A class of fractional parabolic reaction–diffusion systems: analysis and numerics

Fractional diffusion operators arise naturally in the modeling of anomalous transport and nonlocal diffusion phenomena. While several equivalent definitions of the fractional Laplacian coexist in the full space R d , different realizations arise when the problem is posed in a bounded domain; we refer to [3] for an overview. In this talk, we focus on global-in-time existence for a class of parabolic reaction–diffusion systems. The nonlinear reaction terms are assumed to satisfy natural structural conditions ensuring nonnegativity of the solutions and uniform control of the total mass. First, we recall some classical results for systems driven by the usual Laplacian, for which global existence is already a delicate issue and typically requires additional assumptions on the reaction terms; see [4–6]. Then, we investigate to what extent these results remain valid when the classical diffusion is replaced by fractional diffusion. In particular, based on [1, 2], we present global existence results for systems with polynomially growing nonlinearities under a triangular structure, as well as for reversible chemical reactions. Finally, we present some numerical simulations to investigate systems with exponentially growing reaction terms, for which global existence remains an open problem.

References

[1] M. Daoud, E.-H. Laamri, 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).

[2] M. Daoud, A class of parabolic reaction–diffusion systems governed by spectral fractional Laplacians: analysis and numerical simulations, arXiv:2502.13771v2 (2026), submitted to J. Evol. Equ..

[3] M. Daoud, E.-H. Laamri, Fractional Laplacians: a short survey, Discrete Contin. Dyn. Syst. Ser. 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] S.L. Hollis, R.H. Martin, M. Pierre, Global existence and boundedness in reaction–diffusion systems, SIAM J. Math. Anal. 18 (1987), 744–761.

[6] M. Pierre, Global existence in reaction–diffusion systems with control of mass: a survey, Milan J. Math. 78 (2010), 417–455.