In a recent article by Fujishima et al. (J Funct Anal 289:110922, 2025), it was shown that wide classes of semilinear elliptic equations with exponential-type nonlinearities admit singular radial solutions U on the punctured disk in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> which are also distributional solutions on the whole disk. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${u_s}(t,x) \equiv U(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a stationary solution, and there exists also a solution <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${u_r}(t,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> departing from U which is bounded for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. While such non-uniqueness results have been known in higher dimensions by Ni and Sacks (Trans Am Math Soc 287:657–671, 1985), Terraneo (Commun Partial Differ Equ 27:185–218, 2002) and Galaktionov–Vazquez (Commun Pure Appl Math 50:1–67, 1997), only two very specific results have recently been obtained in two dimensions by Ioku–Ruf–Terraneo (Ann Inst H Poincaré C Anal. Non Linéaire 36:2027–2051, 2019) and Ibrahim–Kikuchi–Nakanishi–Wei (Math Ann 380:317–348, 2021).

Non-uniqueness of mild solutions for 2d-heat equations with singular initial data

期刊论文
发表日期：

2026年2月21日

查看全文

Yohei Fujishima,
Norisuke Ioku,
Bernhard Ruf,
Elide Terraneo

摘要

In a recent article by Fujishima et al. (J Funct Anal 289:110922, 2025), it was shown that wide classes of semilinear elliptic equations with exponential-type nonlinearities admit singular radial solutions U on the punctured disk in $\mathbb {R}^2$ $R^{2}$ which are also distributional solutions on the whole disk. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: ${u_s}(t,x) \equiv U(x)$ $u_{s} (t, x) \equiv U (x)$ is a stationary solution, and there exists also a solution ${u_r}(t,x)$ $u_{r} (t, x)$ departing from U which is bounded for $t > 0$ $t > 0$ . While such non-uniqueness results have been known in higher dimensions by Ni and Sacks (Trans Am Math Soc 287:657–671, 1985), Terraneo (Commun Partial Differ Equ 27:185–218, 2002) and Galaktionov–Vazquez (Commun Pure Appl Math 50:1–67, 1997), only two very specific results have recently been obtained in two dimensions by Ioku–Ruf–Terraneo (Ann Inst H Poincaré C Anal. Non Linéaire 36:2027–2051, 2019) and Ibrahim–Kikuchi–Nakanishi–Wei (Math Ann 380:317–348, 2021).