Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Maha Daoud

TL;DR

The paper addresses global existence of strong solutions for systems of fractional parabolic reaction-diffusion equations with spectral fractional Laplacians $(-\Delta)_{Sp}^{s_i}$ and varying orders $s_i\in(0,1)$. It develops a fractional extension of Pierre's duality lemma and leverages maximal-regularity estimates to derive a priori $L^p$ and $L^\infty$ bounds, enabling global existence under polynomial growth and structural assumptions such as quasi-positivity and mass control. Two main results are established: global existence for a reversible three-species reaction under specific relations between the fractional orders and exponents, and global existence for systems with a triangular structure, both extending classical Laplacian results to the fractional spectral setting. Numerical simulations of a fractional Brusselator support the plausibility of global nonnegative solutions and illuminate the open theoretical status by demonstrating stable dynamics under the fractional diffusion operators.

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 $u_i \mapsto d_i (-Δ)_{Sp}^{s_i} u_i$, where $0<s_i<1$. The operator $(-Δ)_{Sp}^{s}$ stands for the commonly called spectral fractional Laplacian. Moreover, the nonlinear reactive 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 $u_i\mapsto d_i (-Δ)^s u_i$, where $(-Δ)^s$ denotes the widely known as regional fractional Laplacian. Furthermore, we use numerical simulations to investigate the global existence of solutions to the fractional version of the so-called ``Brusselator'' system, a theoretical question that remains open to date.

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

TL;DR

The paper addresses global existence of strong solutions for systems of fractional parabolic reaction-diffusion equations with spectral fractional Laplacians and varying orders . It develops a fractional extension of Pierre's duality lemma and leverages maximal-regularity estimates to derive a priori and bounds, enabling global existence under polynomial growth and structural assumptions such as quasi-positivity and mass control. Two main results are established: global existence for a reversible three-species reaction under specific relations between the fractional orders and exponents, and global existence for systems with a triangular structure, both extending classical Laplacian results to the fractional spectral setting. Numerical simulations of a fractional Brusselator support the plausibility of global nonnegative solutions and illuminate the open theoretical status by demonstrating stable dynamics under the fractional diffusion operators.

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 . The diffusion operators are of the form , where . The operator stands for the commonly called spectral fractional Laplacian. Moreover, the nonlinear reactive 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 , where denotes the widely known as regional fractional Laplacian. Furthermore, we use numerical simulations to investigate the global existence of solutions to the fractional version of the so-called ``Brusselator'' system, a theoretical question that remains open to date.

Paper Structure

This paper contains 16 sections, 10 theorems, 114 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectral fractional Laplacian ($SFL$)
  4. Functional framework
  5. First definition of the $(SFL)$
  6. Semigroup properties
  7. Second definition of the $(SFL)$
  8. Evolution Problem governed by the $(SFL)$ and its dual
  9. Fractional evolution problem
  10. Dual of Problem \ref{['ParabolicProblem']}
  11. Global existence for reaction-diffusion systems governed by spectral fractional Laplacians
  12. Local existence
  13. Fractional version of Pierre's duality lemma
  14. Global existence : case of reversible chemical reaction with three species
  15. Global existence : case of triangular structure
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $h\in L^1(0,T; L^p(\Omega))$ and $w_0\in L^p(\Omega)$. Then, Problem ParabolicProblem admits a unique weak solution.

Figures (2)

  • Figure 1: First component of the solution for different values of $(s_1,s_2,d_1,d_2)$ at $t_{\text{final}}=5\times 10^{10}$
  • Figure 2: Second component of the solution for different values of $(s_1,s_2,d_1,d_2)$ at $t_{\text{final}}=5\times 10^{10}$

Theorems & Definitions (23)

  • Definition 2.1: Weak solution
  • Definition 2.2: Strong solution
  • Theorem 2.1: Pazy1983
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 13 more