Complex Asymptotic Behavior Of Solutions To Non-Newtonian Multiparty Seepage Equations

In this paper,we consider the complicated asymptotic behavior of the Cauchy problem of the double-nonlinear diffusion equation ut=div（|▽um|p-2▽um）in Rn×（0,∞）,u（x,0）=u0（x）in RN,where p>1,m>1,m（p-1）-1>0.The doubly nonlinear diffusion equation is derived from many diffusion phenomena,such as soil physics,reaction chemistry,combustion theory,fluid dynamics.The equation is a general form of three classical differential equations.If p=2,m=1,the doubly nonlinear diffusion equation is transformed into a classical heat equation;if p =2,it is transformed into a medium porous equation;if m=1,it is transformed into the evolution p-Laplacian equation.Since doubly nonlinear diffusion equation has degeneration and non-linearity,which reflects some physical reality,the application of nonlinear partial differential equations in practice is mostly concentrated on the complex gradual behavior of its solution,so the research on the complex asymptotic behavior of the solution of the doubly nonlinear diffusion equation is of great significance and has high theoretical value and good application prospect.Asymptotic behavior refers to the nature of the equation solution when time tends to infinity.When we describe the gradual behavior of the solution of the equation through the ω-limit set,the number of elements contained in the ω-limit set is two or more,which indicates that the equation solution has complex asymptotic behavior.The relevant research results on the complex asymptotic behavior of the solution of the general nonlinear diffusion equation are abundant,but there is little attention to the complex asymptotic behavior of the solution of the doubly nonlinear diffusion equation.The main purpose of this paper is to consider the complex asymptotic behavior of the solution of the doubly nonlinear diffusion equation.Firstly,we use the propagation speed estimate and space-time decay estimates to solve the non-linearity and degeneracy of the equation.Then,the equivalence relationship between the solution and the initial value is established to prove that the solution has complex asymptotic behavior in the weighted function space,and the solution of the equation is gradually complex through the equivalence relationship between the solution and the initial value.Finally,we study the complex asymptotic behavior of the solution of the doubly nonlinear diffusion equation in the continuous function space.The full text is divided into five chapters.The first chapter is the introduction,which mainly introduces the source,research status and research significance of doubly nonlinear diffusion equation.The development status of complex asymptotic behaviors such as thermal equation,medium porous equation,evolution p-Laplacian equation and doubly nonlinear diffusion equation is also introduced in detail.The second chapter introduces the basic definitions and concepts of the preparatory knowledge needed in this paper,mainly introduces and proves the propagation speed estimate and space-time decay estimates of the solution of the doubly nonlinear diffusion equation.In the third chapter,we discuss the complex asymptotic behavior of Cauchy problem for doubly nonlinear diffusion equation in weighted function space by using the equivalence relation between the solution of the equation and the initial value.Firstly,the equivalence relationship between the initial value and the ω-limit set of the rescaled solutions is proved by using the propagation speed estimate and space-time decay estimates of the solution of the doubly nonlinear diffusion equation,and it reveals the complex asymptotic behavior of Cauchy problem solution of doubly nonlinear diffusion equation in weighted function space.Then,by establishing the equivalence relation between the initial value and the limit set of the rescaled solutions,the complex asymptotic behavior of the solutions of two special doubly nonlinear diffusion equations is proved.In the fourth chapter,we study the complex asymptotic behavior of the solution of doubly nonlinear diffusion equation in continuous function space when 0＜μ<2N/N[m（p-1）-1]+p and β＞2-μ[m（p-1）-1]/2p.The fifth chapter is the conclusion and prospect of this paper.The research methods and results of this paper are summarized,as well as the prospect of other nonlinear diffusion equations.