Analyzing PDE BVP for ut + ux = 0 with given boundary condition

[Nicolaus]

Homework Statement

1. ut +ux = 0 subject to u(t,x) = x on x^2 + y^2 = 1
   
   Is this a well-posed PDE BVP?
   

Homework Equations

The Attempt at a Solution

This is an easy one to solve: u(t,x) = f(x-t)
I let t(0) = 0 as an initial condition, and so t=s => x= ts + x_o, where x(0) = x_o
s is the variable such that ∂(u(t(s), x(s))/∂s = 0
If I let u(t,x) = x = f(x-t), would this not be well-posed since f must be a function of (x-t)?

 

[Geofleur]

In the problem setup, is the condition really $x^2 + y^2 = 1$, or did you mean $x^2 + t^2 = 1$?