Consider Burger's equation
$$\begin{cases} u_t+uu_x=0 \\ u(x,0) = u_0(x) \end{cases}$$
Can someone explain to me what is meant by "global solution" to this Burger equation? How does that relate to the fact that $u_0$ being monotone increasing?
If $u_0(x) = \alpha x$, for some $\alpha \in \mathbb{R}$, then does that mean that the "global solution" exists only if $\alpha>0$, and from the solution
$$u(x,t) = \frac{\alpha x}{1+ \alpha t}$$
Does it the "global solution" exists for all $t$ except $t=-1/\alpha$? Does it matter if $t<-1/\alpha$ and $t > -1/\alpha$?
By "global solution" they mean a classical solution, one that does not develop shocks.
If the initial data are increasing, the characteristics issued at points $(x,0)$ do not cross, and every point on the $(x,t)$ plane is connected to a unique point on the $t=0$ axis, and the solution is determined by the initial value there.
If the initial condition $u_0$ is decreasing over some interval, you will have, say, a characteristic with velocity $v_1$ issued from $(x_1,0)$ and a characteristic with slope $v_2$ issued from $(x_2,0)$, with $x_1<x_2$ and $v_1>v_2$. If that happens, the characteristics meet at some $T>0$ and it is impossible to continue the solution as a classical one beyond this point.
But you can certainly build a weak solution after that by imposing the Rankine-Hugoniot conditions and the entropy condition along the resulting shock.