# Quasilinear PDE problem

1. Oct 2, 2011

### QuantumJG

1. The problem statement, all variables and given/known data

Solve

$\frac{\partial \phi}{\partial t} + \phi \frac{\partial \phi}{\partial x} - \infty < x < \infty , t > 0$

subject to the following initial condition

$\phi (x,0) = \left\{ \begin{array}{c} 1,\; x<0\\ 1-x,\;0\leq x<1\\ 0,\; x\geq1\end{array}\right.$

2. Relevant equations

see 3

3. The attempt at a solution

Solving the PDE via method of characteristics, the characteristic lines are:

$x = \phi t + s$

$x < 0 : t = x - s$

$0 \leq x < 1 : t = \frac{x-s}{1-s}$

$x \geq 1 : x = s$

My question is that I don't know where to find a shock. All characteristics originating in the region $0 \leq x < 1$ cross over at (1,1), but characteristics also cross over at x = 1.

2. Oct 16, 2011