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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 and Continuous Dynamical Systems, 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), 857-875. 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), 15-28. [3] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 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), 2055-2080. 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), 981-1012. 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), 225-326. [7] A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. 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, vol. 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404. [9] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. [10] A. Ćwiszewski, Periodic solutions of damped hyperbolic equations at resonance: A translation along trajectories approach, Differential Integral Equations, 24 (2011), 767-786. [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), 605-628. 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), 3202-3233. 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, 279, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. [14] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [15] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [16] J. K. Hale, L. T. Magalhaes and W. M. Oliva, Dynamics in Infinite Dimensions, Applied Mathematical Sciences, 47, Springer-Verlag, New York, 2002. doi: 10.1007/b100032. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 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), 527-537. [19] E. Hille and R. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications, 31, American Mathematical Society, Providence, RI, 1957. [20] P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance, Nonlinear Analysis: Theory, Methods and Applications, 85 (2013), 253-278. 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, Methods and Applications, 125 (2015), 167-200. 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), 609-623. [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), 537-578. 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, Vol. I-IV (Tampa, FL, 1992), de Gruyter, Berlin, 1996, 349-358. [25] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations, J. Differential Equations, 48 (1983), 334-349. 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), 733-747. 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), 2089-2096. 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), 351-382. 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), 237-277. doi: 10.1007/BF01766857. [31] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, Berlin, 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), 182-212. 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), 1-41. doi: 10.1090/S0002-9947-1985-0797044-3. [34] M. Schechter, Nonlinear elliptic boundary value problems at resonance, Nonlinear Anal., 14 (1990), 889-903. doi: 10.1016/0362-546X(90)90027-E. [35] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, 1983. [36] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [37] J. Valdo and A. Gonçalves, On bounded nonlinear perturbations of an elliptic equation at resonance, Nonlinear Anal., 5 (1981), 57-60. doi: 10.1016/0362-546X(81)90070-5.

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##### References:
 [1] S. Ahmad, A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc., 323 (1991), 857-875. 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), 15-28. [3] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 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), 2055-2080. 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), 981-1012. 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), 225-326. [7] A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. 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, vol. 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404. [9] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. [10] A. Ćwiszewski, Periodic solutions of damped hyperbolic equations at resonance: A translation along trajectories approach, Differential Integral Equations, 24 (2011), 767-786. [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), 605-628. 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), 3202-3233. 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, 279, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. [14] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [15] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [16] J. K. Hale, L. T. Magalhaes and W. M. Oliva, Dynamics in Infinite Dimensions, Applied Mathematical Sciences, 47, Springer-Verlag, New York, 2002. doi: 10.1007/b100032. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 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), 527-537. [19] E. Hille and R. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications, 31, American Mathematical Society, Providence, RI, 1957. [20] P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance, Nonlinear Analysis: Theory, Methods and Applications, 85 (2013), 253-278. 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, Methods and Applications, 125 (2015), 167-200. 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), 609-623. [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), 537-578. 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, Vol. I-IV (Tampa, FL, 1992), de Gruyter, Berlin, 1996, 349-358. [25] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations, J. Differential Equations, 48 (1983), 334-349. 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), 733-747. 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), 2089-2096. 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), 351-382. 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), 237-277. doi: 10.1007/BF01766857. [31] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, Berlin, 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), 182-212. 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), 1-41. doi: 10.1090/S0002-9947-1985-0797044-3. [34] M. Schechter, Nonlinear elliptic boundary value problems at resonance, Nonlinear Anal., 14 (1990), 889-903. doi: 10.1016/0362-546X(90)90027-E. [35] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, 1983. [36] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [37] J. Valdo and A. Gonçalves, On bounded nonlinear perturbations of an elliptic equation at resonance, Nonlinear Anal., 5 (1981), 57-60. doi: 10.1016/0362-546X(81)90070-5.
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