Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$


$u\_t=\Delta u,$


where $(t,X)\in R \times R^2$. Suppose $\lim\_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of $u\_0(x,y)=u(0,x,y)$ divides $R^2$ into at least four regions? 



Nodal sets under the heat flow
No.
Let $\phi$ be a compactly supported positive smooth function with $\phi(0)>0$.
Now $u\_0(X)=u\_\infty(X)+\phi(X)$ is a possible initial value but the set $\{X;u\_0(X)=0\}$ divides the plane to only three parts.
The function $\phi$ essentially builds a bridge between the first and the third quadrant.


Using a different $\phi$ you can make the initial topology almost anything you like.


