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We consider the evolutionary Hamilton-Jacobi equation \\begin{equation*}\n  \\left\\{\n  \\begin{aligned}\n  &\\partial_t u(x,t)+H(x,u(x,t),\\partial_xu(x,t))=0,\\quad (x,t)\\in M\\times(0,+\\infty),\n  \\\\ &u(x,0)=\\varphi(x),\n  \\end{aligned}\n  \\right. \\end{equation*} where $\\varphi\\in C(M)$ and the stationary one \\begin{equation*}\n  H(x,u(x),\\partial_x u(x))=0, \\end{equation*} where $H(x,u,p)$ is continuous, convex and coercive in $p$, uniformly Lipschitz in $u$. 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We consider the evolutionary Hamilton-Jacobi equation \\begin{equation*}\n  \\left\\{\n  \\begin{aligned}\n  &\\partial_t u(x,t)+H(x,u(x,t),\\partial_xu(x,t))=0,\\quad (x,t)\\in M\\times(0,+\\infty),\n  \\\\ &u(x,0)=\\varphi(x),\n  \\end{aligned}\n  \\right. \\end{equation*} where $\\varphi\\in C(M)$ and the stationary one \\begin{equation*}\n  H(x,u(x),\\partial_x u(x))=0, \\end{equation*} where $H(x,u,p)$ is continuous, convex and coercive in $p$, uniformly Lipschitz in $u$. 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