I am slightly confused about the crossing/noncrossing properties of characteristics for first-order PDEs. Let's start with the case of linear homogeneous first-order PDE, of the general form $$ F(Du,u,x) = {\bf b}(x) \cdot Du(x) + c(x)u(x) = 0.$$ ($F$ is always assumed to be smooth.)
I read on Evans book (on page 109 here) the fairly vague statement
Observe also that the projected characteristics ${\bf x}(\cdot)$ emanating from distinct points on $\Gamma$ cannot cross, owing to uniqueness of solutions of the initial-value problem (17)(a).
Here $\Gamma$ is that part of the boundary on which we have prescribed the boundary condition and (17)(a) is just equation (52) on the same page, i.e., $\dot{\bf x}(s) = {\bf b}({\bf x}(s))$, whose solution is the vector of projected characteristics. The initial condition for this vector equation is provided by $x^0$, a point on the boundary such that the triple $(p^0,z^0,x^0)$ is admissible and noncharacteristic ($p^0$ and $z^0$ serve as initial conditions for the other two equations of the characteristic system but here we are not concerned with them). I say this statement is vague because it is unclear whether it is local or global. Indeed, a solution of the PDE has actually been constructed (by weaving characteristics) only in a neighborhood $V$ of $x^0$, relying (to solve the characteristics system) on the existence&uniqueness theorem for ODEs, from which we have existence&uniqueness for $s \in I$ for some (possibly very small) interval $I$ containing 0. Therefore, I would expect such noncrossing property to certainly hold only within $V$ but I would say that, in general, we cannot extend it to a global property as alluded by the statement. It seems to me this intuition is corroborated by the picture and the discussion of Case 1 just below in connection with the equation ${\bf b(x)} \cdot Du(x) = 0$, on pages 109, 110 (we should note however he is assuming ${\bf b}(\cdot)$ has a zero in the interior of the domain in that context, although he does not detail the consequences of this fact). However, about the general quasilinear case $$F(Du,u,x) = {\bf b}(x,u) \cdot Du(x) + c(x,u) = 0,$$ he says on page 111
In contrast to the linear case, it is possible that the projected characteristic emanating from distinct points in $\Gamma$ may intersect outside V; such an occurrence usually signals the failure of our local solution to exist within all of $U$.
From these words, it seems the first quoted statement should be interpreted as a global one, and this would entail I am wrong in thinking this crossing phenomenon happens also in the linear case, in general. I am really puzzled. Can someone clarify?
