Is there anything known about uniqueness of classical solutions to
$$
\partial\_t u -u\Delta u=0\quad u(0,\cdot)=1
$$
on smooth domains $[0,T]\times D$ without boundary conditions? I know that $u(0,\cdot)=0$ implies that $u=0$ even without boundary conditions.
However, does $u(0,\cdot)=1$ imply that $u=1$? It is not so hard to see that $\partial\_t^k u(0)=0$ for all $k\in \mathbb{N}$ if $u$ is smooth. Hence, for an analytic in time solution, this must be true.



Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$
I found that if one has $\partial\_n u=0$ or $u=1$ on $[0,T]\times \partial D$ it is relatively easy to prove that $u=1$ is the only regular solution.
This can be seen by observing that $e:=u-1$ satisfies
$$
\partial\_t e -e\Delta e -\Delta e =0.
$$
Multiplication by $e$, the identity $\Delta e^2 = 2e\Delta e + |\nabla e|^2$, integration over the domain $D$, and Gronwall's lemma prove $e=0$.


