Topological Methods in Nonlinear Analysis

Attractors for semilinear damped wave equations on arbitrary unbounded domains

Martino Prizzi and Krzysztof P. Rybakowski

Full-text: Open access


We prove existence of global attractors for semilinear damped wave equations of the form $$ \begin{alignat}{2} \eps u_{tt}+\alpha(x) u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u), &\quad &x\in \Omega,t\in[0,\infty[, \\ u(x,t)&=0,&\quad& x\in \partial \Omega,\ t\in[0,\infty[. \end{alignat} $$ on an unbounded domain $\Omega$, without smoothness assumptions on $\beta(\cdot)$, $a_{ij}(\cdot)$, $f(\cdot,u)$ and $\partial\Omega$, and $f(x,\cdot)$ having critical or subcritical growth.

Article information

Topol. Methods Nonlinear Anal., Volume 31, Number 1 (2008), 49-82.

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 semilinear damped wave equations on arbitrary unbounded domains. Topol. Methods Nonlinear Anal. 31 (2008), no. 1, 49--82.

Export citation


  • \ref\no \dfaAB1 W. Arendt and C. J. K. Batty, Exponential stability of a diffusion equation with absorption , Differential Integral Equations, 6(1993), 1009–1024 \ref\no \dfaAB2 ––––, Absorption semigroups and Dirichlet boundary conditions , Math. Ann., 295(1993), 427–448
  • \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 Sect. A, 125(1995), 1051–1062
  • \ref\no \dfaFY D. Fall and Y. You, Global attractors for the damped nonlinear wave equation in unbounded domain , Proceedings of the Fourth World Congress of Nonlinear Analysts(2004), to appear
  • \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 \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
  • \dfaLa1 O. Ladyženskaya, The Boundary Value Problems of Mathematical Physics, Springer–Verlag, New York (1985)
  • \ref\no \dfaLa O. Ladyženskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991)
  • \ref\no \dfaM J. E. Metcalfe, Global Strichartz Estimates for Solutions of the Wave Equation Exterior to a Convex Obstacle, PhD dissertation, Johns Hopkins University, Baltimore (2003)
  • \ref\no \dfaMRW I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations , Nonlinearity, 11(1998), 1369–1393
  • \ref\no \dfaPR1 M. Prizzi and K. P. Rybakowski, Attractors for reaction-diffusion equations on arbitrary unbounded domains , Topological Methods in Nonl. Anal., to appear
  • \ref\no \dfaPR3 M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed semilinear damped wave equations on unbounded domains , Topol. Methods Nonlinear Anal., to appear \ref\no\dfaRa
  • G. Raugel, Global attractors in partial differential equations , Handbook of dynamical systems, Vol. 2, 885–982, North-Holland, Amsterdam(2002)
  • \ref\no \dfaSS H. F. Smith and C. D. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics , Math. Res. Lett., 1(1994), 729–737