Traveling Wave Solutions And Global Attractor For Continuous And Discretized Reaction Diffusion Equations

This dissertation investigates both existence of traveling wave solutions for delayed reaction diffusion systems and lattice differential equations, and global attractor of spatially discretized FitzHugh-Nagumo equations with Dirichlet or Neumann boundary conditions.For delayed reaction diffusion systems, the existence of traveling wavefronts in diffusive and coorperative system with time delays is provided, firstly; the monotone iteration scheme, together with upper-lower solution technique, is applied to establish the existence of traveling wavefronts of delayed reaction diffusion systems with some zero diffusive coefficients. Secondly, Schauder fixed point theorem is applied to some operators to prove the existence of traveling wave solutions in a properly subset equipped with exponential decay norm, which is obtained from a pair of upper and lower solutions for delayed reaction diffusion systems with non-quasimonotoiiicity. Both positive diffusive coefficients and some zero diffusive coefficient cases are considered. Finally, for partial decoupling systems, such as competitive-cooperative model with time delay and delayed epidemic model, we employ the new cross-iteration method, together with Schauder fixed point theorem, to establish the existence of traveling wave solutions. Both partial quasi-monotonicity and partial non-quasimonotonicity cases are considered separately. Similar results are obtained for some delayed lattice differential equations and systems of delayed lattice differential equations.For diffusive Predator-Prey model without time delay, shooting argument is applied, together with unstable manifold theorem and LaSalle's Invariance Principle to prove the existence of traveling wave solutions in R4.For spatially discretized FitzHugh-Nagumo equations with Dirichlet boundary condition or Neumann boundary condition, the existence of global attractor and its Hausdorff dimensions are proved separately. The result shows that the upper bounded estimate is independent of the number of spatially partition. Similar results are obtained for spatial-temporal discretized FitzHugh-Nagumo equations and generalized coupled FitzHugh-Nagumo equations.