On the Local Well-Posedness of Randomly Forced Reaction-Diffusion Equations With L2 Initial Data and a Superlinear Reaction Term

• Authors: Mohammud Foondun, Davar Khoshnevisan and Eulàlia Nualart
• Probability Theory and Related Fields, April 2026

We consider a parabolic stochastic partial differential equation (SPDE) on [0, 1] that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an LlogL growth condition. We prove that the SPDE is well posed when the initial data is in L2[0,1]. This solves a strong form of an open problem.