Singular limit of the generalized Burgers equation with absorption

TL;DR

This paper investigates the behavior of solutions to a generalized Burgers equation with absorption as parameters tend to infinity, revealing non-commutative limits and convergence properties.

Contribution

It establishes the convergence of solutions as parameters go to infinity and demonstrates that the order of limits affects the outcome.

Findings

Solutions converge as m→∞ for fixed p>1

Solutions converge as p→∞ for fixed m>1

Limits do not commute in general

Abstract

We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $\R\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ in $\R$, as $m\to\infty$ for any $p>1$ and $u_0\in L^1(\R)\cap L^{\infty}(\R)$ or as $p\to\infty$ for any $m>1$ and $u_0\in L^{\infty}(\R)$ . We also show that in general $\underset{p\to\infty}\lim\underset{m\to\infty}\lim u_{m,p}\ne\underset{m\to\infty}\lim\underset{p\to\infty}\lim u_{m,p}$.