When we have linear or quasilinear first order pde
$$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$
And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$
I always see on my notes that the solution $u$ is always constant on the characteristics curves. What do they really mean by that? What is the geometric intuition ..?
No, for this equation you get the system $$\dot x=a(x,y,u),\\\dot y=b(x,y,u),\\\dot u=c(x,y,u)$$ which in general leads to a non-constant $u$. $u$ is only constant along characteristics if $c=0$. As the solution surface is a family of curves with a one-dimensional cross-section, the 3 integration constants of the system depend on only one parameter, which allows to incorporate the initial conditions.