# American Institute of Mathematical Sciences

• Previous Article
Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems
• DCDS Home
• This Issue
• Next Article
Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients
June  2016, 36(6): 3227-3250. doi: 10.3934/dcds.2016.36.3227

## Homotopy invariants methods in the global dynamics of strongly damped wave equation

 1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń

Received  May 2015 Revised  October 2015 Published  December 2015

We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and determine the Conley index of the set $K_\infty$, that is the union of the bounded orbits of this equation.
Citation: Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227
##### References:
 [1] S. Ahmad, A nonstandard resonance problem for ordinary differential equations,, Trans. Amer. Math. Soc., 323 (1991), 857. doi: 10.1090/S0002-9947-1991-1010407-9. [2] A. Ambrosetti and G. Mancini, Theorems of existence and multiplicity for nonlinear elliptic problems with noninvertible linear part,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 15. [3] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. [4] J. Arrieta, R. Pardo and A. Rodriguez-Bernal, Equilibria and global dynamics of a problem with bifurcation from infinity,, J. Differential Equations, 246 (2009), 2055. doi: 10.1016/j.jde.2008.09.002. [5] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong'' resonance at infinity,, Nonlinear Anal., 7 (1983), 981. doi: 10.1016/0362-546X(83)90115-3. [6] H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 225. [7] A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287. doi: 10.2140/pjm.2002.207.287. [8] J. W. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, (2000). doi: 10.1017/CBO9780511526404. [9] C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series in Mathematics, (1978). [10] A. Ćwiszewski, Periodic solutions of damped hyperbolic equations at resonance: A translation along trajectories approach,, Differential Integral Equations, 24 (2011), 767. [11] A. Ćwiszewski and P. Kokocki, Krasnosel\cprime skii type formula and translation along trajectories method for evolution equations,, Discrete Contin. Dyn. Syst., 22 (2008), 605. doi: 10.3934/dcds.2008.22.605. [12] A. Ćwiszewski and K. P. Rybakowski, Singular dynamics of strongly damped beam equation,, J. Differential Equations, 247 (2009), 3202. doi: 10.1016/j.jde.2009.09.006. [13] D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications,, Pitman Research Notes in Mathematics Series, (1992). [14] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000). [15] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988). [16] J. K. Hale, L. T. Magalhaes and W. M. Oliva, Dynamics in Infinite Dimensions,, Applied Mathematical Sciences, (2002). doi: 10.1007/b100032. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). [18] P. Hess, Nonlinear perturbations of linear elliptic and parabolic problems at resonance: Existence of multiple solutions,, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 527. [19] E. Hille and R. Phillips, Functional Analysis and Semi-Groups,, Colloquium Publications, (1957). [20] P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance,, Nonlinear Analysis: Theory, 85 (2013), 253. doi: 10.1016/j.na.2013.02.030. [21] P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation,, Nonlinear Analysis: Theory, 125 (2015), 167. doi: 10.1016/j.na.2015.05.012. [22] E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (1970), 609. [23] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,, SIAM Rev., 32 (1990), 537. doi: 10.1137/1032120. [24] A. C. Lazer and P. J. McKenna, Open problems in nonlinear ordinary boundary value problems arising from the study of large-amplitude periodic oscillations in suspension bridges,, World Congress of Nonlinear Analysts '92, (1992), 349. [25] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations,, J. Differential Equations, 48 (1983), 334. doi: 10.1016/0022-0396(83)90098-0. [26] J. Mawhin and J. R. Ward, Bounded solutions of some second order nonlinear differential equations,, J. London Math. Soc., 58 (1998), 733. doi: 10.1112/S0024610798006784. [27] R. Ortega and A. Tineo, Resonance and non-resonance in a problem of boundedness,, Proc. Amer. Math. Soc., 124 (1996), 2089. doi: 10.1090/S0002-9939-96-03457-0. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [29] K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows,, Trans. Amer. Math. Soc., 269 (1982), 351. doi: 10.1090/S0002-9947-1982-0637695-7. [30] K. P. Rybakowski, Nontrivial solutions of elliptic boundary value problems with resonance at zero,, Ann. Mat. Pura Appl., 139 (1985), 237. doi: 10.1007/BF01766857. [31] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (1987). doi: 10.1007/978-3-642-72833-4. [32] K. P. Rybakowski, Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations,, J. Differential Equations, 51 (1984), 182. doi: 10.1016/0022-0396(84)90107-4. [33] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets,, Trans. Amer. Math. Soc., 291 (1985), 1. doi: 10.1090/S0002-9947-1985-0797044-3. [34] M. Schechter, Nonlinear elliptic boundary value problems at resonance,, Nonlinear Anal., 14 (1990), 889. doi: 10.1016/0362-546X(90)90027-E. [35] J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Grundlehren der Mathematischen Wissenschaften, (1983). [36] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, VEB Deutscher Verlag der Wissenschaften, (1978). [37] J. Valdo and A. Gonçalves, On bounded nonlinear perturbations of an elliptic equation at resonance,, Nonlinear Anal., 5 (1981), 57. doi: 10.1016/0362-546X(81)90070-5.

show all references

##### References:
 [1] S. Ahmad, A nonstandard resonance problem for ordinary differential equations,, Trans. Amer. Math. Soc., 323 (1991), 857. doi: 10.1090/S0002-9947-1991-1010407-9. [2] A. Ambrosetti and G. Mancini, Theorems of existence and multiplicity for nonlinear elliptic problems with noninvertible linear part,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 15. [3] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. [4] J. Arrieta, R. Pardo and A. Rodriguez-Bernal, Equilibria and global dynamics of a problem with bifurcation from infinity,, J. Differential Equations, 246 (2009), 2055. doi: 10.1016/j.jde.2008.09.002. [5] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong'' resonance at infinity,, Nonlinear Anal., 7 (1983), 981. doi: 10.1016/0362-546X(83)90115-3. [6] H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 225. [7] A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287. doi: 10.2140/pjm.2002.207.287. [8] J. W. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, (2000). doi: 10.1017/CBO9780511526404. [9] C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series in Mathematics, (1978). [10] A. Ćwiszewski, Periodic solutions of damped hyperbolic equations at resonance: A translation along trajectories approach,, Differential Integral Equations, 24 (2011), 767. [11] A. Ćwiszewski and P. Kokocki, Krasnosel\cprime skii type formula and translation along trajectories method for evolution equations,, Discrete Contin. Dyn. Syst., 22 (2008), 605. doi: 10.3934/dcds.2008.22.605. [12] A. Ćwiszewski and K. P. Rybakowski, Singular dynamics of strongly damped beam equation,, J. Differential Equations, 247 (2009), 3202. doi: 10.1016/j.jde.2009.09.006. [13] D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications,, Pitman Research Notes in Mathematics Series, (1992). [14] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000). [15] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988). [16] J. K. Hale, L. T. Magalhaes and W. M. Oliva, Dynamics in Infinite Dimensions,, Applied Mathematical Sciences, (2002). doi: 10.1007/b100032. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). [18] P. Hess, Nonlinear perturbations of linear elliptic and parabolic problems at resonance: Existence of multiple solutions,, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 527. [19] E. Hille and R. Phillips, Functional Analysis and Semi-Groups,, Colloquium Publications, (1957). [20] P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance,, Nonlinear Analysis: Theory, 85 (2013), 253. doi: 10.1016/j.na.2013.02.030. [21] P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation,, Nonlinear Analysis: Theory, 125 (2015), 167. doi: 10.1016/j.na.2015.05.012. [22] E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (1970), 609. [23] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,, SIAM Rev., 32 (1990), 537. doi: 10.1137/1032120. [24] A. C. Lazer and P. J. McKenna, Open problems in nonlinear ordinary boundary value problems arising from the study of large-amplitude periodic oscillations in suspension bridges,, World Congress of Nonlinear Analysts '92, (1992), 349. [25] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations,, J. Differential Equations, 48 (1983), 334. doi: 10.1016/0022-0396(83)90098-0. [26] J. Mawhin and J. R. Ward, Bounded solutions of some second order nonlinear differential equations,, J. London Math. Soc., 58 (1998), 733. doi: 10.1112/S0024610798006784. [27] R. Ortega and A. Tineo, Resonance and non-resonance in a problem of boundedness,, Proc. Amer. Math. Soc., 124 (1996), 2089. doi: 10.1090/S0002-9939-96-03457-0. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [29] K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows,, Trans. Amer. Math. Soc., 269 (1982), 351. doi: 10.1090/S0002-9947-1982-0637695-7. [30] K. P. Rybakowski, Nontrivial solutions of elliptic boundary value problems with resonance at zero,, Ann. Mat. Pura Appl., 139 (1985), 237. doi: 10.1007/BF01766857. [31] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (1987). doi: 10.1007/978-3-642-72833-4. [32] K. P. Rybakowski, Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations,, J. Differential Equations, 51 (1984), 182. doi: 10.1016/0022-0396(84)90107-4. [33] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets,, Trans. Amer. Math. Soc., 291 (1985), 1. doi: 10.1090/S0002-9947-1985-0797044-3. [34] M. Schechter, Nonlinear elliptic boundary value problems at resonance,, Nonlinear Anal., 14 (1990), 889. doi: 10.1016/0362-546X(90)90027-E. [35] J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Grundlehren der Mathematischen Wissenschaften, (1983). [36] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, VEB Deutscher Verlag der Wissenschaften, (1978). [37] J. Valdo and A. Gonçalves, On bounded nonlinear perturbations of an elliptic equation at resonance,, Nonlinear Anal., 5 (1981), 57. doi: 10.1016/0362-546X(81)90070-5.
 [1] Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617 [2] Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387 [3] M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599 [4] Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629 [5] Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985 [6] Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056 [7] Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243 [8] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [9] Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 [10] Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639 [11] Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511 [12] Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285 [13] Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613 [14] P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692 [15] Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124 [16] Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211 [17] Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135 [18] Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004 [19] Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701 [20] Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249

2018 Impact Factor: 1.143