# A Convection-Diffusion model on a star shaped graph

## TL;DR

This paper investigates a convection-diffusion equation on a star-shaped graph, establishing global solutions and analyzing their long-term behavior, revealing convergence to self-similar heat or Burgers' profiles depending on the nonlinearity.

## Contribution

It proves global well-posedness and characterizes the asymptotic profiles of solutions for a nonlinear convection-diffusion equation on a star graph.

## Key findings

- Solutions are globally well-posed under general conditions.
- For q>2, solutions tend to heat equation self-similar profiles.
- For q=2, solutions converge to Burgers' equation profiles.

## Abstract

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.