Backward uniqueness for the heat equation

Definitions and mathematical statement

Let $D\subset\mathbb R^n$ be a connected open set, $n\geq1$, whose boundary points are regular for the Dirichlet problem: continuous boundary data are attained there by the corresponding Perron harmonic solution. Compare the following two existence properties of $D$.

Property I is the existence of a real-valued, nonzero, bounded temperature $u$ on $D\times[0,1]$. More explicitly, $u$ is continuous there, is twice differentiable in space and once in time in the interior, and satisfies \[ u_t=\Delta_xu\quad\text{on }D\times(0,1),\qquad u(x,1)=0\quad(x\in D),\qquad \sup_{D\times[0,1]}|u|<\infty. \] Property II is the existence of a nonzero real harmonic function $v$ on $D$ and constants $C,R>0$ such that \[ |v(x)|\leq C e^{-|x|^2} \quad(x\in D,\ |x|\geq R). \]

Are Properties I and II equivalent for every such domain? No value is prescribed for either function on the spatial boundary $\partial D$; adding a homogeneous boundary condition would change the question. Property II is a uniform Gaussian decay condition along all directions to infinity within $D$, and is vacuous at infinity when $D$ is bounded. Property I asks whether a bounded nontrivial temperature can become identically zero at a later time. The target is the full domain-dependent equivalence, beyond special geometries such as intervals or cones.

Short English statement

Does failure of backward uniqueness for bounded heat solutions exactly correspond to a Gaussian-decaying harmonic function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700010 (AMR-036-0010), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Backward uniqueness problem. Original source indexed by AMR-036-0010. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Backward uniqueness for the heat equation. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/heatproblem.pdf.