Global Existence and Pathwise Uniqueness for a Stochastic Parabolic-Parabolic Keller-Segel System

Technical Summary: Global Existence and Pathwise Uniqueness for a Stochastic Parabolic-Parabolic Keller-Segel System

Problem Statement
This paper investigates the well-posedness of a stochastic parabolic-parabolic Keller-Segel system in a two-dimensional bounded domain $\mathcal{O} \subset \mathbb{R}^2$. The system models the chemotactic movement of cells (density $u$) toward a chemical substance (concentration $v$) while subject to random environmental fluctuations. The governing equations are:
$$\begin{cases} du = (\Delta u - \nabla \cdot (u \nabla v)) dt + \sigma(u) dW(t), \\ dv = (\Delta v - v + u) dt, \end{cases}$$
with homogeneous Neumann boundary conditions. Here, $W(t)$ is a cylindrical Wiener process, and the noise coefficient $\sigma(u)$ is nonlocal and nonlinear, specifically of the form $\sigma(u)e_k = b_k \|u\|_{L^p(\mathcal{O})}^r u$.

The study addresses a critical gap in the literature: while deterministic parabolic-parabolic Keller-Segel systems in 2D typically exhibit finite-time blow-up if the initial mass exceeds a critical threshold ($4\pi$), and stochastic parabolic-elliptic variants have been studied, the stochastic parabolic-parabolic case with nonlocal nonlinear noise acting only on the cell density equation had not been systematically analyzed. The primary analytical challenges include the fully parabolic coupling of the equations, the lack of coercivity and global Lipschitz continuity in both the chemotactic drift and the noise terms, and the absence of smallness assumptions on the initial data.

Methodology
The authors employ a three-step strategy to establish global existence and pathwise uniqueness for arbitrary initial data:

1. Truncation and Local Existence:
   To handle the lack of global Lipschitz continuity and coercivity, the authors introduce a truncated system (1.4). The truncation function $\theta_N$ depends on a combined energy and gradient dissipation quantity $\|\rho\|_{X_t}^2 = \sup_{s\in[0,t]} \|\rho(s)\|_{L^2}^2 + \int_0^t \|\nabla \rho(s)\|_{L^2}^2 ds$.

   • The chemotactic drift and noise coefficients are "frozen" to satisfy the conditions of Banach's fixed point theorem.
   • Using the variational existence theory for SPDEs (specifically the Gelfand triple framework and monotonicity/coercivity conditions), the authors prove the existence of a unique global strong solution $u_N$ for the truncated system on a fixed time interval.
   • By defining a stopping time $\tau_N$ based on the truncation threshold, they construct a local strong solution for the original system (1.3).

2. Pathwise Uniqueness:
   On a fixed stochastic basis, the authors establish pathwise uniqueness for local strong solutions. This involves deriving energy estimates for the difference $w = u_1 - u_2$ of two solutions.

   • They utilize the Gagliardo-Nirenberg inequality and specific estimates for the chemical concentration $v$ (derived from Lemma 2.2) to control the chemotactic term.
   • A stochastic Gronwall lemma is applied to show that if two solutions share the same initial data, they coincide almost surely up to the minimum of their existence times. This allows for the construction of a maximal local strong solution $(u, \tau^*)$.

3. Global Extension via Lyapunov Functional:
   To prove that the maximal solution does not blow up in finite time (i.e., $\tau^* = \infty$), the authors derive uniform estimates independent of the truncation parameter $N$.

   • They introduce a specialized Lyapunov functional $\Phi(x) = (1+x)^\alpha$ with $\alpha \in (\max\{0, \frac{2-r}{2}\}, \frac{1}{2})$.
   • By applying Itô's formula to $\Phi(\|u\|_{L^2}^2)$, they exploit the specific structure of the nonlocal noise. The second derivative of the concave function $\Phi$ generates a negative trace term that counteracts the aggregation effect of the chemotactic drift.
   • Through careful interpolation and Young's inequalities, they bound the noise and drift terms, showing that the expected value of the Lyapunov functional remains bounded uniformly in $N$.
   • Finally, Markov's inequality is used to demonstrate that the probability of the stopping time $\tau_N$ being less than any finite $T$ vanishes as $N \to \infty$, implying $\tau^* = \infty$ almost surely.

Key Contributions and Results

• Global Existence for Arbitrary Data: The paper establishes the existence of a global pathwise unique strong solution for the stochastic parabolic-parabolic Keller-Segel system with nonlocal nonlinear multiplicative noise. Crucially, this result holds for arbitrary initial data in $L^2(\mathcal{O}) \times W^{1,\infty}(\mathcal{O})$, without requiring the smallness conditions on initial mass that are necessary for the deterministic counterpart.
• Mechanism of Prevention: The analysis demonstrates that the specific nonlocal nonlinear noise structure ($\|u\|_{L^p}^r u$) is sufficient to counteract the chemotactic aggregation, preventing finite-time blow-up. This contrasts with deterministic models where blow-up is inevitable for large masses.
• Technical Novelty: The authors develop a tailored truncation procedure based on the combined energy-gradient norm rather than standard $L^2$ norms, which is essential for handling the fully parabolic coupling. Furthermore, the construction of a specific Lyapunov functional adapted to the nonlocal noise structure provides a new analytical tool for stochastic chemotaxis systems.

Significance
The paper claims that its results sharply contrast with the deterministic two-dimensional Keller-Segel system, where global existence is conditional on the initial mass being below a critical threshold ($4\pi$). By showing that appropriately chosen nonlocal nonlinear noise can ensure global existence regardless of the initial mass, the work highlights the stabilizing role of environmental fluctuations in chemotaxis models. This provides a rigorous mathematical foundation for understanding how stochastic perturbations can prevent the formation of singularities in biological aggregation models, extending previous results from parabolic-elliptic settings to the more complex fully parabolic case.