Critical regularity and dissipativity for stochastic reaction-diffusion equations in Bochner spaces over spaces of continuous functions

1. Problem Statement

The paper investigates the long-time behavior of solutions to the following stochastic reaction-diffusion equation (SRDE) on a smooth bounded domain $\mathcal{O} \subset \mathbb{R}^d$:
$$du = (Au + f(u))dt + \sigma(u)dW, \quad u|_{\partial\mathcal{O}} = 0$$
where:

• $A$ is a second-order self-adjoint elliptic operator.
• $f$ and $\sigma$ are superlinear functions (polynomial growth).
• $W$ is a two-sided trace-class Wiener process.
• The analysis is conducted in the Bochner space $L^q(\Omega; C_0(\mathcal{O}))$ with $q \ge 2$, where $C_0(\mathcal{O})$ denotes the space of continuous functions vanishing on the boundary.

The Core Challenge:
Standard methods for establishing global existence and dissipativity (specifically energy estimates via the Itô formula) face two major obstacles in this setting:

1. Failure of Galerkin Approximations in $L^q$ ($q>2$): In Hilbert spaces ($q=2$), the projection operator $P_N$ commutes with the nonlinearity in energy estimates due to self-adjointness. However, for $q > 2$, $L^q(\mathcal{O})$ is not a Hilbert space. The projection $P_N$ cannot be removed from terms like $\int |u_N|^{q-2}u_N P_N f(u_N) dx$, preventing the application of dissipativity conditions.
2. Insufficient Regularity of Mild Solutions: While mild solutions avoid the projection issue, they typically lack the spatial regularity ($W^{1,q}_0(\mathcal{O})$) required to directly apply the Itô formula to the functional $\|u\|_{L^q}^q$. Specifically, the critical regularity index $\alpha = 1/2$ (corresponding to $W^{1,q}_0$) is difficult to establish for stochastic convolutions due to the singularity of the kernel in the stochastic integral.

2. Methodology

The authors develop a novel framework combining mild solution theory, critical regularity estimates, and a double-index approximation argument.

A. Critical Regularity Estimate for Stopped Processes

To bypass the projection issue, the authors work with mild solutions. To apply the Itô formula, they must prove that the stopped process $u_n(t) = u(t \wedge \tau_n)$ belongs to $W^{1,q}_0(\mathcal{O})$ (i.e., $D(A_q^{1/2})$).

• Decomposition Strategy: They decompose the noise coefficient $\sigma(u_n(s))$ to handle the singularity at the critical exponent $\alpha = 1/2$:
  $$\sigma(u_n(s)) = [\sigma(u_n(s)) - \sigma(u_n(t))] + [\sigma(u_n(t)) - \sigma(0)] + \sigma(0)$$
• Estimation:
  • The first two terms are estimated using the Hölder continuity of the solution and the smoothing properties of the semigroup, canceling the kernel singularity $(t-s)^{-1}$.
  • The constant term $\sigma(0)$ requires a weaker noise intensity condition (Hypothesis H1*) to ensure convergence.
• Result: They establish a global estimate:
  $$\mathbb{E}\|A_q^{1/2} u_n(t)\|_{L^q}^q \le C(t^\kappa + 1)$$
  This rigorously justifies the use of the Itô formula in the non-Hilbert space $L^q(\Omega; L^q(\mathcal{O}))$.

B. Derivation of Dissipativity

Using the critical regularity, the authors apply the Itô formula to $\|u_n(t)\|_{L^q}^q$.

• They derive explicit moment energy estimates and quantitative dissipativity bounds.
• These estimates show that the strong dissipative drift $f(u)$ counteracts the expansive effect of the superlinear noise $\sigma(u)$, preventing blow-up.

C. Double-Index Approximation

The initial regularity estimate requires initial data in a subspace $L^q(\Omega; D(A_q^{1/2}))$ and specific noise conditions. To extend these results to the natural space $L^q(\Omega; C_0(\mathcal{O}))$ with general initial data:

1. Approximation of Initial Data: Approximate $u_0 \in L^q(\Omega; C_0(\mathcal{O}))$ by $u_0^m \in L^q(\Omega; D(A_q^{1/2}))$.
2. Approximation of Noise: Approximate the noise intensity $\mu_j$ by $\mu_j^m$ to satisfy the weaker condition (H1*).
3. Convergence: Prove that the sequence of approximate solutions $\{u^m\}$ is Cauchy in $L^q(\Omega; C_0(\mathcal{O}))$ using uniform energy estimates and Gronwall's inequality, converging to the unique global mild solution.

3. Key Contributions

1. Critical Regularity at $\alpha = 1/2$: The paper establishes, for the first time in this context, global explicit moment estimates for the critical regularity index $\alpha = 1/2$ for standard semilinear SPDEs. This bridges the gap between mild solutions and the applicability of the Itô formula in non-Hilbert spaces.
2. Quantitative Dissipativity in $C_0(\mathcal{O})$: Unlike previous works that provided only qualitative results (existence/uniqueness) in continuous function spaces, this paper provides fully quantitative global dissipativity estimates.
3. Overcoming Non-Hilbert Obstacles: The authors successfully navigate the failure of Galerkin methods in $L^q$ ($q>2$) by utilizing a refined decomposition of the noise term and semigroup smoothing properties, avoiding the projection operator entirely.
4. Explicit Decay Rates: The derived estimates yield explicit exponential decay rates for the moments of the solution.

4. Main Results

Theorem 1.1 (Special Case):
For superlinear drift $f(u) \sim -|u|^{\beta-1}u$ ($b_\beta < 0$) and polynomial noise $\sigma(u) \sim |u|^\gamma$ with $\beta + 1 > 2\gamma$, the mild solution exists globally. Furthermore, for any $q > 2(d+2)$, the solution satisfies the quantitative dissipative estimate:
$$\mathbb{E}\|u(t)\|_{C_0}^q \le C \left( \|u_0\|_{L^{q\beta}}^{q\beta} e^{-\bar{c}(t-1)} + 1 \right), \quad t \ge 1$$
where $\bar{c} > 0$ depends on the dissipativity coefficient of $f$.

Theorem 4.2 (General Case):
Under general polynomial growth and one-sided Lipschitz conditions (H2, H3), the mild solution exists globally in time ($\tau = \infty$ a.s.) and exhibits mean dissipativity in $L^q(\Omega; C_0(\mathcal{O}))$ with explicit exponential decay bounds.

5. Significance

• Theoretical Advancement: This work extends the theory of stochastic reaction-diffusion equations beyond Hilbert spaces ($L^2$) to Banach spaces of continuous functions ($C_0$), which are physically more relevant for modeling concentrations and densities where pointwise values matter.
• Methodological Innovation: The technique of decomposing the noise term to handle the critical singularity in stochastic integrals offers a new tool for analyzing SPDEs in non-Hilbert settings.
• Applications: The quantitative nature of the results (explicit decay rates) is crucial for studying the uniqueness of invariant measures, exponential ergodicity, and the long-term statistical behavior of physical systems modeled by these equations. It improves upon previous results (e.g., by Salins) by providing explicit moment bounds rather than just qualitative existence.