let $u(t,x)$ be a bounded smooth solution of the heat equation $u\_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: u(t,x)>0\}$. Suppose that $(\{t\_1\} \times R^2) \cap V$ has only one open connected component which is also bounded. Is it possible for $(\{t\} \times R^2) \cap V$ to have at least two open connected component for all $t\geq t\_2>t\_1$? 


Can the above be ruled out by strong maximum principle? Note that 
$u(\partial [(\{t\} \times R^2) \cap V] )\subset\{0\}$, for all $t$. 



Strong maximum principle for the heat equation in non-cylindrical domains
This is certainly possible, and it seems rather obvious on physical grounds. Consider an initial condition where you have two hot regions connected by a thin corridor which is also hot, and the surrounding space is cold. It is clear that the connecting corridor will cool rapidly.


