Global Existence for Quadratic Systems of Reaction-Diffusion

Abstract

We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of L log L-entropy type holds. The approach relies on an a priori dimension-independent L

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

“…This is a consequence of the latter result and of a main a priori L 2 ((0, T ) × Ω)-estimate on the solutions implied by (P ) + (M ) and which is interesting for itself. This estimate was noticed in [13,30,35,36] and then widely exploited, see for example [12,6,7,37,40,8,5,10,33]. We will first see that these two results extend to the case when the ϕ i are nonlinear but nondegenerate (that is when ϕ i is bounded from below and from above, see Proposition 2.4).…”

“…We equipp W with the norm : . Using the estimates (10)(11)(12)(13)(14), it is easy to prove that F send W into itself and that some iterate of F is a strict contraction. Whence the existence of a fixed point u n .…”

“…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.…”

Stability of the Derivative of a Canonical Product

“…It is satisfied if, for instance, N = 4 and r 1 = r 3 = c 2 c 4 − c 1 c 3 , r 2 = r 4 = c 1 c 3 − c 2 c 4 [10]. For the existence result, the inequality can be weakened by…”

“…Our main idea to handle (9) is to exploit its entropy structure. We associate to this system the entropy density (10) h(c…”