Let the function $v$ have a nondegenerate minimum at 0. Then there exists a neighborhood $U$ of 0 such that there is a unique function $\varphi$ on $U$ with $\varphi>0$ and $\varphi(0)=0$ that solves the eiconal equation $$ v = |d\varphi|^2$$ This can be obtained by taking the Hamiltonian vector field associated to this equation and applying the stable manifold theorem. See for example here (http://mathoverflow.net/questions/82227/solutions-to-the-eikonal-equation).
My question is if this solution exists on the whole unstable manifold of the gradient field of $v$, or if there is a counterexample. This is not totally obvious to me from the proof.