  In this paper we study a convection-diffusion equation on a star-shaped graph
composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in
C^1(\rr)$ satisfying some additional general conditions. First, we prove the
global well-posedness of the solutions of the system under consideration. Next,
in the particular case that the nonlinear convection is given by
$\partial_x(f(u(t, x))$ with
  $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$,
we analyze the long time behavior of the solutions.
  For $q> 2$ we find that the asymptotic behavior of the solutions is given by
some self-similar profiles of the heat equation on the considered structure.
  In the case $q=2$, the nonnegative/nonpositive solutions converge to the
self-similar profiles of Burgers' equation.
  Explicit representations of the limit profiles are obtained.
