Inertial Manifold For A Nonlocal Dynamical System With Anomalous Diffusion

In this thesis,we mainly study how to prove the existence of inertial manifold for the non-local development equation and coupled system.In the first part,we briefly introduce the definition,developments and applications of the inertial manifold,our research backgrounds and significance,as well as the related knowledge of partial differential equations,operator simigroups,sobolev spaces and infinite-dimensional dynamical systems used in the proof process.In the second part,we consider the non-local development equation with the non-local Laplacian operator(-△)α/2(0<α<2),we can obtain the existence of inertial manifold by the spectral gap condition when 1≤α<2 in 1-dimensional space.In the third part,we study the existence of inertial manifold of coupled systems composed of partial differential equation and ordinary differential equation.By the"spatial averaging principle" and the "abstract invariant manifold theorem",we find the needed Lipschitz function,and then the exponentially tracking property is proved,so as to obtain the existence of the inertial manifold in 3-dimensional space for the coupled system without the spectral gap condition.In the fourth part,we review and look forward to the research,and point out the limitations of the spatial averaging principle,and we cannot prove the existence of the inertial manifold by the spatial average principle when the operator(-△)α/2 in non-local development equation satisfies 0<α<1,because the spectrum does not satisfy some theorems,and we illustrate the relationship between the inertial manifold and control theory.