关键词: 随机反应-扩散方程, 强耗散漂移项, 超线性噪声, Bochner空间, 弱均值吸引子

Abstract:  This paper investigates global solutions and long-time dynamics for the stochastic reaction-diffusion equation $\mathrm{d}u = \left(\Delta u + f(u) + g(x,t)\right)\mathrm{d}t + \sigma(u)\mathrm{d}W$ on a bounded domain, where the drift term $f(u)$, with polynomial growth rate $\beta$, is strongly dissipative and the diffusion term $\sigma(u)$ has growth rate $\gamma$, satisfying $\beta + 1 > 2\gamma$. Under this condition, we establish the existence, uniqueness, and regularity of solutions in Bochner spaces. Our analysis relies only on weak monotonicity conditions and requires no further growth restrictions on $f$ and $\sig$. Moreover, we prove the existence of a weak mean random attractor for the system. These results offer new insights into the balance mechanism between stochastic perturbations and dissipative effects in superlinear regimes.

Key words: Stochastic reaction-diffusion equation, Strongly dissipative drift term, Superlinear noise, Bochner space, Weak mean attractor