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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 $t$.

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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.

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  • $\begingroup$ Thanks Michael. Under the assumption above, I wonder if $(\{t\} \times R^2) \cap V$ can develop two open connected components that stay next to each other (the closures intersect) for all $t>t_2$. $\endgroup$
    – A random mathematician
    Commented Mar 5, 2015 at 0:04
  • $\begingroup$ I have posted a related questions: mathoverflow.net/questions/199295/… $\endgroup$
    – A random mathematician
    Commented Mar 8, 2015 at 23:46

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