A class of parabolic reaction-diffusion systems governed by spectral fractional Laplacians : Analysis and numerical simulations

Abstract

In this paper, we prove the global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of the form $u_i \mapsto d_i (-\Delta)_{Sp}^{s_i} u_i$, where $0 < s_i < 1$. The operator $(-\Delta)_{Sp}^{s}$ stands for the commonly called spectral fractional Laplacian. Moreover, the nonlinear reaction terms are assumed to fulfill natural structural conditions that ensure the nonnegativity of the solutions and provide uniform control of the total mass. We establish the global existence of strong solutions under the assumption that the nonlinearities exhibit at most polynomial growth. Our results extend previous results obtained when the diffusion operators are of the form $u_i \mapsto d_i (-\Delta)^s u_i$, where $(-\Delta)^s$ denotes the widely known regional fractional Laplacian. Furthermore, we present some numerical simulations to address a theoretical question that remains open to date.