When does global solution for Burger's equation exists?

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.