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Stable patterns for fourth-order parabolic equations. (English) Zbl 1020.35027
The authors study the following model equations $$u_t=-\gamma u_{xxxx}+\beta u_{xx}- F'(u),\quad (t,x)\in\bbfR^+\times (0,L),$$ with $\gamma> 0$, $\beta> 0$. Their goal is to study stable stationary states as a function of the parameters $\gamma$, $\beta$, the potential $F$, the intervallength $L$, and the boundary conditions at $x= 0$ and $x\in L$. To this end they develop a new variational gluing method for constructing stable stationary states.

MSC:
35K35Higher order parabolic equations, boundary value problems
35K55Nonlinear parabolic equations
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References:
[1] S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation , J. Differential Equations 62 (1986), 427--442. · Zbl 0581.58026 · doi:10.1016/0022-0396(86)90093-8
[2] T. Bartsch and M. Clapp, Bifurcation theory for symmetric potential operators and the equivariant cup-length , Math. Z. 204 (1990), 341--356. · Zbl 0682.58037 · doi:10.1007/BF02570878 · eudml:183775
[3] J. B. van den Berg, Uniqueness of solutions for the extended Fisher-Kolmogorov equation , C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 447--452. · Zbl 0913.34052 · doi:10.1016/S0764-4442(97)89790-X
[4] --. --. --. --., The phase-plane picture for a class of fourth-order conservative differential equations , J. Differential Equations 161 (2000), 110--153. · Zbl 0952.34026 · doi:10.1006/jdeq.1999.3698
[5] --------, Branches of heteroclinic, homoclinic and periodic solutions in a fourth-order bi-stable system , M.Sc. thesis, Leiden University, Netherlands, 1996.
[6] --------, Dynamics and equilibria of fourth order differential equations , doctoral thesis, Leiden University, Netherlands, 2000. · Zbl 1136.37301
[7] G. J. B. van den Berg, L. A. Peletier, and W. C. Troy, Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation , Arch. Rational Mech. Anal. 158 (2001), 91--153. · Zbl 0983.34032 · doi:10.1007/s002050100132
[8] J. B. van den Berg and R. C. Vandervorst, Second order Lagrangian twist systems: Simple closed characteristics , Trans. Amer. Math. Soc. 354 (2002), 1393--1420. \CMP1 873 011 JSTOR: · Zbl 0998.37014 · doi:10.1090/S0002-9947-01-02882-3 · http://links.jstor.org/sici?sici=0002-9947%28200204%29354%3A4%3C1393%3ASOLTSS%3E2.0.CO%3B2-V&origin=euclid
[9] B. Buffoni, A. R. Champneys, and J. F. Toland, Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system , J. Dynam. Differential Equations 8 (1996), 221--279. · Zbl 0854.34047 · doi:10.1007/BF02218892
[10] B. Buffoni and E. Séré, A global condition for quasi-random behavior in a class of conservative systems , Comm. Pure Appl. Math. 49 (1996), 285--305. · Zbl 0860.58027 · doi:10.1002/(SICI)1097-0312(199603)49:3<285::AID-CPA3>3.0.CO;2-9
[11] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type , Applicable Anal. 4 (1974), 17--37. · Zbl 0296.35046 · doi:10.1080/00036817408839081
[12] B. D. Coleman, M. Marcus, and V. J. Mizel, On the thermodynamics of periodic phases , Arch. Rational Mech. Anal. 117 (1992), 321--347. · Zbl 0788.73015 · doi:10.1007/BF00376187
[13] G. T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation , Phys. Rev. Lett. 60 (1988), 2641--2644.
[14] E. J. Doedel and others, auto97, continuation and bifurcation software for ordinary differential equations (with HomCont), 1997, available from ftp://ftp.cs.concordia.ca, directory: pub/doedel/auto I. Ekeland, On the variational principle , J. Math. Anal. Appl. 47 (1974), 324--353. · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0
[15] E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index , J. Funct. Anal. 26 (1977), 48--67. · Zbl 0363.47029 · doi:10.1016/0022-1236(77)90015-5
[16] R. W. Ghrist, J. B. van den Berg, and R. C. Vandervorst, Closed characteristics of fourth-order twist systems via braids , C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 861--865. · Zbl 0983.37077 · doi:10.1016/S0764-4442(00)01736-5
[17] --------, Morse theory on spaces of braids and Lagrangian dynamics , · Zbl 1017.37031 · doi:10.1007/s00222-002-0277-0 · http://arxiv.org/abs/math.DS/0105082
[18] J. K. Hale, Asymptotic Behavior of Dissipative Systems , Math. Surveys Monogr. 25 , Amer. Math. Soc., Providence, 1988. · Zbl 0642.58013
[19] D. Henry, Geometric Theory of Semilinear Parabolic Equations , Lecture Notes in Math. 840 , Springer, Berlin, 1981. · Zbl 0456.35001
[20] --. --. --. --., Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations , J. Differential Equations 59 (1985), 165--205. · Zbl 0572.58012 · doi:10.1016/0022-0396(85)90153-6
[21] W. D. Kalies, J. Kwapisz, J. B. VandenBerg, and R. C. A. M. VanderVorst, Homotopy classes for stable periodic and chaotic patterns in fourth-order Hamiltonian systems , Comm. Math. Phys. 214 (2000), 573--592. · Zbl 0980.37017 · doi:10.1007/PL00005537
[22] W. D. Kalies, J. Kwapisz, and R. C. A. M. VanderVorst, Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria , Comm. Math. Phys. 193 (1998), 337--371. · Zbl 0908.34034 · doi:10.1007/s002200050332
[23] W. D. Kalies and R. C. A. M. VanderVorst, Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation , J. Differential Equations 131 (1996), 209--228. · Zbl 0872.34033 · doi:10.1006/jdeq.1996.0161
[24] W. D. Kalies, R. C. A. M. VanderVorst, and T. Wanner, Slow motion in higher-order systems and $\Gamma$-convergence in one space dimension , Nonlinear Anal. 44 (2001), 33--57. · Zbl 0976.35005 · doi:10.1016/S0362-546X(99)00245-X
[25] J. Kwapisz, Uniqueness of the stationary wave for the extended Fisher-Kolmogorov equation , J. Differential Equations 165 (2000), 235--253. · Zbl 0965.34039 · doi:10.1006/jdeq.1999.3750
[26] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401--441. · Zbl 0496.35011
[27] K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations , SIAM J. Math. Anal. 26 (1995), 1199--1224. · Zbl 0841.35014 · doi:10.1137/S0036141093250827
[28] V. J. Mizel, L. A. Peletier, and W. C. Troy, Periodic phases in second-order materials , Arch. Rational Mech. Anal. 145 (1998), 343--382. · Zbl 0931.74006 · doi:10.1007/s002050050133
[29] L. A. Peletier and W. C. Troy, Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: Kinks , Differential Integral Equations 8 (1995), 1279--1304. · Zbl 0826.34056
[30] --. --. --. --., A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation , Topol. Methods Nonlinear Anal. 6 (1995), 331--355. · Zbl 0862.34030
[31] --. --. --. --., Chaotic spatial patterns described by the extended Fisher-Kolmogorov equation , J. Differential Equations 129 (1996), 458--508. · Zbl 0862.34012 · doi:10.1006/jdeq.1996.0124
[32] --. --. --. --., Spatial patterns described by the extended Fisher-Kolmogorov equation: Periodic solutions , SIAM J. Math. Anal. 28 (1997), 1317--1353. · Zbl 0891.34048 · doi:10.1137/S0036141095280955
[33] --------, Spatial Patterns: Higher Order Models in Physics and Mechanics , Progr. Nonlinear Differential Equations Appl. 45 , Birkhäuser, Boston, 2001.
[34] L. A. Peletier [Pelet’e], R. C. [K.] A. M. Van der Vorst, and W. C. Troy [V. K. Troĭ], Stationary solutions of a fourth-order nonlinear diffusion equation (in Russian), Differentsial’nye Uravneniya 31 (1995), 327--337.; English translation in Differential Equations 31 (1995), 301--314. · Zbl 0856.35029
[35] M. A. Peletier, Non-existence and uniqueness results for fourth-order Hamiltonian systems , Nonlinearity 12 (1999), 1555--1570. · Zbl 0934.37050 · doi:10.1088/0951-7715/12/6/308
[36] B. Sandstede, Stability of multiple-pulse solutions , Trans. Amer. Math. Soc. 350 (1998), 429--472. JSTOR: · Zbl 0887.35020 · doi:10.1090/S0002-9947-98-01673-0 · http://links.jstor.org/sici?sici=0002-9947%28199802%29350%3A2%3C429%3ASOMS%3E2.0.CO%3B2-7&origin=euclid
[37] J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability , Phys. Rev. A 15 (1977), 319--328.
[38] P. D. Woods and A. R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation , Phys. D 129 (1999), 147--170. · Zbl 0952.37009 · doi:10.1016/S0167-2789(98)00309-1
[39] W. Zimmermann, Propagating fronts near a Lifshitz point , Phys. Rev. Lett. 66 (1991), 1546.