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

$u_t=\Delta u$,

where $(t,X)\in R \times R^2$. Suppose $u_{\infty}(x,y):=\lim_{t \rightarrow \infty} u(t,x,y)=\frac{x y}{1+x^2+y^2}$. Not that the nodal line of $u_{\infty}$ ($\{(x,y)\in R^2: u_{\infty}(x,y)=0\}$) 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?