The Study Of The Dynamic Approximation For Fractional Stochastic Reaction-diffusion Equations And The Well-posedness Of Semilinear Thermoelastical Equations In Three Dimensions

In this paper,we investigate two classical models of evolution equations,one is the fractional stochastic reaction-diffusion equation,the other is the semilinear thermoelastic system.In recent decades,these two models serve as references for the study of a wide range of phenomena such as chemical engineering,environmental and life sciences,aeronautics and nuclear engineering.We firstly consider the following fractional stochastic reaction-diffusion equationsIn Chapter 3,we mainly investigate the Wong-Zakai approximations induced by a stationary process and the dynamical behavior of the equations above when h(t,x,u)is nonlinear in u.More precisely,we firstly prove the existence and uniqueness of the tempered pullback attractor of the Wong-Zakai approximations of the fractional stochastic reaction-diffusion equations.Then we will prove in the cases that h(t,x,u)is linear in u or independent of u the upper semi-continuity of the attractors of the Wong-Zakai approximations respectively.For the thermoelastic system,we plan to investigate the stochastic 3-D semilinear thermoelastic system.However,since some properties of the deterministic 3-D semilinear thermoelastic system are still unknown,we turn to consider the following 3-D semilinear thermoelastic system in Chapter 4 of this article:In the case that the nonlinear term of the momentum equation is defocusing energy critical,we establish the global existence of weak solution in a bounded domain in R~3.The proof is based on the existence and uniqueness of the strong solutions of linear thermoelastic system and the method of energy estimates from wave equation.Based on this,we will consider the stochastic 3-D semilinear thermoelastic system and related problems in our future work.