Nonlocal logistics and nonlinear productions in an attraction-repulsion chemotaxis model: analysis of the global well-posedness

1. Problem Statement

The paper investigates the global well-posedness (existence, uniqueness, and boundedness) of a three-component chemotaxis system describing the interaction between cell density and two chemical signals: an attractant ($v$) and a repellent ($w$). The model incorporates a nonlocal logistic-type source term to regulate population growth and prevent blow-up.

The system is defined on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) with zero-flux boundary conditions:

$$\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w) + au^\alpha - bu^\alpha \int_\Omega u^\beta, \\ \tau v_t = \Delta v - v + f(u), \\ \tau w_t = \Delta w - w + g(u), \end{cases}$$

Key Components:

• $u(x,t)$: Cell density.
• $v(x,t)$: Attractive chemical signal (chemoattractant).
• $w(x,t)$: Repulsive chemical signal (chemorepellent).
• $\chi, \xi > 0$: Chemotactic sensitivity coefficients for attraction and repulsion, respectively.
• Nonlocal Source: The term $au^\alpha - bu^\alpha \int_\Omega u^\beta$ represents nonlinear growth ($\alpha$) and nonlocal competition/damping ($\beta$). Unlike local logistic terms, the death rate depends on the spatial average of the population.
• Production Functions: $f(u) \sim u^\ell$ and $g(u) \sim u^\rho$ describe the production rates of the attractant and repellent, respectively.
• $\tau \in \{0, 1\}$: Determines the nature of the signal equations.
  • $\tau = 0$: Parabolic-Elliptic case (signals adjust instantaneously to cell density).
  • $\tau = 1$: Fully Parabolic case (signals have their own time evolution).

Objective: To establish conditions on the exponents $\alpha, \beta, \ell, \rho$ that guarantee the global existence and uniform boundedness of classical solutions, thereby preventing finite-time blow-up (singularity formation).

2. Methodology

The authors employ a rigorous analytical framework combining energy methods, regularity theory, and interpolation inequalities.

• Local Existence: The paper first establishes the existence of a unique, non-negative local classical solution using standard fixed-point arguments and parabolic regularity theory. A maximal existence time $T_{max}$ is defined, with the blow-up criterion being $\limsup_{t \to T_{max}} \|u(\cdot, t)\|_{L^\infty} = \infty$.
• A Priori Estimates: The core of the proof involves deriving uniform-in-time $L^k(\Omega)$ estimates for the cell density $u$ for sufficiently large $k$.
  • Energy Functional: The authors define an energy functional $\Phi_\tau(t) = e^{\tau t} \int_\Omega u^k$.
  • Differentiation & Integration by Parts: They compute the time derivative of $\Phi_\tau(t)$, utilizing the Neumann boundary conditions to handle diffusion and chemotaxis terms.
  • Handling Chemotaxis Terms: The cross-diffusion terms ($\nabla \cdot (u\nabla v)$ and $\nabla \cdot (u\nabla w)$) are handled using integration by parts, converting them into terms involving $\Delta v$ and $\Delta w$. These are then bounded using the elliptic/parabolic equations for $v$ and $w$.
  • Nonlocal Damping: The critical term $-b \int_\Omega u^\beta \int_\Omega u^{k+\alpha-1}$ is used to counteract the growth terms.
• Functional Inequalities:
  • Gagliardo-Nirenberg Inequalities: Used to interpolate between $L^p$ norms and gradient norms ($\|\nabla u^{k/2}\|_{L^2}$).
  • Young's Inequality: Used to absorb higher-order growth terms into the dissipative gradient terms or the nonlocal damping term.
  • Elliptic/Parabolic Regularity: Lemmas are used to bound the signals $v$ and $w$ (and their Laplacians) in terms of $u$, ensuring that the signal dynamics do not destabilize the system.
• Case Analysis: The proofs are split based on the value of $\tau$ (Elliptic vs. Parabolic) and the relative magnitudes of the exponents (growth-dominated vs. signal-dominated regimes).

3. Key Contributions

1. Extension of Existing Models: The paper generalizes previous results (specifically extending work in [5]) by:
   • Introducing a repulsive signal ($w$) alongside the attractant.
   • Incorporating nonlocal logistic sources ($u^\alpha \int u^\beta$) instead of local ones.
   • Allowing general nonlinear production functions ($f(u) \sim u^\ell, g(u) \sim u^\rho$) rather than linear ones.
2. Dual Regime Analysis: The authors provide distinct, sharp conditions for both the Parabolic-Elliptic ($\tau=0$) and Fully Parabolic ($\tau=1$) cases, highlighting how the time-evolution of chemical signals affects the stability thresholds.
3. Mechanism of Stabilization: The work rigorously demonstrates that nonlocal damping ($\beta$) combined with chemorepulsion ($\xi$) is sufficient to suppress the aggregation caused by chemotaxis, even when local growth ($\alpha$) and signal production ($\ell, \rho$) are aggressive.

4. Main Results

The paper establishes global boundedness under specific relationships between the exponents.

Theorem 1.1: Parabolic-Elliptic Case ($\tau = 0$)

Global boundedness holds if:

• Case A (Growth-dominated): $\beta > \frac{n(\alpha-1)}{2}$ and $\ell \leq \min\{\alpha-1, \rho\}$.
  • Interpretation: If attractant production is weak relative to growth and repulsion, the nonlocal competition ($\beta$) is the primary regulator.
• Case B (Signal-dominated): $\beta > \frac{n\ell + 2(\ell-\alpha+1)}{2}$ and $\alpha-1 < \min\{\ell, \rho\}$.
  • Interpretation: If attractant production is strong, the repulsion exponent $\rho$ must be high enough to provide a spatial buffer, and $\beta$ must scale strictly to compensate for the intense signaling.

Theorem 1.3: Fully Parabolic Case ($\tau = 1$)

Global boundedness holds if:

• Case A (Proliferation-dominated): $\beta > \frac{n(\alpha-1)}{2}$ and $\alpha-1 \geq \max\{\rho, \ell\}$.
  • Interpretation: When signaling is mild, nonlocal damping regulates local birth rates.
• Case B (High-Intensity Signaling): $\beta > \frac{n \max\{\rho, \ell\} + 2(\max\{\rho, \ell\} - \alpha + 1)}{2}$ and $\alpha-1 < \min\{\rho, \ell\}$.
  • Interpretation: In regimes where either the attractant or repellent production is highly potent, the nonlocal death rate $\beta$ must be significantly larger to counteract the cross-diffusive fluxes.

Significance of Results:

• The results show that nonlocal damping can compensate for arbitrarily large local growth ($\alpha$), provided $\beta$ is sufficiently large. This contrasts with local logistic models where blow-up can occur for large $\alpha$.
• The presence of a repellent ($\xi > 0$) acts as a crucial partner to nonlocal damping in maintaining structural stability.

5. Significance and Impact

• Biological Relevance: The model captures complex biological scenarios (e.g., immune response, tumor invasion) where cells respond to both attractive and repulsive cues in a spatially averaged environment. The findings suggest that spatially distributed competition (nonlocal death) is a robust mechanism for preventing population collapse or uncontrolled aggregation.
• Mathematical Advancement: The paper resolves the well-posedness of a complex, coupled system with mixed parabolic-elliptic structures and nonlocal nonlinearities. It provides a comprehensive set of conditions that unify and extend previous literature on chemotaxis with logistic sources.
• Future Directions: The authors note that while global existence is proven, future work could explore the asymptotic behavior (steady states) and the specific thresholds for pattern formation within these stable regimes.

In summary, this paper provides a rigorous mathematical proof that nonlocal logistic damping and chemorepulsion are sufficient mechanisms to ensure the global existence and boundedness of solutions in attraction-repulsion chemotaxis models, even under aggressive growth and signal production conditions.