Research On A Class Of Qualitative Theory To A Fourth-order Degenerate Parabolic Partial Differential Equation

Under the action of surface tension,the fluid thin film can move one the surface,this fluid thin film phenomenan can be seen everywhere in our lives,such as:paint dries gradually,rain water flows down along the glass, the phenomenan of the liquid flow and diffusion on the surface. these phenomena derives a math problem that is the thin film development evolution equations, in general, most of this kind of equations are higher-order, degradation, and nonlinear parabolic differential equations, it is not like the second-order parabolic partial differential equations which have the extremum principle can be used to solve many properties of the equations, so we need to find other ways to study the qualitative theories of such equations.In this article, we study the theory of the function ht+(hnhxxx)x=0Because the equation is degenerate when h?0, firstly, we should find an approximate function with good principles, study the solution of approximate function, by approximation the original equation to find the principles of the solution of the original equation.In general, higer-order nonlinear parabolic differential equations could not find the classical solution under long time, so in this article, we prove the forth-order degenerate nonlinear parabolic equation existence a nonnegative weak solution. Then we compute some bounded estimate to prove when the initial condition is positive, the solution are always maintained its positive under long time. Finally, we use the equation with a lower order term which have a positive classical solution to validation the complerely Laugensen functional's monotonicity.