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)
The null condition and global existence to nonlinear wave equations. (English) Zbl 0599.35105
Nonlinear systems of partial differential equations in applied mathematics, Proc. SIAM-AMS Summer Semin., Santa Fe/N.M. 1984, Lect. Appl. Math. 23, Pt. 1, 293-326 (1986).

[For the entire collection see Zbl 0579.00008.]

The global existence of solutions to the nonlinear wave equation u=F(u,u ' ,u '' ) in the three dimensional space case is discussed. Since John’s example on blow-up of solution to nonlinear wave equation in three dimensional space case shows that generally the global existence is not valid, the author introduces a ”null condition” on the nonlinear function F, which can ensure the existence of global solution of Cauchy problem with sufficiently small data. Meanwhile, some decay estimates of the solution are also obtained.

Reviewer: S.Chen

MSC:
35L70Nonlinear second-order hyperbolic equations
35L15Second order hyperbolic equations, initial value problems
35L05Wave equation (hyperbolic PDE)
35B40Asymptotic behavior of solutions of PDE