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

Abstract

International audienceWe prove here global existence in time of weak solutions for some reaction-diffusion systems with natural structure conditions on the nonlinear reactive terms which provide positivity of the solutions and uniform control of the total mass. The diffusion operators are nonlinear, in particular operators of the porous media type u i → −d i ∆u m i i. Global existence is proved under the assumption that the reactive terms are bounded in L 1. This extends previous similar results obtained in the semilinear case when the diffusion operators are linear of type u i → −d i ∆u i

Global regularity and convergence to equilibrium of reaction–diffusion systems with nonlinear diffusion

“…The existence of global weak solutions to (S) subject to homogeneous Dirichlet boundary conditions and under the assumptions (G)-(M)-(P) was recently obtained in [LP17]. The proof of the following Theorem 1.1 on the existence of weak solutions to (S) subject to Neumann boundary conditions uses similar arguments to [LP17] and is postponed to Section 5. where the constant C depends on the L 2 -norm of the initial data, the constants λ i in (M), the diffusion coefficients d i > 0 and the domain Ω.…”

“…[Smo94,Leu09] . Up to the best of our knowledge, system (S) under the general structural assumptions (G)-(M)-(P) was only studied very recently in [LP17], where the authors showed the global existence of weak solutions. Therefore, the present paper serves as the first result to show the boundedness of weak solutions by assuming stronger conditions on porous medium exponents.…”

Stationary reaction–diffusion systems in L1

“…to some u ∈ L 1 (Ω) +m . Moreover, by definition of h n i (see (13) ) and the continuity property of h i assumed in (H1), h n i (·, u n ) converges µ-a.e. to h(·, u).…”

“…convergence is sufficient to pass to the limit. The difficulty is then to prove precise estimates independent of n, in terms of η, in order to control the other terms as it done in the parabolic case (see [16], [17], [13]). This approach through cut-off functions T k is precisely developed in Section 4 to prove Theorem 1.2 and we refer the reader to this other approach without giving more details here.…”

“…Analysis of systems of this kind may be found for instance in [12], [7], [18], [13], [9], [5], [6], etc. In fact Theorem 1.2 applies to quite general systems like, for instance, those obtained by multiplying the above 3 × 3 (resp.…”