Global in time solutions to stochastic reaction-diffusion systems with superlinear reactions satisfying a triangular control of mass

1. Problem Statement

The paper addresses the global existence and uniqueness of mild solutions for systems of stochastic reaction-diffusion equations (SRDEs) perturbed by multiplicative noise. The system is defined on a bounded domain $D \subset \mathbb{R}^d$ with $C^2$ boundary:

$$\begin{cases} \partial_t u_i - d_i \Delta u_i = f_i(u) + \sum_{k=1}^r \sigma_{ik}(u) \dot{W}_k(t, x) & \text{in } (0, T) \times D, \\ u_i = 0 \text{ or } \partial_n u_i = 0 & \text{on } (0, T) \times \partial D, \\ u_i(0, x) = u_{i0}(x), \end{cases}$$

where $u = (u_1, \dots, u_m)$ represents the concentrations of $m$ species, $d_i > 0$ are diffusion coefficients, and $\dot{W}$ is a space-time Gaussian noise (white in time, colored in space).

The Core Challenge:
In the deterministic setting, global existence for such systems is well-understood under "mass control" assumptions (where reaction terms satisfy specific growth bounds on their sums). However, in the stochastic setting, standard existence proofs typically rely on strong dissipativity conditions (e.g., $f_i(s) \sim -s^{2k+1}$) or global Lipschitz continuity. These standard assumptions exclude many physically relevant models, such as:

• Chemical reaction networks governed by the law of mass action (e.g., reversible reactions $A+B \rightleftharpoons C+D$).
• Systems with superlinear growth that are not strongly dissipative but possess a "triangular" structure where reaction terms cancel each other out when summed.

The question of whether global solutions exist for stochastic systems with triangular mass control and superlinear reactions (without strong dissipativity) has remained open.

2. Methodology

The authors employ a novel approach that avoids the standard fixed-point arguments requiring strong dissipativity. Instead, they leverage the structural interplay between the reaction terms.

A. Truncation and Localization

To handle the superlinear growth of $f$ and $\sigma$, the authors define a sequence of truncated systems with globally Lipschitz coefficients ($f_n, \sigma_n$). This guarantees the existence of unique global mild solutions $u_n$ for the truncated systems. The goal is to prove that the explosion time $\tau_\infty$ of the original system is infinite almost surely. This is achieved by establishing uniform moment bounds (independent of the truncation parameter $n$) for the solutions $u_n$.

B. Decomposition of the Solution

The solution $u_{i,n}$ is decomposed into a deterministic part $v_{i,n}$ and a stochastic convolution $Z_{i,n}$:
$$u_{i,n}(t, x) = v_{i,n}(t, x) + Z_{i,n}(t, x)$$
where $Z_{i,n}$ satisfies the stochastic integral equation driven by the noise, and $v_{i,n}$ satisfies a deterministic PDE with the truncated reaction terms.

C. Exploiting Triangular Mass Control

The key innovation lies in handling the reaction terms $f_i$. The authors assume a Triangular Mass Control structure (Assumption 3):
$$\sum_{j=1}^k f_j(u) \leq C_k \left(1 + \sum_{j=1}^m u_j\right) \quad \text{for } k=1, \dots, m.$$
This structure allows them to sum the equations for $v_{i,n}$ in a specific order. By summing the first $k$ equations, they derive an inequality where the reaction terms are bounded by the total mass of the system.

D. Parabolic PDE Estimates and Induction

The authors utilize Lemma 3 (a duality result from Pierre's survey on deterministic systems) to bound the $L^p$ norms of the deterministic parts $v_{i,n}$.

1. Non-negativity: They first prove that solutions remain non-negative (Lemma 1), which is crucial for the mass control inequalities to hold.
2. Stochastic Estimates: They use the factorization method and BDG inequalities to bound the stochastic convolution $Z_{i,n}$ in terms of the $L^p$ norms of $u_{i,n}$.
3. Inductive Argument:
   • For the $2 \times r$ system, they derive a closed inequality for the $L^p$ norm of the sum of components.
   • For the general $m \times r$ system, they use mathematical induction. They show that if the $L^p$ bounds hold for the first $k$ components, the triangular structure allows them to bound the $(k+1)$-th component, eventually closing the estimate for the entire system.

E. Uniform Bounds and Global Existence

By combining the estimates for $v_{i,n}$ and $Z_{i,n}$, they derive an inequality of the form:
$$E \left[ \sup_{t \in [0, T]} \|u_n(t)\|_{L^\infty}^p \right] \leq C_T (1 + \|u_0\|_{L^\infty}^p)$$
Crucially, the constant $C_T$ is independent of $n$. This uniform bound implies that the probability of the solution exploding in finite time is zero (via Markov's inequality), thus proving global existence.

3. Key Assumptions

The results rely on the following specific structural assumptions:

1. Quasipositivity (Assumption 2): $f_i(u) \geq 0$ if $u_i = 0$ and $u_j \geq 0$. This ensures non-negativity of concentrations.
2. Triangular Mass Control (Assumption 3): The sum of the first $k$ reaction terms is bounded linearly by the total mass. This generalizes mass conservation and includes reversible chemical reactions.
3. Polynomial Growth (Assumption 4): Reaction terms grow polynomially (allowing superlinear growth).
4. Noise Structure: The noise is multiplicative, satisfies a linear growth condition, and vanishes when any component is zero (Assumption 6), preserving non-negativity. The noise covariance satisfies specific singularity conditions (Assumptions 9-11) related to the heat kernel.

4. Key Results

• Theorem 5.1 (2-component system): Proves global existence for $m=2$ under the stated assumptions.
• Theorem 6.1 (General system): Proves global existence for any finite number of components $m$ via induction.
• Uniform Moment Bounds: The paper establishes that for any $p \geq 1$ and time horizon $T$, the $p$-th moments of the supremum norm of the solution are uniformly bounded with respect to the truncation parameter $n$.
• Global Mild Solution: Consequently, the original stochastic system (1) admits a unique global mild solution that exists for all time $t \in [0, \infty)$ with probability one.

5. Significance and Contributions

1. Bridging Deterministic and Stochastic Theory: The work successfully extends the well-known global existence results for deterministic mass-controlled systems (Pierre's framework) to the stochastic setting. Previously, stochastic results required strong dissipativity, which ruled out many natural chemical models.
2. Handling Superlinear Reactions: The paper demonstrates that triangular mass control is sufficient to prevent blow-up even in the presence of superlinear reactions and multiplicative noise, without requiring the reaction terms to be strongly dissipative.
3. Applicability to Chemical Kinetics: The results directly apply to complex chemical reaction networks (e.g., reversible reactions like esterification or ligand binding) that were previously intractable in the stochastic setting.
4. Novel Proof Technique: The authors introduce a method that bypasses fixed-point arguments based on contraction mappings. Instead, they use the structural cancellation properties of the reaction terms combined with parabolic PDE estimates to close the existence argument.
5. Resolution of an Open Problem: The paper answers the open question regarding the global existence of solutions for stochastically perturbed mass-controlled systems, a problem that had remained unresolved since the work of Agresti (who showed delay of blow-up but not global existence without stringent decay assumptions).

In summary, this paper provides a rigorous mathematical foundation for modeling stochastic chemical reaction-diffusion systems with realistic, non-dissipative, superlinear kinetics, ensuring that these models are well-posed globally in time.