Systèmes de réaction–diffusion sans propriété de Fredholm

Competition between transport phenomena in a reaction–diffusion–convection system

“…Therefore, we can imagine the reactiondiffusion-convection (RDC) system as an extension of the reaction-diffusion model, in which the addiction of a convective term will result in a sort of competition with diffusion for the new states of stability for the system Q.Ouyang e J.-M. Flesselles, 1996. However, a range of values of Grashof numbers Gr i will exist, for which the RDC coupling corresponds to a kind of inertia respect to the diffusive regime: the rotating solution imposed by diffusion still assumes a preminent role. As Grashof numbers increase, the system will reach a critical point in which the instability due to the convective motions assumes a prevaricating character and the system dynamics will evolve to new regimes driven by hydrodynamics, with a route to chaos similar to the transition to turbulence observed in fluids Belk e Volpert, 2004;Ducrot e Volpert, 2005;Gaponenko e Volpert, 2003;Gollub e Benson, 1980;Guzmán e Amon, 1994;McLaughlin e Orszag, 1982;Molenaar, Clercx e Heijst, 2005. In figure 30 we illustrate the effect induced by convection on the shape of spiral wave concentration: as Grashof number increases, distortion and also breaking of the spiral wave occurs Budroni et al, 2009;Jahnke, Skaggs e Winfree, 1989, resulting in spatio-temporal chaos Biktashev, Holden e Tsyganov, 1998;Pérez-Villar et al, 2002;Ramos, 2001;Sandstede, Scheel e Wulff, 1999. Although the effects of hydrodynamics in a oscillating RDC system has been extensively studied, a comprehensive understanding on the relative, separate role played by diffusion and convection in terms of stability of periodic solutions is missing. Experimental attempts to shed light on these aspects (see for example Rossi et al, 2005) have pointed out the difficulty to control the species diffusivity without affecting other variables and suggest that a numerical approach is in this context not only convenient but necessary.…”

On the existence of stationary solutions for some nonlinear heat equations

“…Nonlinear non Fredholm elliptic problems were treated in [19] and [22]. Potential applications to the theory of reaction-diffusion equations were explored in [9,10]. Non Fredholm operators arise also when studying wave systems with an infinite number of localized traveling waves (see [1]).…”

Solvability in the sense of sequences for some non Fredholm operators related to the superdiffusion

“…In the present article we will study some properties of 1 the operators of this kind. Let us recall that elliptic equations with non-Fredholm operators were treated extensively in recent years (see [18], [19], [20], [21], [22], [23], [25], [26], also [5]) along with their potential applications to the theory of reaction-diffusion equations (see [7], [8]). In the particular case when a = 0 the operator A 2 satisfies the Fredholm property in some properly chosen weighted spaces [1], [2], [3], [4], [5].…”