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 $u_{\infty}(x,y):=\lim_{t \rightarrow \infty} u(t,x,y)=x y$. Note 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?