1. CMB 2011 (vol 55 pp. 3)
|On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations|
We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations.
Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equation
Categories:34E10, 34C10, 35Q35
2. CMB 2009 (vol 53 pp. 163)
|Variants of Arnold's Stability Results for 2D Euler Equations|
We establish variants of stability estimates in norms somewhat stronger than the $H^1$-norm under Arnold's stability hypotheses on steady solutions to the Euler equations for fluid flow on planar domains.
3. CMB 2000 (vol 43 pp. 477)
|The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations |
Many classes of singularly perturbed reaction-diffusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as nonlocal partial differential equations. Results concerning the existence of equilibria, their stability, and the dynamical behavior of localized structures in the interior and on the boundary of the domain are surveyed for these two models. By examining the spectrum associated with the linearization of these problems around certain canonical solutions, it is shown that the nonlocal term can lead to the existence of an exponentially small principal eigenvalue for the linearized problem. This eigenvalue is then responsible for an exponentially slow, or metastable, motion of the localized structure.
Categories:35Q35, 35C20, 35K60