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Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function the problem is well-posed, b) for initial data above this threshold function , there exists no solution, c) for the singular initial datum there is non-uniqueness. The function is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum .
Ioku, Norisuke 1 ; Ruf, Bernhard 2 ; Terraneo, Elide 2
@article{AIHPC_2019__36_7_2027_0,
author = {Ioku, Norisuke and Ruf, Bernhard and Terraneo, Elide},
title = {Non-uniqueness for a critical heat equation in two dimensions with singular data},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {2027--2051},
publisher = {Elsevier},
volume = {36},
number = {7},
year = {2019},
doi = {10.1016/j.anihpc.2019.07.004},
mrnumber = {4020532},
zbl = {1427.35096},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.07.004/}
}
TY - JOUR AU - Ioku, Norisuke AU - Ruf, Bernhard AU - Terraneo, Elide TI - Non-uniqueness for a critical heat equation in two dimensions with singular data JO - Annales de l'I.H.P. Analyse non linéaire PY - 2019 SP - 2027 EP - 2051 VL - 36 IS - 7 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.07.004/ DO - 10.1016/j.anihpc.2019.07.004 LA - en ID - AIHPC_2019__36_7_2027_0 ER -
%0 Journal Article %A Ioku, Norisuke %A Ruf, Bernhard %A Terraneo, Elide %T Non-uniqueness for a critical heat equation in two dimensions with singular data %J Annales de l'I.H.P. Analyse non linéaire %D 2019 %P 2027-2051 %V 36 %N 7 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.07.004/ %R 10.1016/j.anihpc.2019.07.004 %G en %F AIHPC_2019__36_7_2027_0
Ioku, Norisuke; Ruf, Bernhard; Terraneo, Elide. Non-uniqueness for a critical heat equation in two dimensions with singular data. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 2027-2051. doi: 10.1016/j.anihpc.2019.07.004
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