### Lower semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary

DOI: http://dx.doi.org/10.12775/TMNA.2019.118

#### Abstract

#### Keywords

#### References

G.S. Aragão and F.D.M. Bezerra, Upper semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary, J. Math. Anal. Appl. 462 (2017), 871–899.

G.S. Aragão and F.D.M. Bezerra, Continuity of the set equilibria of non-autonomous damped wave equations with terms concentrating on the boundary, Electron. J. Differential Equations 2019 (2019), 1–19.

G.S. Aragão and S.M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. Differential Equations 253 (2012), 2573–2592.

G.S. Aragão and S.M. Oliva, Asymptotic behavior of a reaction-diffusion problem with delay and reaction term concentrated in the boundary, São Paulo J. Math. Sci. 5 (2011), 347–376.

G.S. Aragão, A.L. Pereira and M.C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating in the boundary, J. Dynam. Differential Equations 26 (2014), 871–888.

J.M. Arrieta, A. Jiménez-Casas and A. Rodrı́guez-Bernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Rev. Mat. Iberoam. 24 (2008), 183–211.

A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992.

S.M. Bruschi, A.N. Carvalho, J.W. Cholewa and T. Dlotko, Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations, J. Dynam. Differential Equations 18 (2006), 767–814.

A.N. Carvalho and J.A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, J. Differential Equations 233 (2007), 622–653.

T. Caraballo, A.N. Carvalho, J.A. Langa and F. Rivero, A gradient-like nonautonomous evolution process, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2751–2760.

T. Caraballo, A.N. Carvalho, J.A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: existence and continuity of the pullback attractor, Nonlinear Anal. 74 (2011), 2272–2283.

A.N. Carvalho, J.A. Langa and J.C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergodic Theory Dynam. Systems 29 (2009), 1765–1780.

A.N. Carvalho, J.A. Langa and J.C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer–Verlag, New York, 2012.

I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst. 20 (2008), 459–509.

W.A. Coppel, Dichotomies in Stability Theory, Springer–Verlag, Berlin, 1978.

M. Daoulatli, Rates of decay for the wave systems with time dependent damping, Discrete Contin. Dynam. Systems 31 (2011), 407–443.

C.M. Elliott and I.N. Kostin, Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn–Hilliard equation, Nonlinearity 9 (1994), 687–702.

M.M. Freitas, P. Kalita and J.A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations 264 (2018), 1886–1945.

J.M. Ghidaglia and R. Temam, Regularity of the solutions of second-order evolution equations and their attractors, Annali della Scuola Normale Superiore di Pisa IV 14 (1987), 485–511.

J.K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. 154 (1989), 281–326.

J.K. Hale and G. Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dynam. Differential Equations 2 (1990), 19–67.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer–Verlag, Berlin, 1981.

A. Jiménez-Casas and A. Rodrı́guez-Bernal, Aymptotic behaviour of a parabolic problem with terms concentrated in the boundary, Nonlinear Anal. 71 (2009), 2377–2383.

A. Jiménez-Casas and A. Rodrı́guez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. Appl. 379 (2011), 567–588.

A.Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations 230 (2006), 702–719.

I. Lasieka and A. Stahel, The wave equation with semilinear Neumann boundary conditions, Nonlinear Anal. 15 (1990), 39–58.

I. Lasieka and R. Triggiani, Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary, J. Math. Anal. Appl. 141 (1989), 49–71.

I. Lasieka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Part I: L2 nonhomogeneous data, Ann. Mat. Pura Appl. 157 (1990), 285–367.

U. De Maio, A.K. Nandakumaran and C. Perugia, Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition, Evol. Equ. Control Theory 4 (2015), 325–346.

P. Martinez and J. Vancostenoble, Stabilization of the wave equation by on-off and positive-negative feedbacks, ESAIM Control Optim. Calc. Var. 7 (2002), 335–377.

V. Pata and S. Zelik, A remark on the damped wave equation, Comm. Pure Appl. Anal. 5 (2006), 611–616.

P. Pucci and J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems, Comm. Pure Appl. Ana. 49 (1996), 177–216.

R.P. da Silva, Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equatio, Electron. J. Differential Equations 2012 (2012), 1–8.

A. Stahel, Hyperbolic initial boundary value problems with nonlinear boundary conditions, Nonlinear Anal. 13 (1989), 231–257.

T.J. Xiao and H. Zhang, Optimal decay rates for semilinear wave equations with memory and Neumann boundary conditions, J. Appl. Anal., An International Journal 97 (2017), 1–16.

Z. Zhang, Stabilization of the wave equation with variable coefficients and a dynamical boundary control, Electron. J. Differential Equations 2016 (2016), 1–10.

### Refbacks

- There are currently no refbacks.