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)
Blow-up directions for quasilinear parabolic equations. (English) Zbl 1167.35393
Summary: We consider the Cauchy problem for quasilinear parabolic equations $u_t=\Delta\varphi(u)+f(u)$, with the bounded non-negative initial data $u_0(x) (u_0(x)\not\equiv 0)$, where $f(\xi)$ is a positive function in $\xi>0$ satisfying a blow-up condition $\int_1^{\infty}1/f(\xi)\,d\xi<\infty$. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation d$v/\text{d} t=f(v)$ with the initial data $\Vert u_0\Vert _{L^{\infty}(\Bbb{R}^N)}>0$. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on $u_0$ for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on $u_0$ for blow-up with the least blow-up time, provided that $f(\xi)$ grows more rapidly than $\varphi(\xi)$.

35K55Nonlinear parabolic equations
35K15Second order parabolic equations, initial value problems
35K65Parabolic equations of degenerate type
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI