Abstract
We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Angenent, S.: The zeroset of a solution of a parabolic equation. J. Reine Angew. Math. 390, 79–96 (1988)
Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)
Brunovský, P., Poláčik, P.: On the local structure of \(\omega \)-limit sets of maps. Z. Angew. Math. Phys. 48, 976–986 (1997)
Busca, J., Jendoubi, M.-A., Poláčik, P.: Convergence to equilibrium for semilinear parabolic problems in \(\mathbb{R}^N\). Commun. Part. Diff. Equ. 27, 1793–1814 (2002)
Chen, X., Lou, B., Zhou, M., Giletti, T.: Long time behavior of solutions of a reaction–diffusion equation on unbounded intervals with Robin boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire (2014). doi:10.1016/j.anihpc.2014.08.004
Chen, X.-Y.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1998)
Chen, X.-Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equ. 78, 160–190 (1989)
Collet, P., Eckmann, J.-P.: Space-time behaviour in problems of hydrodynamic type: a case study. Nonlinearity 5, 1265–1302 (1992)
Du, Y., Matano, H.: Convergence and sharp thresholds for propagation in nonlinear diffusion problems. J. Eur. Math. Soc. 12(2), 279–312 (2010)
Du, Y., Poláčik, P.: Locally uniform convergence to an equilibrium for nonlinear parabolic equations on \(\mathbb{R}^N\). Indiana Univ. Math. J. to appear
Eckmann, J.-P., Rougemont, J.: Coarsening by Ginzburg–Landau dynamics. Commun. Math. Phys. 199(2), 441–470 (1998)
Fašangová, E.: Asymptotic analysis for a nonlinear parabolic equation on \({\mathbb{R}}\). Comment. Math. Univ. Carol. 39, 525–544 (1998)
Fašangová, E., Feireisl, E.: The long-time behavior of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions. Proc. Roy. Soc. Edinb. Sect. A 129, 319–329 (1999)
Feireisl, E.: On the long time behavior of solutions to nonlinear diffusion equations on \({\mathbb{R}}^{N}\). NoDEA Nonlinear Differ. Equ. Appl. 4, 43–60 (1997)
Feireisl, E., Poláčik, P.: Structure of periodic solutions and asymptotic behavior for time-periodic reaction–diffusion equations on \(\mathbb{R}\). Adv. Differ. Equ. 5, 583–622 (2000)
Fife, P.C., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)
Földes, J., Poláčik, P.: Convergence to a steady state for asymptotically autonomous semilinear heat equations on \({R}^{N}\). J. Differ. Equ. 251, 1903–1922 (2011)
Gallay, T., Slijepčević, S.: Energy flow in extended gradient partial differential equations. J. Dyn. Differ. Equ. 13, 757–789 (2001)
Gallay, T., Slijepčević, S.: Distribution of energy and convergence to equilibria in extended dissipative systems. J. Dyn. Differ. Equ. (2014). doi:10.1007/s10884-014-9376-z
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Berlin (1995)
Matano, H., Poláčik, P.: Dynamics of nonnegative solutions of one-dimensional reaction–diffusion equations with localized initial data. (in preparation)
Muratov, C.B., Zhong, X.: Threshold phenomena for symmetric decreasing solutions of reaction–diffusion equations. NoDEA Nonlinear Differ. Equ. Appl. 20, 1519–1552 (2013)
Poláčik, P.: Threshold solutions and sharp transitions for nonautonomous parabolic equations on \(\mathbb{R}^N\). Arch. Ration. Mech. Anal. 199, 69–97 (2011). Addendum: www.math.umn.edu/~polacik/Publications
Poláčik, P.: Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on \(\mathbb{R}\). J. Evol. Equ. (2014). doi:10.1007/s00028-014-0260-4
Poláčik, P., Yanagida, E.: Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM. J. Math. Anal. 46, 3481–3496 (2014)
Zlatoš, A.: Sharp transition between extinction and propagation of reaction. J. Am. Math. Soc. 19, 251–263 (2006)
Acknowledgments
Supported in part by NSF Grant DMS–1161923.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to John Mallet-Paret on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Poláčik, P. Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations. J Dyn Diff Equat 28, 605–625 (2016). https://doi.org/10.1007/s10884-014-9421-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-014-9421-y
Keywords
- Bistable reaction–diffusion equation
- Localized initial data
- Asymptotic behavior
- Nonconvergent solutions
- Threshold solutions