Properties Of Solutions To Nonlinear Nonlocal Evolution Equations With Anomalous Diffusion

In this paper,we mainly studied the initial value problems of nonlinear nonlocal par-tial evolution equations.Analyzed the solutions of two kinds of initial value problems,discussed and proved the existence of time local weak solutions and time global weak so-lutions.Also,we proved that the solutions of the second kind of initial value problems have L¹-contraction.In chapter two,we analyze the solution of the initial value problem(1.24)for a class of nonlinear nonlocal partial evolution equations.For the undetermined parameters m and?in the initial value problem(1.24),we consider the existence of time local weak solution and time global weak solution of the initial value problem(1.24)when m?1.For m=1,we discussed?=2 and??(0,2).For m>1,we use the Gagliardo-Nirenberg inequality,Gronwall lemma,Hardy-Littlewood-Sobolev inequality,H¨older inequality and Young inequality to prove that the initial value problem(1.12)has both time local solution and time global solution when m and?are in the corresponding range.In chapter three,inspired by the background and significance of the variable ex-ponential Laplacian(-?)^?(x),we analyze the solution of the initial value problem of a class of variable exponential nonlocal evolution equations(1.27).We first consider the time local solution and its properties of the initial value problem(1.27),then we prove that the classical solution is global and has L¹-contraction.Secondly,we analyze the classical solution of the initial value problem of the nonlinear nonlocal evolution equation with variable exponent.Similarly,we established three lemmas to explain the time local solution and it's properties of the initial value problem respectively.Then we proved that the classical solution of the initial value problem is global and has L¹-contraction.