Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments

This paper establishes the existence, uniqueness, and global asymptotic stability of positive entire solutions for a parabolic-parabolic chemotaxis model with logistic sources in heterogeneous bounded domains by identifying specific parameter regions where the system converges to a unique steady state regardless of initial conditions.

The Gist

Imagine a bustling city where two types of residents live: The People (represented by $u$) and The Scent (represented by $v$).

In this city, the "People" have a special ability: they can smell the "Scent" and move toward it. This is called chemotaxis. If the scent is strong in one area, the people flock there. However, the people also produce the scent themselves. So, as more people gather, the scent gets stronger, which attracts even more people. It's a feedback loop.

But this city isn't empty. It has rules:

1. Food is limited: If too many people crowd together, they start fighting for resources (competition).
2. Cooperation exists: Sometimes, the total number of people in the whole city helps them survive, or sometimes it hurts them, depending on the specific rules of the city.
3. The environment changes: The city isn't uniform. Some neighborhoods have better food, some have worse. The rules change from day to day and place to place.

The Problem: Chaos vs. Order

The mathematicians in this paper asked a big question: If we start with a random mix of people and scent, will the city eventually settle down into a predictable, stable pattern?

In the past, scientists studied this in "perfect" cities (where everything is the same everywhere). They found that if the "smell sensitivity" (how strongly people follow the scent) isn't too high, the city settles down. But in a real, messy, changing city (heterogeneous environment), nobody knew if the city would ever stop fluctuating or if it would have a single, unique way of organizing itself.

The Solution: Finding the "Sweet Spot"

The author, Tahir Bachar Issa, figured out exactly when this chaotic city becomes stable and predictable.

Think of the "smell sensitivity" ($\chi$) as the volume knob on a radio.

• Volume too low: The people don't move much; they just sit where they are.
• Volume too high: The people get too excited by the scent. They rush to one spot, crowd it, fight, and the whole system becomes unstable and chaotic.
• The Sweet Spot: The author found a specific range for the volume knob. If the sensitivity is kept within this range, the city naturally finds a unique, stable rhythm.

The "Unique Rhythm" (The Main Discovery)

The paper proves two amazing things:

1. There is only one "True" way the city can live. No matter how you start the city (whether you dump all the people in the north corner or scatter them randomly), if you wait long enough, they will all arrange themselves into the exact same pattern of population and scent. There are no "alternative realities" or multiple stable states.
2. It's unshakeable. If you throw a rock into the pond (add a few extra people or change the scent slightly), the city will eventually calm down and return to that exact same unique pattern.

How Did They Prove It? (The Analogy of the "Shadow")

To prove this, the author used a clever mathematical trick called the "Method of Eventual Comparison."

Imagine you have a "Ghost City" (the ideal, stable solution) and a "Real City" (the one you are actually watching).

• At first, the Real City might be very different from the Ghost City.
• The author created a "Shadow" that bounds the Real City. He proved that the Real City is trapped between an "Upper Shadow" and a "Lower Shadow."
• He then showed that these shadows are like a pair of closing jaws. As time goes on, the Upper Shadow gets lower and the Lower Shadow gets higher.
• Eventually, the shadows squeeze together so tightly that the Real City has no room to wiggle—it must become identical to the Ghost City.

Why Does This Matter?

In the real world, this applies to:

• Bacteria: How they form colonies in a petri dish with uneven nutrients.
• Immune Systems: How white blood cells swarm to fight an infection in a complex body.
• Tumor Growth: How cancer cells spread and organize themselves.

The paper tells us that even in a messy, changing world, biological systems have a natural tendency to find a single, stable order, provided they aren't too sensitive to their own signals. It's a mathematical guarantee that life, under the right conditions, seeks balance.

Technical Summary

1. Problem Statement

The paper investigates the long-term asymptotic behavior of a fully parabolic-parabolic chemotaxis system with a logistic-type source in a bounded, heterogeneous environment. The system models the interaction between a mobile species (population density $u$) and a chemical substance (density $v$) produced by the species.

The mathematical model is given by:
$$\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + u \left( a_0(t, x) - a_1(t, x)u - a_2(t, x)\int_{\Omega} u \right), & x \in \Omega \\ \tau v_t = \Delta v - \lambda v + \mu u, & x \in \Omega \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0, & x \in \partial \Omega \end{cases}$$
Key Features:

• Heterogeneity: The coefficients $a_0, a_1, a_2$ depend on both time $t$ and space $x$, representing a non-uniform environment.
• Nonlocal Competition/Cooperation: The term $a_2(t, x)\int_{\Omega} u$ introduces a nonlocal effect where the total mass of the population influences local growth. If $a_2 > 0$, it represents global competition; if $a_2 < 0$, it represents global cooperation.
• Full Parabolicity: Unlike many previous studies that assume the chemical diffuses infinitely fast (parabolic-elliptic limit, $\tau=0$), this model retains the time derivative $\tau v_t$ with $\tau > 0$, making the analysis significantly more complex.
• Objective: The paper aims to establish the uniqueness and global asymptotic stability of positive entire solutions (solutions defined for all $t \in \mathbb{R}$) under specific parameter constraints.

2. Methodology

The author employs a combination of a priori estimates, comparison principles, and iterative contraction arguments. The methodology is structured as follows:

A. Preliminary Bounds and Existence

Building on previous work (specifically reference [17]), the paper assumes the existence of global bounded classical solutions and positive entire solutions under a specific hypothesis (H1) regarding the competition coefficients.

• Boundedness: Establishes uniform bounds for $u$ and $v$ in $L^\infty$ and $W^{2,\infty}$ norms after a transient time.
• Regularity: Uses fractional power spaces of the Laplacian to derive uniform bounds on the gradient of the entire solution $u^*$, specifically $\|u^*\|_{C^1(\bar{\Omega})}$. This is crucial for handling the nonlinear chemotactic term.

B. Transformation to Ratio Variables

To analyze stability, the author introduces the ratio variables:
$$w(t, x) = \frac{u(t, x)}{u^*(t, x)}, \quad \phi(t, x) = \frac{v(t, x)}{v^*(t, x)}$$
where $(u^*, v^*)$ is a known positive entire solution. The system is transformed into a new system for $(w, \phi)$ (Equation 3.1). The goal is to show that $w \to 1$ and $\phi \to 1$ as $t \to \infty$.

C. The "Eventual Comparison" Method

The core of the proof relies on constructing a two-species competition system (Equation 3.5) that acts as a supersolution and subsolution for the ratio $w$.

• The author derives a differential inequality for $w$ involving a perturbation term $C$ (related to the difference in the chemical gradient $\nabla(v^* - v)$).
• By Lemma 2.3, this perturbation term decays exponentially or is bounded by the $L^\infty$ norm of the difference $|u - u^*|$.
• The comparison system (3.5) is a scalar ODE system for upper and lower bounds ($\bar{w}$ and $\underline{w}$) of $w$.

D. Iterative Contraction Argument

The proof of stability (Theorem 1.2) uses an induction argument:

1. Base Case: Show that for large $t$, $w$ is bounded within a small neighborhood of 1, determined by the chemotactic sensitivity $\chi$ and the competition parameters.
2. Inductive Step: Assume the error $|u - u^*|$ is bounded by a term proportional to $(C|\chi|)^n$. Using the decay properties of the chemical equation and the comparison principle, the author shows that the error for the next step is proportional to $(C|\chi|)^{n+1}$.
3. Convergence: If the chemotactic sensitivity $\chi$ is sufficiently small such that $C|\chi| < A_0 - A_1 - A_2$, the error term converges to zero as $n \to \infty$.

3. Key Contributions

1. Extension to Heterogeneous Environments: The results generalize previous uniqueness and stability theorems (e.g., Winkler [42]) from homogeneous, convex domains to bounded, non-convex, heterogeneous domains with time-dependent coefficients.
2. Full Parabolic System: The paper addresses the fully parabolic case ($\tau > 0$), which is mathematically more challenging than the parabolic-elliptic limit due to the lack of instantaneous smoothing for $v$.
3. Nonlocal Effects: The analysis explicitly incorporates the nonlocal logistic term ($\int_\Omega u$), distinguishing between local competition and global cooperation/competition scenarios.
4. Explicit Parameter Regions: The paper provides explicit conditions (Hypothesis H2) on the model parameters ($a_1, a_2, \chi, \mu, \lambda$) that guarantee uniqueness and stability. Specifically, it requires the local competition strength ($a_1$) to dominate the nonlocal effects and the chemotactic aggregation.

4. Main Results

Theorem 1.2 (Uniqueness and Stability):
Under hypotheses (H1) and (H2), there exists a threshold $\chi_1 > 0$ such that for any chemotactic sensitivity $|\chi| \leq \chi_1$:

• Uniqueness: The system possesses a unique positive entire solution $(u^*(t, x), v^*(t, x))$.
• Global Asymptotic Stability: For any non-negative initial data $(u_0, v_0)$ with $u_0 \not\equiv 0$, the global classical solution $(u(t), v(t))$ converges uniformly to the entire solution as $t \to \infty$:
  $$\lim_{t \to \infty} \left( \sup_{t_0 \in \mathbb{R}} \|u(t+t_0) - u^*(t+t_0)\|_\infty + \sup_{t_0 \in \mathbb{R}} \|v(t+t_0) - v^*(t+t_0)\|_\infty \right) = 0$$

Hypothesis (H2) Condition:
The stability relies on the condition:
$$\eta a_{1, \inf} > |\Omega| M \left( (a_2^+)_{\sup} + (a_2^-)_{\sup} \right)$$
This implies that the local self-limitation (competition) must be strong enough to overcome the destabilizing effects of nonlocal interactions and chemotactic aggregation.

5. Significance

• Biological Relevance: The results confirm that in realistic, spatially and temporally varying environments, chemotactic populations can still settle into a unique, stable dynamic pattern if the internal competition is sufficiently strong. This prevents blow-up or chaotic oscillations in the presence of nonlocal resource competition.
• Mathematical Rigor: The paper resolves open questions regarding the uniqueness of entire solutions for fully parabolic chemotaxis systems in heterogeneous settings. The use of the "eventual comparison" method adapted for non-convex, heterogeneous domains is a significant technical advancement.
• Robustness: The findings suggest that the qualitative behavior of the system (stability) is robust against environmental heterogeneity, provided the chemotactic sensitivity remains below a critical threshold determined by the competition parameters.

In summary, this paper provides a rigorous mathematical framework ensuring that complex chemotaxis models with nonlocal interactions and environmental variations possess a single, stable long-term state, provided that local competition dominates chemotactic aggregation.