Implicit solution of a PDE

**gammafunction** · January 17th 2009, 01:53 AM

Hi! Can somebody help me with this exercise?
Let $F: \mathbb{R} \rightarrow \mathbb{R}^n, \, g: \mathbb{R}^n \rightarrow \mathbb{R}$ be smooth. Consider the following IVP on $\mathbb{R}^n \times (0, \infty):$

$u_t+F'(u) \cdot Du=0$
$u(x,0)=g(x)$

Under which condition on $F$ and $g$ does $u(x,t)=g(x-tF'(u(x,t)))$ define an implicit solution? When does this condition fail?

I am pretty aimless and would be thankful for any ideas.

**gammafunction** · January 17th 2009, 04:12 PM

I think it is just about calculating the derivatives but i do not get on here...

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