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In this paper we study a convection-diffusion equation on a star-shaped graphcomposed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\inC^1(\rr)$ satisfying some additional general conditions. First, we prove theglobal 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 bysome self-similar profiles of the heat equation on the considered structure.  In the case $q=2$, the nonnegative/nonpositive solutions converge to theself-similar profiles of Burgers' equation.  Explicit representations of the limit profiles are obtained.

A Convection-Diffusion model on a star shaped graph