Nonlinear Higher-order Parabolic Partial Differential Equations

The motion of a liquid under the influence of surface tension, such as a fluid coating a solid surface, is a phenomenon we experience every day. The underlying dynamics of such phenomenon depend heavily on the surface chemistry. Many industrial processes rely on the ability to control these interactions. And such contact line problem has generated some very interesting and difficult mathemat-ical problems associated with the model equations. In this dissertation, the thin viscous flows over an inclined plane is considered. It is proved that this model has finite speed of propagation for the solutions. Furthermore, the wellposedness of the weak solution to the sixth-order nonlinear uniformly parabolic equation which arises in the description of thin, epitaxially strained films in the presence of wetting intersections with the substrate is obtained. And the finite-time blow-up is shown. The main achievements contained here are as follows:1. Considering the thin liquid film equationthis is a fourth-order nonlinear degenerate parabolic equation which arises in the description of thin viscous flows over an inclined plane with slopeθ∈(0,π/2), where the function h(x, t) represents the thickness of the film,α> 0 andβ> 0 are constants proportional to g sin 9 and g cosθrespectively (g is the gravitational acceleration), and the exponent n> 0 characterizes the condition assumed on the fluid-solid interface (the case n= 3 corresponds to contact without slip, while n∈(0,3) corresponds to the motion of the fluid with partial slip). We prove that the finite speed propagation property for the strong solutions of this equa-tion holds if the exponent n satisfies 4/3≤n<2. Our approach for proving is perturbation method combined with the local energy method which relies on local energy/entropy estimates and differential inequalities. And the upper bound for the speed of the support of this solution is obtained.One of the characteristics of the physically correct free boundary value prob-lem is the finite speed of propagation. Our results indeed confirm that the model we considered is physically correct one.2. Considering the thin solid film equationthis is a sixth-order nonlinear uniformly parabolic equation which arises in the de-scription of thin, epitaxially strained films in the presence of wetting intersections with the substrate with constants g,p and q depending on some characteristic of the physical domain. Under periodic boundary conditions we prove the existence, uniqueness and regularity of the weak solution by means of Galerkin method and energy method. Also, we show the finite-time blow-up under the Dirichlet bound-ary conditions using the method of the first eigenvalue and obtain the estimates for the blow-up time.