On The Existence And Local Structure Of Global Attractors For Two Classes Of Reaction-Diffusion Equations

In this doctoral thesis, we are concerned with the existence and local propertiesof global attractors for the following two classes of Reaction-Diffusion equations:andFor the first equation, if f(u) is arbitrary polynomial nonlinear andλ> 0 is any positive constant, we obtain the existence of global attractors in spaces L^q(Ω,) and H₀¹(Ω,). After that, we estimate the lower bound on dimensions of the attractors. In abstract framework, applying Z₂ index theories, we give a new method to estimate the lower bound on dimensions of global attractors for odd dynamical systems. In application, we need to find a bounded symmetric subset of a Hilbert space, whoseω-limit set doesn't contain zero, and then, we can prove that the Z₂ index of global attractors except zero is bigger than that of the bounded symmetric subset, thus, the lower bound on dimensions is got by use of the properties of Z₂ index of compact subset in Hilbert spaces.As for the second equation, because of the absence of uniqueness for weak solution, we have to use the multivalued semiflow theories to study the long time behavior of this equation. At first, we give a new abstract result for the existence of global attractors (which is compact, attracting and minimum) for generalized semiflow without upper-semicontinuity condition. In the abstract result, we useω-limit compactness instead of those given in other references. After that, we apply the new abstract results to equation (â…¡), and obtain the existence of global attractors in L²(Ω).