Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

Abstract

International audienceThe goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or - in mathematical terms-on the L(1)-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well

Improved conditions for single-point blow-up in reaction–diffusion systems

“…Also we recall that non-equidiffusive parabolic systems are often much more involved, both in terms of behavior of solutions and at the technical level (cf. [15] and [16,Chapter 33]). As for the general problem (1.2), we shall be able to handle a large class of nonlinearities which need not follow a precise power behavior.…”

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

“…Existence of regular bounded solutions on (0, +∞) may be found for example in [27,17,29,28,20,19,38,11,10,23,42,15,18,3,4] and in several other articles whose references may be found in the survey [33] or in the book [39]. However, it is well-known that the solutions may blow up in L ∞ (Ω) in finite time as proved in [35,36] where explicit finite time blow up in L ∞ (Ω) are given.…”

“…We are interested in looking for extensions to these nonlinear diffusions of the two following main results proved in the semilinear case : -first the global existence result of weak solutions for (1) when (P ), (M ) hold and when moreover an a priori L 1 -estimate holds for the nonlinear reactive part, namely (see [31], [32] and the survey [33]). …”

A reaction-diffusion system and its shadow system describing harmful algal blooms

“…We first show a pr i or i estimate for w D (t ) in a similar way to the ones in [15] and [19]. Adding the first equation and the third one of (1.2) multiplied by 1 |Ω| and by r |Ω| , respectively, and integrating it over Ω, we have…”