Complex Asymptotic Behavior Of Solutions To The Dirichlet Problem Of The Non-Newtonian Seepage Equation

In this paper,we consider the complicated asymptotic behavior of the Dirichlet problem of the fast diffusion non-Newtonian filtration equation,Non-Newtonian filtration equation,also known as the developed p-Laplace equation,comes from the diffusion phenomenon widely existing in nature,and is applied to many fields such as biochemistry,biological population dynamics,seepage theory,phase transition theory and image processing non-Newtonian filtration equation are quasilinear,degenerate or other singularity.In practice,this kind of equations can better reflect some physical phenomena than linear equations and equations without degenerate or other singularity.On the other hand,its degenerate or other singularity makes the research content more abundant,Therefore,the study of this kind of parabolic equation is very necessary and has high theoretical value and good application prospect.The asymptotic behavior of solutions of parabolic partial differential equations is to describe the properties of solutions when time tends to infinity.For example,the solution will converge to a fixed function or explode as time tends to infinity.Generally,the complex asymptotic behavior is characterized by the number of elements in the ?-limit set.Only when the number of elements in the ?-limit set is greater than or equal to 2 can it have complex asymptotic behavior.The complex asymptotic behavior of the solution of Cauchy problem of second-order parabolic partial differential equation has attracted extensive interest,but people have not paid enough attention to the complex asymptotic behavior of the solution of Dirichlet problem of second-order parabolic partial differential equation.The main purpose of this paper is to consider the complex asymptotic behavior of the solution of Dirichlet problem of non-Newtonian filtration equation.We will first consider the complex asymptotic behavior of the solution of a special model,and then take the solution of the special model as a comparison function to obtain the complex asymptotic behavior of the solution in the general case.The full text is divided into six chapters.The first chapter is the introduction,which mainly introduces the research status of non-Newtonian seepage equations and other nonlinear partial differential equations,specifically introduces the thermal equations,Newtonian seepage equations,non-Newtonian seepage equations,non-Newtonian polytropic seepage equations,and introduces the development status of the existence and uniqueness of their solutions,asymptotic behavior and so on.The second chapter introduces the basic theorems and related inequalities of the preparatory knowledge needed in this paper.In Chapter 3,a special model is introduced,that is,the problem is considered on the ?:=B:=BR(0),and the initial boundary value is still radially symmetric.We first consider the regularization problem of the model,and then use the comparison principle and extreme value principle to obtain the different complex asymptotic behavior of the solution when the initial value of the special model satisfies different conditions.The fourth chapter introduces the different complex asymptotic behavior of the appropriate solution of the problem,when the initial value satisfies different conditions under the general model,which is as follows:(1)Suppose that p/2-p0,such that for all r>0,u satisfying(?);(2)Suppose that ?1=?2=n-p/p-1,then there exists c1,c2 such that u satisfying c1|X|-n-p/p-1?u(x,t)?c2|x|?|-n-p/p-1;(3)Suppose that n-p/p-1