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

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 $$L\log L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>log</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> growth condition. We prove that the SPDE is well posed when the initial data is in $$L^2[0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . This solves a strong form of an open problem.