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Consider the PDE:

$\partial_tu+u\partial_xu=u$

With boundary conditions:

$u(0,x)=u_0(x)=-\tanh(x)$

Let $X(t,y,z)$ and $U(t,y,z)$ be the solutions to ODE Cauchy Problem:

$\frac{dX}{dt}=U(t)$

$\frac{dU}{dt}=U(t)$

With initial conditions:

$X(0,y,z)=y$

$U(0,y,z)=z$

  1. Show that the function implicitly given by

$u(t,X(t,y,u_0(y)))=U(t,y,u_0(y))$,

is a $C^1$ solution to the PDE for a small time

  1. 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"?

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  • 1
    $\begingroup$ Do you know what that theorem says? $\endgroup$ – Umberto P. Mar 13 at 19:26

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