On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term

Mohammud Foondun; Davar Khoshnevisan; Eulalia Nualart

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arXiv · 2510.00214

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term

Mohammud Foondun·Davar Khoshnevisan·Eulalia Nualart

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Abstract

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 $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

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Mohammud Foondun, Davar Khoshnevisan, Eulalia Nualart. 2025-09-30. On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term. https://arxiv.org/abs/2510.00214

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