The Existence And Asymptotic Behavior Of Solutions For A Nonclassical Reaction-diffusion Equations

In this paper, we investigate the existence and the asymptotic behaviors of the solutionfor the following non-classical reaction-diffusion equations with super critical nonlinearity,where domain is a smooth boundary.Equations (1), which appear as non-classical reaction-diffusion equations in fluidmechanics, solid mechanics and heat conduction theory, have a very broad background inapplying. Equations (1)，in whichΔu andΔu tare dissipative terms, is different from theclassical reaction-diffusion equations essentially. As with a termΔut, The regularity ofsystem (1) is not as high as reaction-diffusion equations. Thus the compactness of the systemcan’t be obtained by Sobolev Embedding Theorem. Since the nonlinear external force issubcritical or supercritical exponential growth, it brings some difficulties. For thenon-autonomous case, we only assume that the forcing term g(x, t) is translation bounded,rather than translation compact or regular. As the assumptions of g above, we can not obtainthe compactness of the corresponding symbol space.In chapter 3, we apply the Galerkin method and combine with method of energyestimate to discuss the existence and uniqueness of the global strong solution and continuousdependence of the solutions on initial value in the non-autonomous case, where the growthexponent of the nonlinear term without any limit.In chapter 4, we mainly discuss the long- time behaviors of the strong solution of thenon-autonomous case in bounded domain. By texting uniform ( C )condition, we get that thecorresponding processes of the solutions of equtions (1) are uniformω-limit compact.Sequentially, we obtain the existence of uniform attractor of system (1) in space, where theforce term is not translation compact.