On the effect of randomization on supercritical heat equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915582671060992 |
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| author | Luongo, Eliseo |
| author_facet | Luongo, Eliseo |
| contents | Recently, in \cite{glogic2025non}, it has been shown that the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH}
\partial_t u -Δu = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$ has non-unique local solutions in $L^q(\mathbb{R}^d)$ for $q < d(p-1)/2$ provided that $p < p_{JL}$, where $p_{JL}$ 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 \eqref{Eq:Heat_abstract} is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of \eqref{Eq:Heat_abstract} with additive noise. Second, we discuss how randomizing the initial conditions of \eqref{Eq:Heat_abstract} affects its well-posedness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the effect of randomization on supercritical heat equations Luongo, Eliseo Probability Analysis of PDEs Recently, in \cite{glogic2025non}, it has been shown that the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -Δu = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$ has non-unique local solutions in $L^q(\mathbb{R}^d)$ for $q < d(p-1)/2$ provided that $p < p_{JL}$, where $p_{JL}$ 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 \eqref{Eq:Heat_abstract} is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of \eqref{Eq:Heat_abstract} with additive noise. Second, we discuss how randomizing the initial conditions of \eqref{Eq:Heat_abstract} affects its well-posedness. |
| title | On the effect of randomization on supercritical heat equations |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2510.24268 |