1.1. The solution is constant along the characteristic lines.**]{}

Let $\,\, 0 \leq  \lambda \leq  t  \,\,$ be some given parameter and $\,\,
u({\scriptstyle \bullet},\, {\scriptstyle \bullet}) \,\,$ a solution of equation (1.1.1). Then function $\,\,  u({\scriptstyle \bullet},\, {\scriptstyle \bullet})
\,\,$ is constant along the characteristic lines, [*i.e.*]{}

(1.1.5) $\qquad \displaystyle 
u(x-a \lambda,\, t- \lambda) \,\,= \,\, u(x,\,t) \,\,,\qquad \forall \,\,\, 
x ,\, t ,\, \lambda \,. \,$

$\bullet \qquad \,\,\, $ The [**proof of Proposition 1.1**]{} is obtained as follows. We consider a fixed point $\,\,
(x,\,t) \,\,$ in space-time $\,\, \R \times [0,\, +\infty[ \,\,$ and the auxiliary function $\,\, [0,\, t] \, \ni \lambda \longmapsto v(\lambda) \,=\, u (x-a \lambda,\,
t- \lambda) \,. \,$ We have, due to the usual chain rule for derivation of operators :

$ \displaystyle 
{{{\rm d}v}\over{{\rm d}\lambda}} \,\,=\,\, \Bigl[ (-a) \, {{\partial u}\over{\partial
x}} \,-\,  {{\partial u}\over{\partial t}} \, \Bigr] \, (x-a \lambda,\, t- \lambda)
\,\,= \,\,0 \,\,$ if function $\,\, u({\scriptstyle \bullet},\, {\scriptstyle
\bullet}) \,\,$ is solution of the advection equation (1.1.1). Then $\,\, v(\lambda)
\,\,$ does not depend on variable $\,\, \lambda \,\,$ and we have in particular $\,\,
v(\lambda) \,=\, v(0) \,,\,$ which exactly expresses the relation (1.1.5). We 