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Our concern in this paper is to shed some additional light on the mechanism and the effect caused by the so called cross-diffusion. We consider a two-species reaction–diffusion (RD) system. Both “fluxes” contain the gradients of both unknown solutions. We show that–for some parameter range– there exist two different type of periodic stationary solutions. Using them, we are able to divide into parts the (eight-dimensional) parameter space and indicate the so called Turing domains where our solutions exist. The boundaries of these domains, in analogy with “bifurcation point”, called “bifurcation surfaces”. As it is commonly believed, these solutions are limits as tgoes to infinity of the solutions of corresponding evolution system. In a forthcoming paper we shall give a detailed account about our numerical results concerning different kind of stability. Here we also show some numerical calculations making plausible that our solutions are in fact attractors with a large domain of attraction in the space of initial functions.

Abstract.: Geometric properties of shape functions of self-similar solution to the equation are studied, and q are positive numbers. These shapes-the solutions of the corresponding nonlinear ODE-are of very different nature. The properties usually depend on three critical values of q (1, 3/2 and 2). For the range 1

The Cauchy problem for a class of nonlinear diffusion-reaction equations is studied. The equations may be classified as being of degenerate parabolic type. It is shown that under certain conditions solutions of the problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded above, although the support of the initial data function is not. We also provide some estimates of the behavior of the free boundary.