zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An initial value approach to rotationally symmetric harmonic maps. (English) Zbl 1109.58021

Summary: We study the effect of the varying y ' (0) on the existence and asymptotic behavior of solutions for the initial value problem

y '' (r)+(n-1)f ' (r)y ' (r) f(r)-(n-1)g(y(r))g ' (y(r)) f(r) 2 =0,y(0)=0,

where f and g are some prescribed functions. Global solutions of this ODE on [0,) represent rotationally symmetric harmonic maps, with possibly infinite energies, between certain class of Riemannian manifolds. By studying this ODE, we show among other things that (i) all rotationally symmetric harmonic maps from n to the hyperbolic space n blow up in a finite interval; (ii) all such harmonic maps from n to n are bounded; and (iii) a trichotomy phenomenon occurs for such harmonic maps from n into itself, viz., they blow up in a finite interval, are the identity map, or are bounded according as the initial value y ' (0)<1, =1, or >1. Finally when n=2, the above equation can be solved exactly by quadrature method. Our results supplement those of A. Ratto and M. Rigoli [J. Differ. Equations 101, No. 1, 15–27 (1993; Zbl 0767.34029)] and A. Tachikawa [Tokyo J. Math. 11, No. 2, 311–316 (1988; Zbl 0686.58013)].

58E20Harmonic maps between infinite-dimensional spaces
34A05Methods of solution of ODE
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions