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

$u_t=\Delta u,$

where $(t,X)\in R \times R^2$. Suppose $u(t,x,y)$ behaves like $e^{-\alpha t} xy$ for large $t$. Note that the nodal line of $u(t,x,y)$ ($\{(x,y)\in R^2: u(t,x,y)=0\}$) divides $R^2$ into four regions. 

Can we prove that the nodal line of $u_0(x,y)=u(0,x,y)$  divides $R^2$ into at least four regions?