow-up solutions. They also studied the Cauchy problem in RN x R+ and showed that the corresponding self-similar family{uk(x, t)} is countable, the com-pact support were oscillatory near the interfaces. The three scientists [13]also studied asymptotic large time behavior of global solutions of (3). The free-boundary prob-lem with zero-contact-angle, zero-moment and zero-flux conditions at the interface allowed continuous families (branches) of radially symmetric self-similar solutions.John W. Barrett, Stephen Langdon,Robert Nurnberg [14] considered a finite ele-ment approximation of the sixth order nonlinear degenerate parabolic equationIn this paper the author developed and analyzed a fully practical scheme that works in all space dimensionsThey proved its well-posedness, derived stability bounds,established convergence of our approximation and presented some numerical computations in one and two space dimensions.M. Xu and S. Zhou [17] considered the following problem with p> 1 in RN With the Stelov averages they had energy estimation By employing the difference and variation methods they had the existence of weak solutions. With the interpolation inequality they had some regularities of the solutions.Zhang Chao [22] did some researches on the equation They had the solutions{uh} of time discrete problem through the minimum of the functional. Then they estimated them and had the existence of the solutions. Moreover they used the test function to get the uniqueness. Under the prior estimation and intercalate inequalities they showed some regularities of solutions.Lidia Ansini and Lorenzo Giacomelli [4] studied the doubly nonlinear thin film equation First they changed it into a Neumman boundary problem Using Galerkin method, energy estimate and monotonous method they had the exis-tence of weak solutions in boundary domain. Then let domain (-a, a) tend to infinity, so the long time behavior was derived.C. Liu, J. Yin, H. Gao [23] consider the equation It is relevant to capillary driven flows of thin films of power-law fluids. Here the authors only considered the conditions in R2. First they used the method of time discrete to construct the approaching solution, then did some estimations and let them to the infinity, so the solution u was derived. With the energy estimation, poincar6 inequality and Friedrichs inequality they had the uniqueness and long time behavior.Thin film with nonstandard growth conditions equations are the new topic in re-cent years. It arouses much interest with the development of elastic mechanics, electro rheological fluid dynamics and image processing, etc.p(χ)-growth conditions can be regarded as a very important class of nonstandard growth conditions. The difficulties are the nonlinear and short of maximum principle.S. N. Antontsev and S. I. Shmarev [24] studied the Dirichlet problem for the degenerate parabolic equationIt describes a motion of an ideal baro tropic gas through a porous medium. Because of the mass conservation lawρt+div(ρv)= 0, and Darcy law V=-k(x)Vp, it had the formThe paper only involved the simple one, and the authors showed the existence, unique-ness of solutions, and the long time behavior. C.Liu,T. Li[6]studied the equation (12)existed a uniqueness weak solution and the long time behavior:When 0≤ρ(χ)∈C2(Ω),thenC.Zhang and S.Zhou[7]considered the equation with variable exponent If u0∈H01(Ω),p(χ)∈C+((Ω)),and satisfied log-Holder inequality then they had the uniqueness of weak solutions.If p-＞max{1,(?)},then they got the long time behavior as follows Then they constructed the solutions and because of energy estimate they got the unique-ness.Base on the researches, I do some researches on the equation This is a sixth order thin film equation with variable exponent. I get the existence and uniqueness of solutions. Also I have the long time behavior.