Non-uniqueness of mild solutions for 2d-heat equations with singular initial data
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912329345531904 |
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| author | Fujishima, Yohei Ioku, Norisuke Ruf, Bernhard Terraneo, Elide |
| author_facet | Fujishima, Yohei Ioku, Norisuke Ruf, Bernhard Terraneo, Elide |
| contents | In a recent article by the authors [15] it was shown that wide classes of semilinear elliptic equations with exponential type nonlinearities admit singular radial solutions $U$ on the punctured disc in $\mathbb R^2$ which are also distributional solutions on the whole disc. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: ${u_s}(t,x) \equiv U(x)$ is a stationary solution, and there exists also a solution ${u_r}(t,x)$ departing from $U$ which is bounded for $t > 0$. While such non-uniqueness results have been known in higher dimensions by Ni--Sacks [33], Terraneo [40] and Galaktionov--Vazquez [16], only two very specific results have recently been obtained in two dimensions by Ioku--Ruf--Terraneo [22] and Ibrahim--Kikuchi--Nakanishi--Wei [21]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of mild solutions for 2d-heat equations with singular initial data Fujishima, Yohei Ioku, Norisuke Ruf, Bernhard Terraneo, Elide Analysis of PDEs In a recent article by the authors [15] it was shown that wide classes of semilinear elliptic equations with exponential type nonlinearities admit singular radial solutions $U$ on the punctured disc in $\mathbb R^2$ which are also distributional solutions on the whole disc. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: ${u_s}(t,x) \equiv U(x)$ is a stationary solution, and there exists also a solution ${u_r}(t,x)$ departing from $U$ which is bounded for $t > 0$. While such non-uniqueness results have been known in higher dimensions by Ni--Sacks [33], Terraneo [40] and Galaktionov--Vazquez [16], only two very specific results have recently been obtained in two dimensions by Ioku--Ruf--Terraneo [22] and Ibrahim--Kikuchi--Nakanishi--Wei [21]. |
| title | Non-uniqueness of mild solutions for 2d-heat equations with singular initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.10966 |