发表机构
Indian Institute of Technology Goa（印度理工学院果阿分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究由乘性噪声驱动、具粗糙初始数据的多维半线性随机演化方程，在系数全局Lipschitz时建解的适定性等，局部Lipschitz时证最大局部解相关性质，条件针对特定范数，可用于多种方程及处理奇异和非光滑初始数据。
AI 中文摘要
我们研究由乘性噪声驱动且具有粗糙初始数据的多维半线性随机演化方程。假设漂移和扩散系数取值于负分数阶空间，且在初始时刻可能呈现时间奇点。主要结果在系数为全局Lipschitz时建立了解的适定性和路径时空正则性；在系数为局部Lipschitz时证明了最大局部解的存在性、唯一性和路径正则性。局部Lipschitz条件针对特定时间加权范数给出。这使我们能将理论结果应用于线性随机偏微分方程以及具有非全局Lipschitz非线性的模型，如随机Burgers、Allen - Cahn、Fisher - KPP、Burgers - Fisher方程和Ginzburg - Landau系统。此外，我们的框架能处理奇异初始数据，如d = 1时的狄拉克测度，以及d ∈ {2, 3}时的非光滑初始数据。
英文摘要
We study multidimensional semilinear stochastic evolution equations driven by multiplicative noise and subject to rough initial data. The drift and diffusion coefficients are assumed to take values in negative fractional order spaces and may exhibit temporal singularities at the initial time. Our main results establish well-posedness and pathwise spatio-temporal regularity of the solutions when these coefficients are globally Lipschitz, and we prove the existence, uniqueness, and pathwise regularity of maximal local solutions when the coefficients are locally Lipschitz. The local Lipschitz condition is formulated with respect to a specific time-weighted norm. This enables the application of our theoretical results to linear stochastic partial differential equations, as well as to a broad class of models with non-globally Lipschitz nonlinearities. These include the stochastic Burgers, Allen--Cahn, Fisher--KPP, and Burgers--Fisher equations, alongside the Ginzburg--Landau system and models featuring exponential growth, such as the stochastic solid fuel ignition model.
CommentsExpanded the introduction with recent literature. Revised the proof of Theorem 2.4. Clarified parameter ranges in Lemmas 3.3 & 3.5, and made assumptions explicit in Lemmas 4.1 & 4.2. Updated SPDE notations globally for consistency. Added a new application example (Sec 4.3) and revised Sections 4.6 & 4.7 (stochastic Burgers and Burgers-Fisher equations). Minor typos fixed
