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> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$u=u(t,x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is defined for $$(t,x)\in (0,\infty )\times \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> and $$\dot{W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:math> denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u (0) is bounded and measurable, and b and $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.