Symmetry And Monotonicity Of Solutions To Several Elliptic And Parabolic Equations

In this thesis,we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations and Hamilton-Jacobi equations;we also generalize the Hopf lemma for anti-symmetric functions involving non-local elliptic operators,and apply it in moving plane method to research the symmetry of positive solutions for a class of fractional Kirchhoff equations.The thesis consists of four chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations.We first obtain a series of needed key ingredi-ents such as narrow region principles,various asymptotic maximum principles and asymptotic strong maximum principles for antisymmetric functions.Then we illus-trate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space.Namely,for the following equation(?)tu+(-?)su=f(t,u),(x,t)??×(0,?),(E1)where s ?(0,1)is a constant.We show that no matter what the initial data are,the positive solutions of problem(E1)with ?=B1(0)or RN will eventually approach to radially symmetric functions.The main results in this chapter have been published in(Adv.Math.,377(2021),107463).In Chapter Three;by using the important tool of constructing sub-solution used in the asymptotic moving plane method,we prove the Hopf lemma for anti-symmetric functions involving fractional elliptic operators.Our result generalizes the result of C.Li and W.Chen in(Proc.Amer.Math.Soc.,2019)and greatly simplifies the proof process.As an application,we use it in the moving plane method to consider a class of fractional Kirchhoff equation(a+b?Rn|(-?)s/2u|2dx)(-?)su=f(x,u),u>0,x??,(E2)where a?0,b>0,s?(0,1)are constants.Under some assumptions on.f,we prove the symmetry and monotonicity of solutions for(E2)both in bounded domain and in Rn.The main results in this chapter have been been published in(Comm.Pure Appl.Anal.,20(2021),1431-1445).In Chapter Four,by using the asymptotic moving plane method developed in Chapter Two,we study the following fractional Hamilton-Jacobi equation(?)tu+(-?)su=H(t,x,u,?u),t>0,x?Rn,(E3)where s ?(0,1)is a constant.Under some suitable assumptions on H,we prove that the positive solutions of(E3)possess some kind of asymptotic monotonicity in the x1-direction.