Periodic Traveling Wave Solutions for Diffusion Equations with Time Delayed and Non-Local Responding Reaction
Department of Mathematical Sciences, UAH
October 9, 2007
102N Technology Hall
Recently a new type of diffusion equation has emerged in the literature in which the reaction term depends not only on the current and local state but also the past states as well as non-local interaction. For this type of equation, the existence of certain types of traveling waves, that is, solutions of the form u(x; t) = U(x + ct) that connect two equilibrium points, has recently been studied. In many biological and physical systems, the associated reaction equation supports a periodic solution (predator-prey models, for example). In this dissertation, it is shown that such a periodic solution gives rise to a periodic traveling wave solution for the corresponding diffusion system. To prove this result, one can use a recently developed technique in which the equation for a traveling wave solution is reduced to a singularly perturbed functional differential equation, which in turn is transformed into a regularly perturbed integral equation. Upon defining an appropriate linear operator L on a Banach space, this integral equation is basically reduced to the study of an equation of the form Lx = F(x; μ): Solving this equation requires the study of the null space and range of the operator L; the use of the Liapunov-Schmidt method of decomposing the domain of L; and the application of the Implicit Function Theorem as well as a generalized implicit function theorem.
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