Topological Methods in Nonlinear Analysis

Attractors for singularly perturbed damped wave equations on unbounded domains

Martino Prizzi and Krzysztof P. Rybakowski

Full-text: Access by subscription


For an arbitrary unbounded domain $\Omega\subset\mathbb R^3$ and for $\varepsilon> 0$, we consider the damped hyperbolic equations \begin{equation} \varepsilon u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}=f(x,u), \tag{$(\text{\rm H}_\varepsilon)$} \end{equation} with Dirichlet boundary condition on $\partial\Omega$, and their singular limit as $\varepsilon\to0$. Under suitable assumptions, (H$_\varepsilon)$ possesses a compact global attractor $\mathcal A_\varepsilon$ in $H^1_0(\Omega)\times L^2(\Omega)$, while the limiting parabolic equation possesses a compact global attractor $\widetilde{\mathcal A_0}$ in $H^1_0(\Omega)$, which can be embedded into a compact set ${\mathcal A_0}\subset H^1_0(\Omega)\times L^2(\Omega)$. We show that, as $\varepsilon\to0$, the family $({\mathcal A_\varepsilon})_{\varepsilon\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(\Omega)\times H^{-1}(\Omega)$.

Article information

Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 1-20.

First available in Project Euclid: 13 May 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Prizzi, Martino; Rybakowski, Krzysztof P. Attractors for singularly perturbed damped wave equations on unbounded domains. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 1--20.

Export citation


  • \ref\no \dfaACDR J. M. Arrieta, J. W. Cholewa, T. Dłotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains , Nonlinear Anal., 56(2004), 515–554
  • \ref\no \dfaBV A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations , J. Math. Pures Appl., 62(1983), 441–491
  • \ref\no \dfaBa J. M. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dynam. Systems, 10 (2004), 31–52
  • \ref\no \dfaCH T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford (1998) \ref\no
  • \dfaDCh J. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge (2000)
  • \ref\no \dfaF E. Feireisl, Attractors for semilinear damped wave equations on $\R^3$ , Nonlinear Anal., 23(1994), 187–195 \ref\no \dfaF1 ––––, Asymptotic behaviour and attractors for semilinear damped wave equations with a supercritical exponent , Proc. Roy. Soc. Edinburgh, 125A(1995), 1051–1062
  • \ref\no \dfaGT J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations , J. Math. Pures Appl., 66(1987), 273–319 \ref\no
  • \dfaGo J. A. Goldstein, Semigroups of Linear Operators and applications, Oxford University Press, New York (1985)
  • \ref\no\dfaGraPa1 M. Grasselli and V. Pata, On the damped semilinear wave equation with critical exponent. Dynamical systems and differential equations (Wilmington, NC, 2002) , Discrete Contin. Dynam. Syst. Suppl. (2003), 351–358 \ref\no\dfaGraPa2 ––––, Asymptotic behavior of a parabolic-hyperbolic system , Comm. Pure Appl. Anal., 3 (2004), 849–881
  • \ref\no \dfaHa J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence (1988)
  • \ref\no \dfaHR J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation , J. Differential Equations, 73(1988), 197–214
  • \ref\no\dfaHar A. Haraux, Two remarks on hyperbolic dissipative problems , Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983–1984), 161–179, Pitman, Boston(1985)
  • \ref\no\dfaPaZe V. Pata and S. Zelik, A remark on the damped wave equation , Comm. Pure Appl. Anal., 5 (2006), 609–614
  • \ref\no \dfaPR3 M. Prizzi and K. P. Rybakowski, Attractors for reaction-diffusion equations on arbitrary unbounded domains , Topol. Methods Nonlinear Anal., 30 (2007), 252–270 \ref\no \dfaPR ––––, Attractors for semilinear damped wave equations on arbitrary unbounded domains , Topol. Methods Nonlinear Anal., 31 (2008), 49–82 \ref\no\dfaRa
  • G. Raugel, Global attractors in partial differential equations , Handbook of dynamical systems, 2, 885–982, North-Holland, Amsterdam(2002)
  • \ref\no \dfaW B. Wang, Attractors for reaction-diffusion equations in unbounded domains , Physica D, 179(1999), 41–52