Solutions of Lattice Differential Equations over Inhomogeneous Media

Abstract

This thesis investigates one-dimensional spatially-discrete reaction-diffusion equations with a diffusion term that involves nearest-neighbor coupling and with a reaction-term that is a smooth-cubic nonlinearity. Specifically, we consider two nontrivial examples of lattice differential equations (LDEs) on Z that are related to the (homogeneous) lattice Nagumo equation. The LDEs that we consider are used to model natural phenomena defined over an inhomogeneous medium, namely: (1) a lattice Nagumo equation with a negative diffusion coefficient. Such is still a well-posed problem in the LDE setting and has been shown to arise from a discrete model of phase transition for shape memory alloys. This thesis shows that the anti-diffusion lattice Nagumo equation has a period-2 traveling wavefront solution that is stable and unique. Utilizing the concrete expressions for the nonlinearities, we obtain criteria on the (d, a)-parameter plane that guarantee a display of bistable and monostable dynamics. Where there's bistable dynamics, we study the propagation failure phenomenon; where there's monostable dynamics, we compute a minimum wave speed for the traveling waves. (2) a lattice Nagumo equation that has a single diffusion-defect in the middle of Z, which may occur due to deviations in the diffusive property of the medium. This thesis shows that such an equation has a time-global solution which behaves as two fronts coming from the both sides of Z. A key idea for the existence proof is a characterization of the asymptotic behavior of the solutions for negative time in terms of an appropriate super-solution, sub-solution pair.