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p}(x,u,Du)\\right)=0,\\quad &&(x,t)\\in M\\times(0,T],\\\\ &m(0)=m_0, \\end{aligned}\\right. \\end{equation*} where the system comprises a stationary Hamilton-Jacobi equation in the contact case and an evolutionary continuity equation.\n  Then, for any fixed $\\lambda>0$, let $(u^\\lambda,m^\\lambda)$ be a solution of the system \\begin{equation*} \\left\\{ 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