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Well-Posedness for Fractional Reaction-Diffusion Systems with Mass Dissipation in $\mathbb R^N$

Phuoc-Tai Nguyen, Bao Quoc Tang

TL;DR

The paper develops a global well-posedness theory for fractional-reaction-diffusion systems on $\mathbb{R}^N$ with mass dissipation and nonnegativity. It proves global existence and uniform-in-time bounds for quadratic nonlinearities, independent of the fractional order $\alpha$, by exploiting a Hölder-regularized framework for the nonlocal diffusion. For more general (super-quadratic) nonlinearities, a fractional Lp-Lq regularization combined with improved duality estimates yields a bootstrap that reaches $L^\infty$, under intermediate sum conditions with growth exponent $\rho$ depending on $\alpha$ and $N$. A non-divergence fractional diffusion Hölder regularity theory is developed to support the regularity and a key auxiliary estimate, and the mass-dissipation condition is shown to be equivalent to a mass-conservation formulation via augmentation. Overall, the work extends global well-posedness and boundedness results for mass-dissipating reaction-diffusion systems to nonlocal diffusion in unbounded domains, with explicit dependence of admissible nonlinear growth on the fractional order.

Abstract

The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate total mass. We first show that if the nonlinearities are at most quadratic then there exists a unique global bounded solution regardless of the fractional order. This is done by combining a regularizing effect of the fractional diffusion operator and the Hölder continuity of a non-local inhomogeneous parabolic equation. When the nonlinearities might be super-quadratic, but satisfy some intermediate sum conditions, we prove the global existence of bounded solutions by adapting the well-known duality methods to the case of fractional diffusion. In this case, the order of the intermediate sum conditions depends on the fractional order. These results extend the existing theory for mass dissipated reaction-diffusion systems to the case of non-local diffusion and unbounded domains.

Well-Posedness for Fractional Reaction-Diffusion Systems with Mass Dissipation in $\mathbb R^N$

TL;DR

The paper develops a global well-posedness theory for fractional-reaction-diffusion systems on with mass dissipation and nonnegativity. It proves global existence and uniform-in-time bounds for quadratic nonlinearities, independent of the fractional order , by exploiting a Hölder-regularized framework for the nonlocal diffusion. For more general (super-quadratic) nonlinearities, a fractional Lp-Lq regularization combined with improved duality estimates yields a bootstrap that reaches , under intermediate sum conditions with growth exponent depending on and . A non-divergence fractional diffusion Hölder regularity theory is developed to support the regularity and a key auxiliary estimate, and the mass-dissipation condition is shown to be equivalent to a mass-conservation formulation via augmentation. Overall, the work extends global well-posedness and boundedness results for mass-dissipating reaction-diffusion systems to nonlocal diffusion in unbounded domains, with explicit dependence of admissible nonlinear growth on the fractional order.

Abstract

The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate total mass. We first show that if the nonlinearities are at most quadratic then there exists a unique global bounded solution regardless of the fractional order. This is done by combining a regularizing effect of the fractional diffusion operator and the Hölder continuity of a non-local inhomogeneous parabolic equation. When the nonlinearities might be super-quadratic, but satisfy some intermediate sum conditions, we prove the global existence of bounded solutions by adapting the well-known duality methods to the case of fractional diffusion. In this case, the order of the intermediate sum conditions depends on the fractional order. These results extend the existing theory for mass dissipated reaction-diffusion systems to the case of non-local diffusion and unbounded domains.
Paper Structure (15 sections, 31 theorems, 280 equations)

This paper contains 15 sections, 31 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Problem setting and state of the art
  3. Main results and key ideas
  4. Preliminaries
  5. Systems with quadratic growth rates
  6. Regularization with Hölder continuity
  7. Global existence and boundedness
  8. Systems with intermediate sum conditions
  9. $L^p-L^q$-regularization of the fractional Laplacian
  10. Duality methods
  11. Global existence and boundedness
  12. Hölder continuity of a non-divergence fractional diffusion equation
  13. Appendix
  14. Fractional heat semigroup in the $L^p$ setting
  15. Fractional heat semigroup in the Hölder setting

Key Result

Theorem 1.2

Assume F, P, M and nonlinearities have at most quadratic growth rates, i.e. there exists a constant $C>0$ such that Then for any non-negative initial data ${\bm{u}}_0 = (u_{1,0},\ldots,u_{m,0}) \in (L^{1}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N))^m$, there exists a unique global non-negative mild solution to sys which is bounded uniformly in time, i.e.

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2: Quadratic growth rates
  • Remark 1.3
  • Theorem 1.4: Intermediate sum condition
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 3.1
  • ...and 53 more