Long-time Behavior Of Solution For Doubly Nonlinear Parabolic Equation

There are significant applications of doubly nonlinear parabolic equation occurred in mental compositions. However, the long time behavior and propertiesof its solution are known uncompletely. The aim of this thesis is to study the existence and the long time behavior of the solution for doubly nonlinear parabolic equations with the following formhere bothαand f are nonlinear functions with polynomial growth of arbitrary order,Ωis a bounded smooth domain in R^N. Because of the nonlinear termα, the study of this problem becomes more difficult.Firstly, the existence of solution for a general doubly nonlinear P-Laplacian equation is obtained by using Faedo-Galerkin approximation method. Secondly,by the traditional energy estimate method, regularity of the solution of system (*) in various sense is proved under certain assumptions of the nonlinearαand f. Thirdly, combining sign function and maximum principal, we obtain uniqueness of the solution and continuously depending upon initial state.In order to study the long time behavior of the solution, the theory of norm-to-weak continuous semigroup(in [19]) is introduced. Firstly, using Legendre transform, the existence of global attractor in (L^r+2(Ω), L²(Ω)) of semigroup S(t) is obtained. Secondly, by constructing a new semigroup T(t), and asymp-totic a prior estimate method, we obtain the existence of the (L^r+2/r+1(Ω),L^q/r+1(Ω)) global attractors for the semigroup T(t). Thirdly, making use of the relationbetween semigroup T(t) and S(t), the existence of global attractor in (L^r+22(Ω),L^q(Ω)) is verified. For the case of the space dimension N≤2, the existence of global attractor in (L^r+2(Ω), H₀¹(Ω)) is obtained. Comparing with the result in [10,82], the assumptions of the nonlinear a are more common.