A Generalized Thin Film Equation With Convection

Thin film equations, as a class of important higher order nonlinear diffusion equations, come from the diffusion phenomena on the surface of crystal. In the last decades, especially in recent twenty years, the study in this direction attracts a large number of mathematicians both in China and abroad. Besides, the thin film equations with convection have drawn great interest of many researchers [18][23]-[25].The study of thin film equations began at 1990, F. Bernis and A. Friedman [13] considered the following initial-boundary value problem firstlywhere n≥1. The authors showed the existence of generalized solution for the problem. Moreover, the authors also studied the regularity and monotoncity of support of the solutions of the problem.The thin film equationmodels thin viscous flows on solid surfaces. Many researchers have studied the thin film equation widely and obtained the existence of compactly-supported weak solutions and nonnegtive solutions, see [14]-[16]. Yin and Gao [17] disscussed the finite speed of propagation of perturbations of Cauchy problem for 0 < n < 1. F. Ber-nis [22] also proved that the interface is Holder continuous, when 1/23 + uⁿ, n∈(0,3), n accounts for different forms of the slip condition at the liquid-solid interface. The equation described the capillarity driven evolution of a Newtonian liquid film over a dry solid substrace. The authors replaced m(u) by m∈C([0,∞)), which is increasing, and m{u)～uⁿ as u↓0 for some n∈(0,3). The authors proved that the problem (2) admits weak solutions and satisfies u(x,0) = u₀(x) for any nonnegative u₀ with F(u₀) <∞, whenθ_e > 0,Ω= (-α,α),α> 0, whereJ. R. King [4] extended the above equations and derived the thin film equation firstlywhere u(x, t) denotes the height from the surface of the oil to the surface of the solid. The equation is a typical higher order equation and has a sharp physical background and a rich theoretical connotation. It is relevant to capillary driven flows of thin films of power-law fluids. J. R. King [4] studied the Cauchy problem of the equation in one-dimension, exploiting local analysis about the edge of the support and special closed form solutions such as traveling waves, separable solutions and instantaneous source solutions, see also [2]. Ansini and Giacomelli [1] studied the doubly nonlinear thin film equation (3) in one-dimensioncal case, and obtained the existence of solutions to the free-boundray problem on a multi-step approximating procedure. S. I. Betelu and M. A. Fontelos [3] investigated the spreading of thin liquid films of power-law rtheology They contracted an explicit travelling wave solution and similarity solutions. They showed that when the nonlinearity exponent for the rheology is larger than one, the problem admits solutions with compact support and moving fronts.When N = 2 and p > 2, Liu, Yin and Gao [5] investigated the existence, uniqueness and asymptotic behavior of generalized solutions for the equation (3) with n = 0 to initial-boundary problem. Xu and Zhou [6] extended the equation (3). They studied the following thin film equationand established the existence of weak solutions. In addition, they presented some results on the regularity.Felix Otto and Michael Westdickenberg [24] studied the thin film equation with convectionThe equation models the dynamics of a thin film of fluid on a vertical plane under the influence of gravity. It has the property that the thin film height u^εstays nonnegative for positive times if the initial data is. They proved that, asε→0, the functions u^εconverged to the entropy solution of the scalar conservation law.Lorenzo Giacomelli and Andrey Shishkov [18] studied the thin film equation with convectionThe equation arises in the description of the evolution of the height y= u(x, t) of a Newtonian liquid film over a solid surface (y = 0) in lubrication approximation. Authors studied the finite speed of propagation property for zero contact-angle solutions of above problem. Author' approach is based on energy/entropy methods shaped upon suitable existensions of Stampacchia Lemma. In the case of strong slippage, they obtained bounds in term of the initial mass for both the fast and slow interfaces, and for both short and large times, which author expect to be sharp. In the case of weak slippage, obtain partial results for short times, which include a quantitive bound for moderate growths of the convective term.In this paper, we try to extend the equation (5) with a convection term. We consider the following fourth-order parabolic initial-boundary value problem:whereΩ(?)(?)^N is a bounded domain with smooth boundary (?)Ω, andα, p be a positive number withα, p > 1. The cylinder Q =Ω×(0, T], the lateral surfaceΓ= (?)Ω×(0, T],βis a constant vector of (?)^N. Since the principal part in the equation is nonlinear which bring some difficulties somehow. This equation is something quite like the p-Laplacian equations, but many methods used in the p-Laplacian equation such as the methods based on maximum principle are no longer valid for this equation. Therefore, new tools to study problem (7) are needed. Under some assumptions, the existence of weak solutions is established by the difference, variation and monotonicity methods. The uniqueness and regularity of weak solutions are also discussed.