A convection–diffusion model on a star-shaped graph

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