Pullback Exponential Attractors For A Class Of Nonclassical Diffusion Equations

The attractor theory is very important to describe the long time behavior of dissi-pative dynamical systems derived from evolutionary equations.The existence of global attractors and exponential attractors for autonomous diffusion equations has been studied extensively.In the past few years,many mathematicians have devoted themselves to the study of non-autonomous partial differential equations and their generating process.Pull-back exponential attractor is a suitable notion used to describe the long time behavior of non-autonomous dynamical systems.Reaction-diffusion equations are widely used to describe a variety of phenomena in natural science.However,the classical reaction-diffusion equation does not contain each aspect of the reaction-diffusion problem.It neglects the viscidity,elasticity,and pressure factors of medium in the process of solid diffusion and so forth.In the reference[1],E.C.Aifantis constructed some mathematical models by some concrete examples,which contains the viscidity,elasticity,and pressure factors of medium.That is the following nonclassical diffusion equation:u_t?(?)u_t?(?)u_t?f(u)=g(x,t).This equation is a special form of the nonclassical diffusion equation used in fluid mechanics,solid mechanics,and heat conduction theory.The main purpose of this thesis is to prove the existence of a pullback exponential attractor for the nonclassical diffusion equation above.Since the above equation contains the term(?)u_t,it is essentially different from the usual reaction-diffusion equation.The classical reaction-diffusion equation has a certain“regularity”that describes the smooth-ness of the function.Nevertheless,the nonclassical diffusion equation has no such kind of“regularity”,which is similar to hyperbolic equations.But it is not a real hyperbolic equation.This brings difficulty to proving the existence of pullback exponential attractors for the nonclassical diffusion equations.The method given in reference[2]is no longer valid in this thesis.To overcome this difficulty,we adopt the method given in reference[3]which decomposes the equation into two parts,one of which satisfies the regularity property and the other satisfies the contraction property.