# Liouville theorems for self-similar solutions of heat flows

Journal of the European Mathematical Society (2009)

- Volume: 011, Issue: 1, page 207-221
- ISSN: 1435-9855

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topLi, Jiayu, and Wang, Meng. "Liouville theorems for self-similar solutions of heat flows." Journal of the European Mathematical Society 011.1 (2009): 207-221. <http://eudml.org/doc/277485>.

@article{Li2009,

abstract = {Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $(\mathbb \{R\}^m,e^\{|x|^2/2(m-2)\}/ds^2_0)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds2^_0$ is the
Euclidean metric in $\mathbb \{R\}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.},

author = {Li, Jiayu, Wang, Meng},

journal = {Journal of the European Mathematical Society},

keywords = {harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow; harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow},

language = {eng},

number = {1},

pages = {207-221},

publisher = {European Mathematical Society Publishing House},

title = {Liouville theorems for self-similar solutions of heat flows},

url = {http://eudml.org/doc/277485},

volume = {011},

year = {2009},

}

TY - JOUR

AU - Li, Jiayu

AU - Wang, Meng

TI - Liouville theorems for self-similar solutions of heat flows

JO - Journal of the European Mathematical Society

PY - 2009

PB - European Mathematical Society Publishing House

VL - 011

IS - 1

SP - 207

EP - 221

AB - Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $(\mathbb {R}^m,e^{|x|^2/2(m-2)}/ds^2_0)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds2^_0$ is the
Euclidean metric in $\mathbb {R}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.

LA - eng

KW - harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow; harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow

UR - http://eudml.org/doc/277485

ER -

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