zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation. II. (English) Zbl 0924.35055

Authors’ abstract: The blow-up of solutions of the Cauchy problem,

u t =u xx +|u| p-1 uin×(0,),u(x,0)=u 0 (x)in,

is studied. Let Λ k be the set of functions on which change sign k times. It is shown that for

p k =1+2 k+1,k=0,1,2,,

any solution with u 0 Λ k blows up in finite time if 1<pp k , whereas a global solution with u 0 Λ k exists if p>p k . It is also shown that if u 0 decays more slowly than |x| -2/(p-1) as |x|, then the solution blows up in finite time regardless of the number of sign changes.

Part I, see Math. Ann. 307, No. 4, 663-675 (1997; Zbl 0872.35046).

MSC:
35K15Second order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations