# 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)
Global attractors for damped semilinear wave equations. (English) Zbl 1056.37084
The nonlinear damped wave equation $$u_{tt}+\beta u_t-\Delta u+f(u)=0$$ is considered on a bounded domain $\Omega\subset {\Bbb R}^n$ imposing Dirichlet boundary conditions. For the nonlinearity it is assumed that $\liminf_{\vert u\vert \to\infty}f(u)/u>-\lambda_1$, with $\lambda_1$ the first eigenvalue of $-\Delta$. In addition, the growth condition $\vert f(u)\vert \le C(1+\vert u\vert ^{n/(n-2)})$ is supposed if $n\ge 3$, whereas $f$ may grow exponentially for $n=2$. The main result of the paper asserts that the equation has a connected global attractor in $H_0^1(\Omega)\times L^2(\Omega)$, identifying $u$ with $(u, u_t)$. It is further shown that for each global orbit in the attractor the $\alpha$- resp. $\omega$-limit set is a connected subset of the critical points of the Lyapunov functional $V(u, u_t)=\int_\Omega\{(1/2)u_t^2+(1/2)\vert \nabla u\vert ^2+F(u)\}\,dx$, where $F'=f$. If the set of critical points is totally disconnected, then the solutions do not only approach the attractor as a set, but they converge to an individual critical point as $t\to\pm\infty$. The proofs rely on the application of suitable abstract results concerning the existence of attractors.

##### MSC:
 37L30 Attractors and their dimensions, Lyapunov exponents 35L70 Nonlinear second-order hyperbolic equations 37L05 General theory, nonlinear semigroups, evolution equations 35B41 Attractors (PDE)
Full Text: