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?