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On the elliptic problem Δu-|∇u| q +λu p =0


Chipot, M; Weissler, F (1988). On the elliptic problem Δu-|∇u| q +λu p =0. In: Ni, W M; Peletier, L A; Serrin, J. Nonlinear diffusion equations and their equilibrium states I (Berkeley, CA, 1986). New York: Springer, 237-243.

Abstract

We consider regular solutions of the elliptic problem (1) $\Delta u-|\nabla u|^q+\lambda u^p=0$, and $u>0$ for $x\in\Omega $, $u=0$ on $x\in\partial \Omega $, where $\Omega \subset \bold R^n$ is a smooth, bounded domain, $u\colon \overline{\Omega }\rightarrow \bold R$ is $C^2$, and $q, p, \lambda$ are parameters satisfying $q, p>1$ and $\lambda >0$. The motivation for studying problem (1) comes from a related parabolic problem, $(2)\ v_t=\Delta v-|\nabla v|^q+v^p\ (t>0,\ x\in\Omega )$, $v(0, x)=\varphi(x)\ge 0\ (x\in\Omega )$, $v(t, x)=0\ (t>0$, $x\in\partial \Omega )$, where now $v=v(t, x)$. Previously, we investigated whether there exist initial values $\varphi$ for which the resulting solution of (2) blows up in finite time. It turns out that a natural candidate for such a $\varphi$ is a solution of (1). More precisely, assume either $(3)\ 1<q< 2p/(p+1)$, $n/2<(p+1)/(p-1)$, $\lambda>0$ is sufficiently small, or $(4)\ q=2p/(p+1)$, $p$ is sufficiently large, $0<\lambda\le 2/(p+1)$. It follows that if $\varphi$ is a solution of (1), then the resulting solution of (2) blows up in finite time.
Thus we are led to investigate under what conditions there exist solutions of (1) and whether these solutions are unique.

We consider regular solutions of the elliptic problem (1) $\Delta u-|\nabla u|^q+\lambda u^p=0$, and $u>0$ for $x\in\Omega $, $u=0$ on $x\in\partial \Omega $, where $\Omega \subset \bold R^n$ is a smooth, bounded domain, $u\colon \overline{\Omega }\rightarrow \bold R$ is $C^2$, and $q, p, \lambda$ are parameters satisfying $q, p>1$ and $\lambda >0$. The motivation for studying problem (1) comes from a related parabolic problem, $(2)\ v_t=\Delta v-|\nabla v|^q+v^p\ (t>0,\ x\in\Omega )$, $v(0, x)=\varphi(x)\ge 0\ (x\in\Omega )$, $v(t, x)=0\ (t>0$, $x\in\partial \Omega )$, where now $v=v(t, x)$. Previously, we investigated whether there exist initial values $\varphi$ for which the resulting solution of (2) blows up in finite time. It turns out that a natural candidate for such a $\varphi$ is a solution of (1). More precisely, assume either $(3)\ 1<q< 2p/(p+1)$, $n/2<(p+1)/(p-1)$, $\lambda>0$ is sufficiently small, or $(4)\ q=2p/(p+1)$, $p$ is sufficiently large, $0<\lambda\le 2/(p+1)$. It follows that if $\varphi$ is a solution of (1), then the resulting solution of (2) blows up in finite time.
Thus we are led to investigate under what conditions there exist solutions of (1) and whether these solutions are unique.

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Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:regular solutions; existence; uniqueness; support
Language:English
Date:1988
Deposited On:28 Oct 2009 13:37
Last Modified:14 Sep 2016 13:39
Publisher:Springer
Series Name:Mathematical Sciences Research Institute Publications.
Number:12
ISSN:0940-4740
ISBN:0-387-96771-0
Official URL:http://www.springer.com/mathematics/analysis/book/978-0-387-96771-4
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0699.35102
http://www.recherche-portal.ch/primo_library/libweb/action/search.do?fn=search&mode=Advanced&vid=ZAD&vl%28186672378UI0%29=isbn&vl%281UI0%29=contains&vl%28freeText0%29=0-387-96771-0

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