Singular limit of the generalized Burgers equation with absorption

Abstract

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

12 pages

References in corpus (1)

• The Hele-Shaw asymptotics for mechanical models of tumor growth