Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations
Abstract
At the Fujita exponent p=1+2/N we construct a strictly positive mild solution of ut−Δu=|u|p−1u in ℝN whose initial trace in L1 is zero. In similarity variables the solution approaches a Gaussian profile, and |U(t)|1∼ CN(log(1/t))−N/2, |U(t)|∞∼ cNt−N/2(log(1/t))−N/2, with explicit constants CN and cN. In particular, uniqueness fails at the borderline logarithmic rate, and the solution lies outside every stronger logarithmic uniqueness class; this answers two questions raised by Miyamoto [J. Evol. Equ. 21 (2021), 151--166, Remark~1.6]. We also prove that, up to time translation, this is the only ancient solution of fixed sign of the rescaled equation that decays in a Gaussian weighted supremum norm. On a bounded smooth domain the whole-space solution can be localized at finitely many interior points, with independently chosen signs, on top of an arbitrary bounded initial datum. The resulting solutions are classical for t>0 and continuous at t=0 in L1. Consequently, uniqueness fails for every bounded datum, and positivity is lost for every bounded nonnegative datum, u0≡1 included. These results provide affirmative answers to Open Problems~3 and~4 posed by Brezis and Cazenave in [J. Anal. Math. 68 (1996), 277--304].
Related
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BibTeX
@misc{torebek2026nonuniqueness,
title = {{Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations}},
author = {Berikbol T. Torebek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40315},
eprint = {2610.40315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40315},
}
RIS
TY - UNPB
TI - Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations
AU - Berikbol T. Torebek
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40315
UR - https://arxiv.org/abs/2610.40315
AB - At the Fujita exponent p=1+2/N we construct a strictly positive mild solution of u_t−Δu=|u|^p−1u in ℝ^N whose initial trace in L^1 is zero. In similarity variables the solution approaches a Gaussian profile, and |U(t)|_1∼ C_N(log(1/t))^−N/2, |U(t)|_∞∼ c_Nt^−N/2(log(1/t))^−N/2, with explicit constants C_N and c_N. In particular, uniqueness fails at the borderline logarithmic rate, and the solution lies outside every stronger logarithmic uniqueness class; this answers two questions raised by Miyamoto [J. Evol. Equ. 21 (2021), 151--166, Remark~1.6]. We also prove that, up to time translation, this is the only ancient solution of fixed sign of the rescaled equation that decays in a Gaussian weighted supremum norm. On a bounded smooth domain the whole-space solution can be localized at finitely many interior points, with independently chosen signs, on top of an arbitrary bounded initial datum. The resulting solutions are classical for t>0 and continuous at t=0 in L^1. Consequently, uniqueness fails for every bounded datum, and positivity is lost for every bounded nonnegative datum, u_0≡1 included. These results provide affirmative answers to Open Problems~3 and~4 posed by Brezis and Cazenave in [J. Anal. Math. 68 (1996), 277--304].
ER -