## Solve Hamilton–Jacobi PDE

$$\frac{\partial S}{\partial t} + \frac{1}{2}\left((\nabla S)^2 + (x, \Omega^2 x) \right)= 0$$ $$S|_{t=0} = (k,x)$$ Where $x \in \mathbf{R}^n,\ \Omega^2$ - Positive-definite matrix, $k$ is constant vector

Where should I start?

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 Assuming S is a scalar-valued function, this is a scalar-valued, first order PDE, so you could examine its characteristic ODE. – Victor Dods Dec 29 2011 at 11:03