Critical regularity and dissipativity for stochastic reaction-diffusion equations in Bochner spaces over spaces of continuous functions
In this paper, we consider the stochastic reaction-diffusion equation $\mathrm{d}u = (\mathcal{A} u + f(u))\mathrm{d}t + \sigma(u)\mathrm{d}W$ on a smooth bounded domain $\mathcal{O}$ with homogeneous Dirichlet boundary conditions. We investigate the long-time behavior of solutions with a strongly dissipative drift nonlinearity and superlinear multiplicative noise in the Bochner space $L^q(\Omega; C_0(\overline{\mathcal{O}}))$, $q \ge 2$. Here $\mathcal{A}$ is a second-order self-adjoint elliptic operator and $W$ is a two-sided trace-class Wiener process. The standard Galerkin method fails to yield energy estimates in $L^q(\Omega; L^q(\mathcal{O}))$ via the It\^o formula for $q > 2$, owing to the interference of projection operators when dealing with nonlinear terms; meanwhile, the classical theory of mild solutions lacks sufficient spatial regularity to apply the It\^o formula directly. To overcome these difficulties, we consider mild solutions and establish a critical regularity estimate for the corresponding stopped process $u_n(t)$ in $W_0^{1,q}(\mathcal{O})$, which rigorously justifies the use of the It\^o formula in the non-Hilbert space $L^q(\Omega; L^q(\mathcal{O}))$. As a result, we derive explicit moment energy estimates and quantitative dissipativity bounds, yielding global existence, uniqueness, and exponential asymptotic decay of solutions in $L^q(\Omega; C_0(\overline{\mathcal{O}}))$. Unlike previous qualitative results in continuous function spaces, our framework provides a fully quantitative theory of global dissipativity.