<p>Recently, in Glogić et al. (Non-uniqueness of mild solutions to supercritical heat equations. arXiv:2501.17032 (2025)), it has been shown that the focusing power nonlinearity heat equation <Equation ID="Equ1"> <EquationNumber>NLH</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} \partial _t u -\Delta u = |u|^{p-1}u, \quad p>1, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mi>p</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in dimensions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> has non-unique local solutions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^q(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q < d(p-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo><</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> provided that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p < p_{JL}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <msub> <mi>p</mi> <mrow> <mi mathvariant="italic">JL</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p_{JL}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mrow> <mi mathvariant="italic">JL</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in (NLH) is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of (NLH) with additive noise. Second, we discuss how randomizing the initial conditions of (NLH) affects its well-posedness.</p>

On the effect of randomization on supercritical heat equations

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2026年5月6日

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Eliseo Luongo

摘要

Recently, in Glogić et al. (Non-uniqueness of mild solutions to supercritical heat equations. arXiv:2501.17032 (2025)), it has been shown that the focusing power nonlinearity heat equation NLH $\begin{aligned} \partial _t u -\Delta u = |u|^{p-1}u, \quad p>1, \end{aligned}$ $\begin{matrix} \partial_{t} u - Δ u = {| u |}^{p - 1} u, p > 1, \end{matrix}$ in dimensions $d \ge 3$ $d \geq 3$ has non-unique local solutions in $L^q(\mathbb {R}^d)$ $L^{q} (R^{d})$ for $q < d(p-1)/2$ $q < d (p - 1) / 2$ provided that $p < p_{JL}$ $p < p_{JL}$ , where $p_{JL}$ $p_{JL}$ denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in (NLH) is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of (NLH) with additive noise. Second, we discuss how randomizing the initial conditions of (NLH) affects its well-posedness.