×

zbMATH — the first resource for mathematics

Threshold solutions and sharp transitions for nonautonomous parabolic equations on \({\mathbb{R}^N}\). (English) Zbl 1262.35130
Author’s abstract: “This paper is devoted to a class of nonautonomous parabolic equations of the form \(u _{t } = \varDelta u + f(t, u)\) on \({\mathbb{R}^N}\). We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as \(t \rightarrow \infty \), are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.”

MSC:
35K58 Semilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B07 Axially symmetric solutions to PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aronson D.G. (1968) Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 22: 607–694 (Addendum: 25, 221–228 (1971)) · Zbl 0182.13802
[2] Aronson D.G., Weinberger H. (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30: 33–76 · Zbl 0407.92014
[3] Bartsch T., Poláčik P., Quittner P. Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations. J. Eur. Math. Soc. (to appear) · Zbl 1215.35041
[4] Berestycki H., Lions P.-L. (1983) Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82: 313–345 · Zbl 0533.35029
[5] Bidaut-Véron M.-F.: Initial blow-up for the solutions of a semilinear parabolic equation with source term. In: \(\backslash\)’Equations aux dérivées partielles et applications, pp. 189–198. Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998 · Zbl 0914.35055
[6] Brunovský , Poláčik P., Sandstede B. (1992) Convergence in general periodic parabolic equations in one space dimension. Nonlinear Anal. 18: 209–215 · Zbl 0796.35009
[7] Busca J., Jendoubi M.-A., Poláčik P. (2002) Convergence to equilibrium for semilinear parabolic problems in $${\(\backslash\)mathbb{R}\^N}$$ . Commun. Partial Differ. Equ. 27: 1793–1814 · Zbl 1021.35013
[8] Chen X., Fila M., Guo J.-S. (2008) Boundedness of global solutions of a supercritical parabolic equation. Nonlinear Anal. 68: 621–628 · Zbl 1128.35057
[9] Cortázar C., del Pino M., Elgueta M. (1999) The problem of uniqueness of the limit in a semilinear heat equation. Commun. Partial Differ. Equ. 24: 2147–2172 · Zbl 0940.35107
[10] Du, Y., Matano, H.: Convergence and sharp thresholds for propagation in nonlinear diffusion problems. J. Eur. Math. Soc. (to appear) · Zbl 1207.35061
[11] Fašangová E. (1998) Asymptotic analysis for a nonlinear parabolic equation on $${\(\backslash\)mathbb{R}}$$ . Comment. Math. Univ. Carolinae 39: 525–544 · Zbl 0963.35080
[12] Fašangová E., Feireisl E. (1999) The long-time behavior of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions. Proc. R. Soc. Edinb. Sect. A 129: 319–329 · Zbl 0933.35101
[13] Feireisl E., Petzeltová H. (1997) Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Differ. Integral Equ. 10: 181–196 · Zbl 0879.35023
[14] Feireisl E., Poláčik P. (2000) Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on R. Adv. Differ. Equ. 5: 583–622 · Zbl 0987.35079
[15] Fife P.C., McLeod J.B. (1977) The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65: 335–361 · Zbl 0361.35035
[16] Flores J.G. (1988) On a threshold of codimension 1 for the Nagumo equation. Commun. Partial Differ. Equ. 13: 1235–1263 · Zbl 0665.35034
[17] Húska J. (2006) Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: divergence case. Indiana Univ. Math. J. 55: 1015–1044 · Zbl 1112.35082
[18] Húska, J.: Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: nondivergence case. Trans. Am. Math. Soc. (to appear) · Zbl 1155.35052
[19] Húska J., Poláčik P. (2008) Exponential separation and principal Floquet bundles for linear parabolic equations on $${\(\backslash\)mathbb{R}\^N}$$ . Disc. Cont. Dyn. Syst. 20: 81–113 · Zbl 1170.35021
[20] Húska J., Poláčik P., Safonov M.V. (2007) Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Lineaire 24: 711–739 · Zbl 1139.35046
[21] Jones C. (1983) Spherically symmetric solutions of a reaction-diffusion equation. J. Differ. Equ. 49: 142–169 · Zbl 0523.35059
[22] Kanel’ Ja.I. (1964) Stabilization of the solutions of the equations of combustion theory with finite initial functions. Mat. Sb. (N.S.) 65((107): 398–413
[23] Lunardi A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Berlin · Zbl 0816.35001
[24] Mierczyński, J.: Flows on order bundles, unpublished · Zbl 0877.35052
[25] Mierczyński J. (1998) Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions. J. Math. Anal. Appl. 226: 326–347 · Zbl 0921.35064
[26] Mierczyński, J., Shen, W.: Spectral theory for random and nonautonomous parabolic equations and applications. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 139. CRC Press, Boca Raton, 2008 · Zbl 1387.35007
[27] Poláčik P., Tereščák I. (1993) Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations. J. Dyn. Differ. Equ. 5: 279–303 Erratum: 6(1), 245–246 (1994) · Zbl 0786.58002
[28] Poláčik P. (2005) Symmetry properties of positive solutions of parabolic equations on $${\(\backslash\)mathbb{R}\^N}$$ : I. Asymptotic symmetry for the Cauchy problem. Commun. Partial Differ. Equ. 30: 1567–1593 · Zbl 1083.35045
[29] Poláčik P. (2006) Symmetry properties of positive solutions of parabolic equations on $${\(\backslash\)mathbb{R}\^{N}}$$ : II. Entire solutions. Commun. Partial Differ. Equ. 31: 1615–1638 · Zbl 1128.35051
[30] Poláčik P., Quittner P. (2006) A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation. Nonlinear Analysis, TMA 64: 1679–1689 · Zbl 1092.35045
[31] Poláčik P., Quittner P., Souplet Ph. (2007) Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II. Parabolic equations. Indiana Univ. Math. J. 56: 879–908 · Zbl 1122.35051
[32] Quittner, P., Souplet, Ph.: Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts. Birkhäuser, Basel, 2007 · Zbl 1128.35003
[33] Zlatoš A. (2006) Sharp transition between extinction and propagation of reaction. J. Am. Math. Soc. 19: 251–263 · Zbl 1081.35011
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.