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<p>We prove that mild solutions to the stochastic heat equation with superlinear accretive forcing and polynomially growing multiplicative noise cannot explode under two sets of assumptions. The first set of assumptions allows both the deterministic forcing and multiplicative noise terms to grow polynomially, as long as the multiplicative noise is sufficiently larger. The second set of assumptions imposes an Osgood condition on the deterministic forcing and allows the multiplicative noise to grow polynomially. In both cases, the multiplicative noise cannot grow faster than <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(u^{\frac{3}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </math></EquationSource> </InlineEquation>, as this would cause explosion.</p>

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Global solutions to the stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative white noise coefficient

  • 期刊论文

  • 发表日期：

    2025年2月20日

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  • M. Salins

摘要

We prove that mild solutions to the stochastic heat equation with superlinear accretive forcing and polynomially growing multiplicative noise cannot explode under two sets of assumptions. The first set of assumptions allows both the deterministic forcing and multiplicative noise terms to grow polynomially, as long as the multiplicative noise is sufficiently larger. The second set of assumptions imposes an Osgood condition on the deterministic forcing and allows the multiplicative noise to grow polynomially. In both cases, the multiplicative noise cannot grow faster than \(u^{\frac{3}{2}}\) u 3 2 , as this would cause explosion.