On the Well-Posedness of Stochastic Partial Differential Equations with Locally Lipschitz Coefficients

Abstract We consider the stochastic partial differential equation (SPDE) $$\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:msubsup> <mm