Weak solution for Burgers' equation I have the following IVP:
$$
u_t + u u_x = 0,\qquad 
u(x,0) = \left\lbrace
\begin{aligned}
   &0 && \text{if } x<-3 \\
   &0.5 && \text{if } {-3}<x<-2 \\
   &1   && \text{if } {-2}<x<0 \\
   &1-x && \text{if } 0\leq x<1 \\
   &0   && \text{if } x\geq 1
\end{aligned}\right.
$$
Question is to find weak solution say at $t=1,2$
try
Well, the characteristics are given by
$$ \frac{dt}{ds} = 1, \frac{d x }{ds} = u, \frac{u}{ds} = 0 $$
and we choose the initial curve to be $(r,0,f(r))$ where $f(r) = u(r,0)$
We see that the characteristics are of the form $x= f(r) s +r $ and $t = s$
when trying to graph the characteristics, there at the endpoints of the intial data we have lines that cross but I dont understand how to find weak solution. what would be the strategy?

Since your initial data are discontinuous and also decreasing on part of the domain, your solution contains rarefaction waves and shock waves. The rarefaction waves appear at $x=-3$ and $x=-2$ at time $t=0$ and the shock appears at some positive time when the characteristics collide. To extend the solution beyond the collision you need the Rankine-Hugoniot conditions on the shock, giving you the right speed of the shock line. You can read about the topic here
https://web.stanford.edu/class/math220a/handouts/conservation.pdf
