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<p>We consider a parabolic stochastic partial differential equation (SPDE) on [0,&#xa0;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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L\log L\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>log</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> growth condition. We prove that the SPDE is well posed when the initial data is in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2[0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This solves a strong form of an open problem.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月13日

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  • Mohammud Foondun,
  • Davar Khoshnevisan,
  • Eulalia Nualart

摘要

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