Global existence and stability in a class of chemotaxis systems with lethal interactions, nonlinear diffusion and production

Technical Summary: Global Existence and Stability in a Class of Chemotaxis Systems with Lethal Interactions

Problem Formulation
This paper investigates a class of chemotaxis systems modeling the interaction between bacterial populations (specifically E. coli) and a self-produced toxic chemical substance (Hydrogen peroxide, $H_2O_2$). The system accounts for lethal interactions where the chemical acts as both a chemorepellent and a mortality agent. The study focuses on two distinct mathematical formulations within a smooth, bounded domain $\Omega \subset \mathbb{R}^n$ ($n \ge 1$) subject to homogeneous Neumann boundary conditions:

1. Fully Parabolic System ($\tau=1$): Both the population density $u$ and chemical concentration $v$ evolve according to parabolic equations.
2. Parabolic-Elliptic System ($\tau=0$): The population density $u$ follows a parabolic equation, while the chemical concentration $v$ is governed by an elliptic equation (quasi-steady-state approximation).

The governing system (1.1) incorporates:

• Nonlinear Diffusion: Represented by $d_1 \nabla \cdot (D(u)\nabla u)$, where $D(s) \ge (s+1)^\alpha$.
• Nonlinear Chemotactic Repulsion: Modeled by $\chi \nabla \cdot (S(u)\nabla v)$, where $0 \le S(s) \le s(s+1)^\beta$.
• Logistic Growth and Lethality: A logistic growth term $ru(1-u^{\kappa-1})$ and a mortality term $-\mu uv$.
• Chemical Dynamics: Production proportional to $u^m$, decay rate $b$, and an external source $f(x,t)$.

Methodology
The authors employ a rigorous analytical framework combining several advanced techniques in partial differential equations (PDEs):

• Local Existence and Regularity: Standard fixed-point arguments and parabolic regularity theory are used to establish the local existence of classical solutions.
• A Priori Estimates: To prove global existence, the authors derive $L^p$-bounds for the solution $u$. This involves testing the equations with specific powers of $(u+1)$ and utilizing the Gagliardo-Nirenberg inequality to handle nonlinear terms.
• Maximal Sobolev Regularity: This theory is applied to the chemical equation to control the gradient of $v$ and the Laplacian $\Delta v$, which is crucial for handling the chemotactic cross-diffusion terms.
• Lyapunov Functionals: To analyze long-term behavior, carefully constructed energy functionals (Lyapunov functionals) are utilized. These functionals combine entropy-like terms for the population and $L^2$-norms for the chemical concentration.
• Moser-Alikakos Iteration: This technique is employed to upgrade $L^p$ bounds to $L^\infty$ bounds, ensuring the uniform boundedness of solutions.

Key Contributions and Results

1. Global Existence and Boundedness (Theorem 1.1):
   The paper establishes the existence of unique, globally bounded classical solutions for arbitrary spatial dimensions $n \ge 1$. The primary condition for global existence is a constraint on the exponents governing diffusion, chemotaxis, and production:
   $$\beta + m < \alpha + \frac{2}{n}$$
   This inequality ensures that the nonlinear diffusion ($\alpha$) is sufficiently strong to counteract the destabilizing effects of nonlinear chemotactic repulsion ($\beta$) and chemical production ($m$). Under this condition, the solutions satisfy:
   $$\|u(\cdot, t)\|_{L^\infty(\Omega)} + \|v(\cdot, t)\|_{W^{1,\infty}(\Omega)} \le C$$
   for all $t > 0$.

2. Asymptotic Stability of Coexistence States (Theorem 1.2):
   When the external supply $f$ is constant ($f_c$) and the interaction parameters satisfy $\mu f_c < br$ (allowing for a positive equilibrium), the system converges exponentially to a spatially homogeneous coexistence steady-state $(u^*, v^*)$. This stability holds provided:

   • $\kappa \ge 2$ and $m=1$.
   • The chemotactic sensitivity is sufficiently small: $\chi^2 < \frac{4d_1 d_2 \mu}{a u^*}$.
   • The diffusion dominates the chemotactic sensitivity in a specific sense: $2\beta \le \alpha$.

3. Asymptotic Stability of Extinction States (Theorem 1.3):
   In the regime where the lethal interaction is dominant ($\mu f_c \ge br$), the system converges exponentially to a "vanishing state" where the bacterial population goes extinct ($\bar{u}=0$) and the chemical concentration stabilizes at $\bar{v} = f_c/b$. This result assumes $\kappa=2$ and $m=1$.

Significance and Claims
The authors position this work as the first study to address lethal interactions in chemotaxis systems that simultaneously incorporate nonlinear diffusion and nonlinear production mechanisms.

• Extension of Keller-Segel Theory: The paper claims to extend the critical exponent analysis found in classical Keller-Segel models (where the critical threshold is typically $2/n$). By introducing the inequality $\beta + m < \alpha + 2/n$, the authors demonstrate how nonlinear diffusion ($\alpha$) can stabilize the system even in higher dimensions or with stronger chemotactic sensitivity, effectively recovering and extending previous results by Liu, Tao, and Winkler.
• Biological Relevance: The mathematical results provide a theoretical basis for understanding how E. coli populations might persist or collapse in the presence of self-generated toxins. The analysis suggests that nonlinear diffusion acts as a crucial stabilizing mechanism, preventing finite-time blow-up (population explosion) and ensuring that the system settles into either a stable coexistence or a controlled extinction state, depending on the balance between growth rates and lethal interactions.
• Robustness: The results hold for both fully parabolic and parabolic-elliptic settings, offering a comprehensive view of the system's dynamics under different time-scale assumptions for the chemical signal.

The paper concludes that under the derived structural assumptions, the complex interplay between chemotactic repulsion, nonlinear diffusion, and lethal interactions does not lead to unbounded growth, but rather leads to well-defined, stable long-term behaviors.