# PDE and Implicit Function Theorem

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"?

• Do you know what that theorem says? – Umberto P. Mar 13 at 19:26