Topological Methods in Nonlinear Analysis

Dimension of attractors and invariant sets of damped wave equations in unbounded domains

Martino Prizzi

Full-text: Open access


Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation $$ \begin{alignat}{2} u_{tt}+\alpha u_t+\beta(x)u-\Delta u& =f(x,u), &\quad&(t,x)\in[0,+\infty[\times\Omega, \\ u&=0,&\quad &(t,x)\in[0,+\infty[\times\partial\Omega, \end{alignat} $$ in $H^1_0(\Omega)\times L^2(\Omega)$ has finite Hausdorff and fractal dimension. Here $\Omega$ is a regular, possibly unbounded, domain in $\mathbb R^3$ and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff and fractal dimension of $\mathcal I$ in terms of the structure parameters of the equation.

Article information

Topol. Methods Nonlinear Anal., Volume 41, Number 2 (2013), 267-285.

First available in Project Euclid: 21 April 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Prizzi, Martino. Dimension of attractors and invariant sets of damped wave equations in unbounded domains. Topol. Methods Nonlinear Anal. 41 (2013), no. 2, 267--285.

Export citation


  • W. Arendt and C.J.K. Batty, Exponential stability of a diffusion equation with absorption , Differential Integral Equations, 6 (1993), 1009–1024 \ref\key 2 ––––, Absorption semigroups and Dirichlet boundary conditions , Math. Ann., 295 (1993), 427–448 \ref\key 3
  • A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam (1991) \ref\key 4
  • F.E. Browder, Estimates and existence theorems for elliptic boundary value problems , Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 365–372 \ref\key 5
  • T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford (1988) \ref\key 6
  • D.E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford (1987) \ref\key 7
  • E. Feireisl, Attractors for semilinear damped wave equations on $\R^3$ , Nonlinear Anal., 23 (1994), 187–195 \ref\key 8
  • J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York (1985) \ref \key 9
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840 , Springer–Verlag, New York (1981) \ref\key 10
  • T. Kato, Linear evolution equations of hyperbolic type , J. Fac. Sci. Univ. Tokyo Sec. I, 17 (1970), 241–258 \ref\key 11
  • N.I. Karachalios and N.M. Stavrakakis, Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\R^N$ , Discrete Contin. Dynam. Systems, 8 (2002), 939–951 \ref\key 12
  • O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991) \ref\key 13
  • P. Li and Shing-Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. , 88 (1983), 309–318 \ref\key 14
  • J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright , J. Differential Equations, 22 (1976), 331–348 \ref\key 15
  • R. Ma\~ né, On the dimension of the compact invariant sets of certain nonlinear maps , Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), 230–242, Lecture Notes in Math., 898 , Springer, Berlin–New York (1981) \ref\key 16
  • M. Prizzi, Regularity of invariant sets in semilinear damped wave equations , J. Differential Equations, 247 (2009), 3315–3337 \ref\key 17
  • M. Prizzi, Dimension of attractors and invariant sets in reaction diffusion equations , Topol. Methods Nonlinear Anal., 40 (2012), 315–326 \ref\key 18
  • M. Prizzi and K.P. Rybakowski, Attractors for reaction diffusion equations on arbitrary unbounded domains , Topol. Methods Nonlinear Anal., 30 (2007), 251–270 \ref\key 19 ––––, Attractors for semilinear damped wave equations on arbitrary unbounded domains , Topol. Methods Nonlinear Anal., 31 (2008), 49–82 \ref\key 20 ––––, Attractors for singularly perturbed hyperbolic equations on unbounded domains , Topol. Methods Nonlinear Anal., 32 (2008), 1–21 \ref\key 21
  • J.C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, 186 , Cambridge University Press, Cambridge (2011) \ref\key 22
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV – Analysis of Operators, Academic Press, New York (1978) \ref\key 23
  • G. Rozenblum and M. Solomyak \paper \romCLR-estimate for the generators of positivity preserving and positively dominated semigroups, (Russian, Algebra i Analiz, 9 (1997), 214–236)\moreref \transl\nofrills transl., St. Petersburg Math. J., 9 (1998), 1195–1211 \ref\key 24
  • Z. Shengfan, Dimension of the global attractor for damped nonlinear wave equations , Proc. Am. Math. Soc., 127 (1999), 3623–3631 \ref\key 25
  • R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York (1988) \ref\key 26
  • S. Zelik, The attractor for a nonlinear hyperbolic equation in an unbounded domain , Discrete Contin. Dynam. Systems, 7 (2001), 593–641