Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations

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].