Consider the PDE:
With boundary conditions:
Let $X(t,y,z)$ and $U(t,y,z)$ be the solutions to ODE Cauchy Problem:
With initial conditions:
- Show that the function implicitly given by
is a $C^1$ solution to the PDE for a small time
- Estimate the maximal time for which the solution given by point "1" be computed
How can I use the Implicit Function Theorem here in order to show point "1"?