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)
Longtime behavior of the Kirchhoff type equation with strong damping on N . (English) Zbl 1208.35147

The author considers a Kirchhoff type equation with strong damping written as u tt -M(u 2 )Δu-Δu t +u+u t +g(x,u)=f(x), posed in N × + . Here M(s)=1+s m/2 , m1, f belongs to L 2 ( N ) and g is a nonlinear function which satisfies at least some p-growth condition with respect to s.

The first result of the paper proves an existence and uniqueness result for a solution of this problem, which is global in time and belongs to C(0,;H 2 )C 1 (0,;H 1 )H 1 (0,;H 2 )H 2 (0,;L 2 ). Assuming some further hypotheses on the first-order partial derivatives of g, the author proves that the underlying continuous semi-group of this problem has a global attractor in H 2 H 1 . The proof of the existence result follows the lines of the paper by M. Nakao and the author [J. Differ. Equations 227, No. 1, 204–229 (2006; Zbl 1096.35024)]. It is based on Banach’s fixed point theorem in an appropriate space-time functional space for T>0.

Then the author proves a uniform bound on u(t) H 2 2 +u t (t) H 1 2 , which implies that the unique solution is global in time. The proof of the main result is based on a decomposition of the semi-group and on the construction of estimates. The paper ends with the proof of some further properties of the semi-group: it has finite fractal and Hausdorff dimension. The author here quotes a result from J. Hale [Asymptotic behavior of dissipative systems (Mathematical Surveys and Monographs, 25. Providence, RI: American Mathematical Society) (1988; Zbl 0642.58013)].

35Q74PDEs in connection with mechanics of deformable solids
35L70Nonlinear second-order hyperbolic equations