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)
Cauchy problem for the generalized Benney-Luke equation. (English) Zbl 1144.81421
Summary: This paper is concerned with the study of the Cauchy problem associated with an $n$-dimensional generalized Benney-Luke equation, $$u_{tt}-m\Delta u-\Delta u_{tt}+\Delta^2u+\alpha(2\nabla u\cdot\nabla u_t+u_t\Delta u)+\beta\nabla (\vert\nabla u\vert^p\nabla u)=0,$$ where $n=1,2,3,4$. We prove the existence and the uniqueness of the global solution of the Cauchy problem for the $\beta\le 0$ case by using energy conservation law and give the existence and the nonexistence of the global solution of the Cauchy problem for the $\beta>0$ case by constructing the stable set and the unstable set.

35Q53KdV-like (Korteweg-de Vries) equations
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI