×

zbMATH — the first resource for mathematics

Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation. (English) Zbl 1411.34037
Authors’ abstract: In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to \[ u^{\prime \prime \prime \prime }+qu^{\prime \prime }+f(u)=0 \] under various hypotheses on the parameter \(q\) and on the potential \(F\left( t\right) =\int_{0}^{t}f\left( s\right) ds,\) generally assumed to be bounded from below. We prove a non-existence result in the case \(q\leq 0\) and an existence result of periodic solution for: 1) almost every suitably small (depending on \(F\)), positive values of \(q\); 2) all suitably large (depending on \(F\)) values of \( q.\) Finally, we describe some conditions on \(F\) which ensure that some (or all) solutions \(u_{q}\) to the equation satisfy \(\left\| u_{q}\right\| _{\infty }\rightarrow 0\) as \( q\downarrow 0\).

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berchio, E.; Ferrero, A.; Gazzola, F.; Karageorgis, P., Qualitative behavior of global solutions to some nonlinear fourth order differential equations, J. Differential Equations, 251, 10, 2696-2727, (2011) · Zbl 1236.34042
[2] Bernis, F., Elliptic and parabolic semilinear problems without conditions at infinity, Arch. Ration. Mech. Anal., 106, 3, 217-241, (1989) · Zbl 0678.35034
[3] Bonheure, D.; Sanchez, L., Heteroclinic orbits for some classes of second and fourth order differential equations, (Handbook of Differential Equations: Ordinary Differential Equations. Vol. III, (2006), Elsevier/North-Holland Amsterdam), 103-202
[4] Ferreira, V.; Moreira dos Santos, E., On the finite space blow up of the solutions of the Swift-Hohenberg equation, Calc. Var. Partial Differential Equations, 54, 1, 1161-1182, (2015) · Zbl 1344.34044
[5] Gazzola, F., Mathematical models for suspension bridges. nonlinear structural instability, Modeling, Simulation and Applications, vol. 15, (2015), Springer Cham · Zbl 1325.00032
[6] Gazzola, F.; Karageorgis, P., Refined blow-up results for nonlinear fourth order differential equations, Commun. Pure Appl. Anal., 14, 2, 677-693, (2015) · Zbl 1325.34046
[7] Gazzola, F.; Pavani, R., Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations, Arch. Ration. Mech. Anal., 207, 2, 717-752, (2013) · Zbl 1278.34036
[8] Kalies, W. D.; Kwapisz, J.; VanderVorst, R. C.A. M., Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria, Comm. Math. Phys., 193, 2, 337-371, (1998) · Zbl 0908.34034
[9] Kalies, W. D.; VanderVorst, R. C.A. M., Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation, J. Differential Equations, 131, 2, 209-228, (1996) · Zbl 0872.34033
[10] Karageorgis, P.; Stalker, J., A lower bound for the amplitude of traveling waves of suspension bridges, Nonlinear Anal., 75, 13, 5212-5214, (2012) · Zbl 1266.34078
[11] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 4, 537-578, (1990) · Zbl 0725.73057
[12] Lazer, A. C.; McKenna, P. J., On travelling waves in a suspension bridge model as the wave speed goes to zero, Nonlinear Anal., 74, 12, 3998-4001, (2011) · Zbl 1228.34065
[13] McKenna, P. J.; Walter, W., Nonlinear oscillations in a suspension bridge, Arch. Ration. Mech. Anal., 98, 2, 167-177, (1987) · Zbl 0676.35003
[14] McKenna, P. J.; Walter, W., Travelling waves in a suspension bridge, SIAM J. Appl. Math., 50, 3, 703-715, (1990) · Zbl 0699.73038
[15] Mosconi, S. J.N., Heteroclinic connections for the Swift-Hohenberg equation with multi-wells potentials, Adv. Nonlinear Stud., 14, 4, 873-894, (2014) · Zbl 1319.34074
[16] Mosconi, S. J.N.; Santra, S., On the existence and non-existence of bounded solutions for a fourth order ODE, J. Differential Equations, 255, 11, 4149-4168, (2013) · Zbl 1294.34041
[17] Peletier, L. A.; Troy, W. C., Spatial patterns. higher order models in physics and mechanics, Progress in Nonlinear Differential Equations and Their Applications, vol. 45, (2001), Birkhäuser Boston, Inc. Boston, MA · Zbl 0872.34032
[18] Radu, P.; Toundykov, D.; Trageser, J., Finite time blow-up in nonlinear suspension bridge models, J. Differential Equations, 257, 11, 4030-4063, (2014) · Zbl 1312.34076
[19] Smets, D.; van den Berg, J. B., Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations, 184, 1, 78-96, (2002) · Zbl 1029.34036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.