Research On The Dynamical Behavior Of Two Classes Of Reaction Diffusion Equations

Reaction-diffusion equations are widely used to describe phenomena in many different fields such as chemistry, biology and physics. What we are concerned here is the asymptotic property of solutions when time goes to infinity.In this thesis, we investigate the dynamical behavior of the following two classes of reaction diffusion equations: andEquation (I) is a mathematical model arising in the theory of superconduc-tivity with diffusion subject to the homogeneous Dirichlet boundary condition. We obtain the existence of global attractor in L2(Cl)n. The energy estimation method is applied to study the asymptotic compactness of the semiflow.Equation (II) is a mathematical model describing oscillating polymerization process in chemistry with diffusion subject to the homogeneous Neumann bound-ary condition. First, we formulate some bounded positively invariant regions via the upper and lower solution method. Then we prove the existence of maximal attractors in L2(?;?) and H1(?;?), respectively. Finally, we establish the ex-istence of global attractors in (L2(?)?L?(?))2 and (L2(?)?L?(?))2. It is important to note that there's something special about the attractivity of the attractors here. We propose a new method to obtain a priori estimate to verify the asymptotic compactness of the semiflow. In this case, the nonlinear terms are not restricted to satisfy some polynomial growth assumptions.