Has the 'transport equation': $$ \partial_t f+ \frac{x}{t}\partial_x f=0 $$ set on $(0,1)^2$ non constant solutions ?
If yes are there solutions satisfying $f(t,x)\leq x$ ?
Due to the origin of the problem, $x$ and $t$ appear to play similar roles.
Any hint would be appreciated.
In this case, the characteristics are the lines passing by zero, we may hence have non-constant solutions by setting a non-constant profile on the opposite sides of the unit square. Doing so we lose the smoothness of f near (0,0). Do any of these solutions satisfy $f(t,x)\leq x$ ? yes, but only $f=0$.