Titre : A class of parabolic fractional reaction-diffusion systems with control of total mass: Theory and numerics
Résumé :
In this talk based on [1, 2], we present some new results about global-in-time existence of strong
solutions to a class of fractional parabolic reaction–diffusion systems posed in a bounded open subset
of R^d. The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide
nonnegativity of the solutions and uniform control of the total mass. The diffusion operators are of
type u_i → d_i(−∆)^{s_i}u_i where 0 < s_i < 1. For more details about this kind of operators, we refer
the interested reader to [3] and references therein. Global existence of strong solutions is proved
under the assumption that the reactive terms are at most of polynomial growth. Our results extend
previous results obtained in [4, 5] where the diffusion operators are of type u_i → −d_i ∆u_i.
Also, we present some numerical simulations in order to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case where the diffusion is driven by the classical Laplacian. See [6, 7, 8].
Références :
[1] M. Daoud, E.-H. Laamri and A. Baalal, A class of parabolic fractional reaction-diffusion systems with control of total mass: Theory and numerics, J. Pseudo-Differ. Oper. Appl. 15(18) (2024). DOI : 10.1007/s11868-023-00576-w.
[2] M. Daoud, A class of parabolic reaction-diffusion systems governed by spectral fractional Laplacians : Analysis and numerical simulations, In final drafting phase.
[3] M. Daoud and E.-H. Laamri, Fractional Laplacians : a short survey, Discrete Contin. Dyn. Syst. – S, 15(1) (2022), 95–116.
[4] E.-H. Laamri, Global existence of classical solutions for a class of reaction-diffusion systems, Acta. Appl. Math. 115 (2011), 153-165.
[5] M. Pierre, Global Existence in Reaction-Diffusion Systems with Control of Mass : a Survey, Milan J. Math. 78 (2010), 417–455.
[6] A. Haraux and A. Youkana, On a result of K. Masuda concerning reaction-diffusion equations, Tôhoku Math. J. 40 (1988), 159–163.
[7] A. Barabanova, On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity, Proc. Amer. Math. Soc. 122 (1994), 827-831.
[8] M.A. Herrero, A.A. Lacey and J.L. Velàzquez, Global existence for reaction-diffusion systems modelling ignition, Arch. Rat. Mech. Anal., 142 (1998), 219–251.
