# American Institute of Mathematical Sciences

September  2014, 13(5): 2059-2093. doi: 10.3934/cpaa.2014.13.2059

## Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit

 1 Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation 2 Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  November 2012 Revised  November 2013 Published  June 2014

We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter $\varepsilon$ at the second derivative with respect to the variable $t$ corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function $f$ and the external forces $g(t)$, this problem possesses the uniform attractor $\mathcal A_\varepsilon$ and that these attractors tend as $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolic equation. Moreover, in case where the limit attractor $\mathcal A_0$ is regular, we give the detailed description of the structure of the uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, and estimate the symmetric distance between the attractors $\mathcal A_\varepsilon$ and $\mathcal A_0$.
Citation: Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
##### References:
 [1] S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., l20 (1967), 207-229. [2] A. V. Babin, Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math., 44 (1995), 207-223. doi: 10.1070/IM1995v044n02ABEH001594. [3] A. V. Babin, Inertial manifolds for traveling-wave solutions of reaction-diffusion systems, Comm. Pure Appl. Math., 48 (1995), 167-198. doi: 10.1002/cpa.3160480205. [4] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Nauka, Moscow 1989; English transl. Stud. Math. Appl., 25, North Holland, Amsterdam, 1992. [5] O. V. Besov, V. P Ilyin and S. M. Nikolskij, Integral Representations of Functions and Embedding Theorems, M.: Nauka, 1996. [6] À. Calsina, X. Mora and J. Solà-Morales, The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limit, J. Diff. Eqns., 102 (1993) 244-304. doi: 10.1006/jdeq.1993.1030. [7] À. Calsina, J. Solà-Morales and M. València, Bounded solutions of some nonlinear elliptic equations in cylindrical domains, J. Dynam. Diff. Eqns., 9 (1997), 343-372. doi: 10.1007/BF02227486. [8] V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math, 192 (2001), 11-47. doi: 10.1070/SM2001v192n01ABEH000534. [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, RI, 2002. [10] G. Dangelmayr, B. Fiedler, K. Kirchgässner and A. Mielke, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Pitman Research, Notes in Mathematics Series, 352, Longman, Harlow, 1996. [11] M. Efendiev and S. Zelik, The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differential Equations, 8 (2003), 673-732. [12] B. Fiedler, A. Scheel and M. I. Vishik, Large patterns of elliptic systems in infinite cylinders, J. Math. Pures Appl., 77 (1998), 879-907. doi: 10.1016/S0021-7824(01)80002-7. [13] A. Yu. Goritskij and M. I. Vishik, Integral manifolds for nonautonomous equation, Rend. Accad. Naz. XL, Mem di Math. e Appl., 115 (1997), 106-146. [14] A. Yu. Goritskij and M. I. Vishik, Local integral manifolds for a nonautonomous parabolic equation, Journal of Math. Sci., 85 (1997), 2428-2439. doi: 10.1007/BF02355848. [15] A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, 1991. [16] A. Ilyin, Averaging for dissipative dynamical systems with rapidly oscillating righ-hand sides, Mat. Sbornik, 187 (1996), 15-58. doi: 10.1070/SM1996v187n05ABEH000126. [17] K. Kirchgässner, Wave-solutions of reversible systems and applications, J. Diff. Eqns., 45 (1982), 113-127. doi: 10.1016/0022-0396(82)90058-4. [18] O. A. Ladyzhenskaya, O. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translated from Russian, AMS, Providence, RI, 1967. [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982. [20] A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Diff. Eqns., 110 (1994), 322-355. doi: 10.1006/jdeq.1994.1070. [21] A. Milke and S. Zelik, Infinite-dimensional trajectory attractors of elliptic boundary value problems in cylindrical domains, Uspekhi Mat. Nauk, 57 (2002) (346), 119-150 (Russian). doi: 10.1070/RM2002v057n04ABEH000550. [22] D. Peterhof, B. Sandstede and A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Diff. Eqns, 140 (1997), 266-308. doi: 10.1006/jdeq.1997.3303. [23] A. Scheel, Existence of fast travelling waves for some parabolic equations: a dynamical systems approach, J. Dyn. Diff. Eqns., 8 (1996), 469-547. doi: 10.1007/BF02218843. [24] B. W. Schulze, M. I. Vishik, I. Witt and S. V. Zelik, The trajectory attractor for a nonlinear elliptic system in a cylindrical domain with piecewise smooth boundary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 125-166. [25] A. Shapoval, Attractors of nonlinear elliptic equations with a small parameter, Differ. Uravn., 37 (2001), 1239-1249, doi: 10.1023/A:1012582031204. [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New-York, 1988. doi: 10.1007/978-1-4684-0313-8. [27] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam-New York, 1978. [28] M. I. Vishik and S. V. Zelik, The trajectory attractor for a nonlinear elliptic system in a cylindrical domain, Math. Sbornik, 187 (1996), 1755-1789. doi: 10.1070/SM1996v187n12ABEH000177. [29] M. I. Vishik and S. V. Zelik, The regular attractor for a nonlinear elliptic system in a cylindrical domain, Math. Sbornik, 190 (1999), 23-58. doi: 10.1070/SM1999v190n06ABEH000411. [30] M. Vishik, S. Zelik and V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 3-46. doi: 10.1070/SM2013v204n01ABEH004290. [31] S. Zelik, The dynamics of fast nonautonomous travelling waves and homogenization, in Proc. of Conf. in Honor of R. Temam 60th annivesary, 7-10 of March 2000, Atlantique Editions de l'Actualite Scientifique, Poitou-Charents, 2001, 131-142. [32] S. Zelik, Global averaging and parametric resonances in damped semilinear wave equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1053-1097. doi: 10.1017/S0308210500004881.

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##### References:
 [1] S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., l20 (1967), 207-229. [2] A. V. Babin, Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math., 44 (1995), 207-223. doi: 10.1070/IM1995v044n02ABEH001594. [3] A. V. Babin, Inertial manifolds for traveling-wave solutions of reaction-diffusion systems, Comm. Pure Appl. Math., 48 (1995), 167-198. doi: 10.1002/cpa.3160480205. [4] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Nauka, Moscow 1989; English transl. Stud. Math. Appl., 25, North Holland, Amsterdam, 1992. [5] O. V. Besov, V. P Ilyin and S. M. Nikolskij, Integral Representations of Functions and Embedding Theorems, M.: Nauka, 1996. [6] À. Calsina, X. Mora and J. Solà-Morales, The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limit, J. Diff. Eqns., 102 (1993) 244-304. doi: 10.1006/jdeq.1993.1030. [7] À. Calsina, J. Solà-Morales and M. València, Bounded solutions of some nonlinear elliptic equations in cylindrical domains, J. Dynam. Diff. Eqns., 9 (1997), 343-372. doi: 10.1007/BF02227486. [8] V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math, 192 (2001), 11-47. doi: 10.1070/SM2001v192n01ABEH000534. [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, RI, 2002. [10] G. Dangelmayr, B. Fiedler, K. Kirchgässner and A. Mielke, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Pitman Research, Notes in Mathematics Series, 352, Longman, Harlow, 1996. [11] M. Efendiev and S. Zelik, The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differential Equations, 8 (2003), 673-732. [12] B. Fiedler, A. Scheel and M. I. Vishik, Large patterns of elliptic systems in infinite cylinders, J. Math. Pures Appl., 77 (1998), 879-907. doi: 10.1016/S0021-7824(01)80002-7. [13] A. Yu. Goritskij and M. I. Vishik, Integral manifolds for nonautonomous equation, Rend. Accad. Naz. XL, Mem di Math. e Appl., 115 (1997), 106-146. [14] A. Yu. Goritskij and M. I. Vishik, Local integral manifolds for a nonautonomous parabolic equation, Journal of Math. Sci., 85 (1997), 2428-2439. doi: 10.1007/BF02355848. [15] A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, 1991. [16] A. Ilyin, Averaging for dissipative dynamical systems with rapidly oscillating righ-hand sides, Mat. Sbornik, 187 (1996), 15-58. doi: 10.1070/SM1996v187n05ABEH000126. [17] K. Kirchgässner, Wave-solutions of reversible systems and applications, J. Diff. Eqns., 45 (1982), 113-127. doi: 10.1016/0022-0396(82)90058-4. [18] O. A. Ladyzhenskaya, O. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translated from Russian, AMS, Providence, RI, 1967. [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982. [20] A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Diff. Eqns., 110 (1994), 322-355. doi: 10.1006/jdeq.1994.1070. [21] A. Milke and S. Zelik, Infinite-dimensional trajectory attractors of elliptic boundary value problems in cylindrical domains, Uspekhi Mat. Nauk, 57 (2002) (346), 119-150 (Russian). doi: 10.1070/RM2002v057n04ABEH000550. [22] D. Peterhof, B. Sandstede and A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Diff. Eqns, 140 (1997), 266-308. doi: 10.1006/jdeq.1997.3303. [23] A. Scheel, Existence of fast travelling waves for some parabolic equations: a dynamical systems approach, J. Dyn. Diff. Eqns., 8 (1996), 469-547. doi: 10.1007/BF02218843. [24] B. W. Schulze, M. I. Vishik, I. Witt and S. V. Zelik, The trajectory attractor for a nonlinear elliptic system in a cylindrical domain with piecewise smooth boundary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 125-166. [25] A. Shapoval, Attractors of nonlinear elliptic equations with a small parameter, Differ. Uravn., 37 (2001), 1239-1249, doi: 10.1023/A:1012582031204. [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New-York, 1988. doi: 10.1007/978-1-4684-0313-8. [27] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam-New York, 1978. [28] M. I. Vishik and S. V. Zelik, The trajectory attractor for a nonlinear elliptic system in a cylindrical domain, Math. Sbornik, 187 (1996), 1755-1789. doi: 10.1070/SM1996v187n12ABEH000177. [29] M. I. Vishik and S. V. Zelik, The regular attractor for a nonlinear elliptic system in a cylindrical domain, Math. Sbornik, 190 (1999), 23-58. doi: 10.1070/SM1999v190n06ABEH000411. [30] M. Vishik, S. Zelik and V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 3-46. doi: 10.1070/SM2013v204n01ABEH004290. [31] S. Zelik, The dynamics of fast nonautonomous travelling waves and homogenization, in Proc. of Conf. in Honor of R. Temam 60th annivesary, 7-10 of March 2000, Atlantique Editions de l'Actualite Scientifique, Poitou-Charents, 2001, 131-142. [32] S. Zelik, Global averaging and parametric resonances in damped semilinear wave equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1053-1097. doi: 10.1017/S0308210500004881.
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