On the local well-posedness of randomly forced reaction-diffusion equations with L² initial data and a superlinear reaction term

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 Llog L L log L growth condition. We prove that the SPDE is well posed when the initial data is in L²[0,1] L 2 [ 0 , 1 ] . This solves a strong form of an open problem.