Delay-partial differential equations with some large diffusion

Abstract

IT IS OFTEN the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v. Its eigenvalues and eigenfunctions will also depend upon v. In many cases (see, for example [l-7]) there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies, the remaining eigenvalues blow up at a certain rate. In this case one expects the dynamics to be dictated by a system of ordinary differential equations. This is indeed the case in several applications and its proof strongly uses the invariant manifold theory. In the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain, whereas the remaining blow up. This is the case, for example, in parabolic equations in thin domains studied by Hale and Raugel [S]. We will not be able to address the general case of thin domains, but only the case of a thin straight channel. However, the property that allows this theory to work is present in the general case; that is, there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up (see also Arrieta [9]). The theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation. The main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem. A physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that, in that direction, spatial changes will not be observed. The same interpretation can be given to a damped wave problem in a thin straight channel. The wave speed can be interpreted as being very large in one direction. This would intuitively imply that there would be no spatial changes for the amplitude of the wave, in that direction. We also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances. In this