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arXiv · 1904.08309

A Convection-Diffusion model on a star shaped graph

Cristian M. Cazacu·Liviu I. Ignat·Ademir F. Pazoto·Julio D. Rossi
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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.

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Cristian M. Cazacu, Liviu I. Ignat, Ademir F. Pazoto, Julio D. Rossi. 2019-04-17. A Convection-Diffusion model on a star shaped graph. https://arxiv.org/abs/1904.08309

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