\begin{subequations}
 \label{eq:level_system}
\begin{align}%\label{heat_equation_1}
    \frac{\partial \phi}{\partial t}&= \phi^{m/(1+m)} \left(\frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2}\right)+(1+m) \phi \left(1-\phi^{1/(1+m)}\right )\label{Eq:PFSLevelSet}\; \; \text{on}   \;\;   \varphi(x,y,t)<0,\\
    &\frac{\partial \varphi}{\partial t}+F \mid \nabla \varphi \mid \, = 0\;\;\text{on}   \;\;   (x,y)\in\mathcal{D}, \label{Eq:PFSLevelSetVelocity}\\
    &\phi(x,y,t)=0 \;\; \text{on}   \;\;  \varphi(x,y,t)=0, \\
    &\phi(x,y,0)=U(x,y)^{m+1} \;\; \text{on}   \;\; \varphi(x,y,0)<0,\\
    &F=-\frac{\kappa}{1+m}\nabla \phi \cdot \hat{n}\;\;  \text{on}   \;\;  \varphi(x,y,t)=0. \label{Eq:PFSLevelSet_Extension}
\end{align}
\end{subequations}
