Long-time Behavior Of Solutions For Diffusion Equations With Memory

In this thesis,we mainly discuss the existence and regularity of the attractors in different phase spaces for the following nonlinear evolution equations with memory:where ε is perturbed parameter.First of all,we propose the concept of bi-product-spaces contractive semigroup,and obtain a general existence criterion to prove the existence of the bi-product-spaces global attractor for the nonlinear evolutionary equation by the properties of contractive semigroup.As an application,when ε=0,we consider the long-time behavior of above system in different phase spaces.In this case,above model becomes reaction-diffusion equation with memory.In order to overcome the difficulties of the memory term and nonlinearity with exponential growth of polynomials of any order,we introduce a new operator decomposition technique,it should be pointed out that this operator decomposition method can not only prove the asymptotic regularity of the solutions,but also obtain the bi-product-space contractive function according to the regularity of the solutions,and thus obtain the existence and regularity results of the attractor.The main results are as follows:（ⅰ）By using the contractive function method,the asymptotic compactness of solutions semigroup in product space L²（Ω）×L_μ²（（?）;H₀¹（Ω））is proved,the existence of global attractor （?）_p in this space is obtained.When the initial value belongs to L²（Ω）×L_μ²（（?）;H 0¹（Ω））,the existence of the bi-product-space global attractors （?）_p and A are obtained by verifying the asymptotic compactness of the semigroups on productspaces L²（Ω）×L_μ²（（?）;H 0¹（Ω））and H 0¹（Ω）× L_μ²（（?）;H 0¹（Ω））respectively;（ⅱ）Using a new operator decomposition technique to prove the asymptotic regularity of the solutions,and thus obtain the contractive function,and further get the existence and regularity of the attractor （?）,i.e.,（?）（?）D（A）×L_μ²（（?）;D（A））,In particular,the relationship among the three types of attractors is also obtained,i.e.,（?）₀=（?）_p=（?）.Next,the contractive function method is extended to the case of time-dependent bi-product-space when ε=ε（t）is time-dependent disturbance parameter.Thus,a general criterion to prove the existence of pullback D-attractors in time-dependent product space is given.In addition,in order to obtain the asymptotic regularity of the solutions for the external forcing term belonging only to the lower regular space H^-1（Ω）,another new operator decomposition method is given.As an application,we consider the long-time behavior of the solutions for the memorizing diffusion equation with ε（t）.The specific results are as follows:（i）By using the Galerkin approximation method,we obtain the well-posedness of the equation,and then the existence of the pullback （?）-attractor in time-dependent space is obtained through the time-dependent product space contractive process method.（ii）By using two new operator decomposition methods,the asymptotic regularity of the solution is proved in the case that g∈L²（Ω）but the nonlinearity satisfies the polynomial growth of any order and in the case that g∈H^-1（Ω）but the nonlinear term satisfies the critical exponential growth,respectively.Then,the regularity of the attractor is obtained;Finally,we propose some new problems related to the diffusion equation with memory.