The Solution For A Class Of Degenerate Quasilinear PDE

Many areas of the mathematical model can be used to describe the partial differential equations(PDE), many of the major physical,the basic mechanics equation itself are PDE. Nearly two hundred years, the most common PDE(heat conduction equation,harmonic equations,wave equation,etc.) in close contact with the need of physics,mechanics,geometry; In the classical theory of partial differential equations category, multiple differential and integral calculus (and ordinary differential equations,complex function,linear algebra, etc.) as the main tool to create a lot still widely used in an effective method.With the application of PDE, many significant natural science and engineering technology issues boil down to the study of nonlinear PDE. And with the depth of research, some of the original are available for the approximation of linear PDE, we must also consider the nonlinear effects. Therefore, the application of PDE is the subject of the study of nonlinear PDE. The method of functional analysis which builts on the basis of Sobolev space, to do with the problem of linear and nonlinear PDE, provide a strong framework and tools which have been widely applied in practice.Subsequently, on the emergence of generalized functions as a symbol, the issue also provides a framework for dealing with PDE, in which many classical methods (such as Fourier analysis outstanding) play a major role. On this basis, after a succession of quasi-linear differential operator,Fourier integral operator,micro-local analysis,the theory of super- functions which can become powerful tools that can greatly change the face of linear PDE, and began to be used to deal with the problem of nonlinear PDE.In this paper we discuss the solutions of two kinds of quasi-linear PDE.Firstly, it popularizes the result of P-Laplacian PDE; And its'approach does rely on the theory of the fixed point index to discuss the poitive solution and its'bifurcation problem of degenerate quasilinear elliptic PDE; In the same time ,we also consider the eigenvalue problem of degenerate quasi-linear elliptic PDE and presents results of the existence of principal eigenvalue and its natures. Secondly, we establish comparson principle on the quasi-linear parabolic PDE , prove the existence of solution for the quasi-linear parabolic PDE by iterating upper and lower solutions , and also discuss the occurrence of blow-up phenomenon.