On the Role of Diffusivity in Some Stable Equilibria of a Diffusion Equation

Abstract

Our main goal in this work is to exhibit a class of diffusivity functionswhich, regardless of the shape of ˝0, allows for the existence of patternsfor (P).By considering (P) as a singularly perturbed problem where the dif-fusivity is given by =k(x), = a small positive parameter, the existence of pat-terns which develop boundary transition layers and spikes was studied in[N1]. The case in which these patterns develop internal transition layerswas studied for a more general equation in [N2], using an approach basedon 1-convergence. See also [N3, N4].Problem (P) was considered in [M] for the case in which k is a constantfunction. Therein the author resorted to the shape of ˝0 to produce pat-terns. Therefore the geometric structure of the patterns found here is dif-ferent from the geometric structure of the ones found in [M].Actually the technique we use here stems from the one presented in [M],where after finding a invariant set for the flow, the monotonicity propertyof the flow as well as property (S) (see [M, p. 431]), which is based onZorn’s lemma, were used.Our approach is different since after finding an invariant set we only usebasic results for the flow and variational techniques to show existence of alocal minimiser of the energy functional in this invariant set. This localminimiser turns out to be a stable stationary solution to (P).The very same analysis still go through for the case in which the trivialequilibrium solutions a and b are positive, i.e., 0<a<b and the reactionterm is replaced by a spatially inhomogeneous term s(x) f(v), where s(x)isa positive continuous function.Problem (P) appears as a mathematical model in many distinct areasand just for the sake of illustration as well as interpreting the results froma physical point of view, let us suppose that v models the time evolutionof the concentration of a diffusing substance in a medium whose diffusivityis given by k(x), x # 0 and under the effect of a source term f, when0<a<b.Then typically, as far as the existence of patterns in higher space dimen-sion is concerned, the diffusivity is small on a closed surface S/0. But thisis not sufficient and another condition on the behavior of k(x), has to beimposed.In [N2N4] this condition relates the slope and concavity of k(x), alongthe normal to S, to the Gaussian and mean curvatures of S. Herein, as longas the areas of the negative and positive bumps of f are the same (theequal-area condition), it is only required that k(x) be large on two smootharbitrary sets, O