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Very slow grow-up of solutions of a semi-linear parabolic equation

Part of: Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  30 March 2011

Marek Fila ,
John R. King ,
Michael Winkler  and
Eiji Yanagida
Marek Fila
Affiliation:
Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava, Slovakia (fila@fmph.uniba.sk)
John R. King
Affiliation:
Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, UK (etzjrk@maths.nottingham.ac.uk)
Michael Winkler
Affiliation:
Fachbereich Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany (michael.winkler@uni-due.de)
Eiji Yanagida
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan (yanagida@math.tohoku.ac.jp)
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Abstract

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We consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.

Keywords

grow-upsemi-linear parabolic equationcomparison principle

MSC classification

Secondary: 35K57: Reaction-diffusion equations 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 381 - 400
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Fila, M., King, J. R., Winkler, M. and Yanagida, E., Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Diff. Eqns 228 (2006), 339–356.CrossRefGoogle Scholar
2.Fila, M., King, J. R., Winkler, M. and Yanagida, E., Grow-up of solutions of a semilinear parabolic equation with a critical exponent, Adv. Diff. Eqns 12 (2007), 1–26.Google Scholar
3.Fila, M., King, J. R., Winkler, M. and Yanagida, E., Linear behaviour of solutions of a super-linear heat equation, J. Math. Analysis Applic. 340 (2008), 401–409.CrossRefGoogle Scholar
4.Fila, M. and Winkler, M., Rate of convergence to a singular steady state of a supercritical parabolic equation, J. Evol. Eqns 8 (2008), 673–692.CrossRefGoogle Scholar
5.Fila, M., Winkler, M. and Yanagida, E., Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Diff. Eqns 205 (2004), 365–389.CrossRefGoogle Scholar
6.Fila, M., Winkler, M. and Yanagida, E., Convergence rate for a parabolic equation with supercritical nonlinearity, J. Dynam. Diff. Eqns 17 (2005), 249–269.CrossRefGoogle Scholar
7.Fila, M., Winkler, M. and Yanagida, E., Slow convergence to zero for a parabolic equation with supercritical nonlinearity, Math. Annalen 340 (2008), 477–496.CrossRefGoogle Scholar
8.Fila, M., Winkler, M. and Yanagida, E., Convergence to self-similar solutions for a semilinear parabolic equation, Discrete Contin. Dynam. Syst. 21 (2008), 703–716.CrossRefGoogle Scholar
9.Galaktionov, V. and Vázquez, J. L., Continuation of blow-up solutions of nonlinearheat equations in several space dimensions, Commun. Pure Appl. Math. 50 (1997), 1–67.3.0.CO;2-H>CrossRefGoogle Scholar
10.Gui, C., Ni, W.-M. and Wang, X., On the stability and instability of positive steady states of a semilinear heat equation in ℝn, Commun. Pure Appl. Math. 45 (1992), 1153–1181.CrossRefGoogle Scholar
11.Gui, C., Ni, W.-M. and Wang, X., Further study on a nonlinear heat equation, J. Diff. Eqns 169 (2001), 588–613.CrossRefGoogle Scholar
12.Hoshino, M. and Yanagida, E., Sharp estimate of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity, Nonlin. Analysis TMA 69 (2008), 3136–3152.CrossRefGoogle Scholar
13.Joseph, D. D. and Lundgren, T. S., Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Analysis 49 (1973), 241–269.CrossRefGoogle Scholar
14.Mizoguchi, N., Growup of solutions for a semilinear heat equation with supercritical nonlinearity, J. Diff. Eqns 227 (2006), 652–669.CrossRefGoogle Scholar
15.Poláčik, P. and Yanagida, E., On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Annalen 327 (2003), 745–771.Google Scholar
16.Poláčik, P. and Yanagida, E., Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation, Diff. Integ. Eqns 17 (2004), 535–548.Google Scholar
17.Quittner, P. and Souplet, Ph., Superlinear parabolic problems: blow-up, global existenceand steady states, Birkhäuser Advanced Texts (Birkhäuser, Basel, 2007).Google Scholar
18.Stinner, C., Very slow convergence to zero for a supercritical semilinear parabolic equation, Adv. Diff. Eqns 14 (2009), 1085–1106.Google Scholar
19.Stinner, C., Very slow convergence rates in a semilinear parabolic equation, NoDEA. Nonlin. Diff. Eqns Applic. 17 (2010), 213–227.CrossRefGoogle Scholar
20.Wang, X., On the Cauchy problem for reaction-diffusion equations, Trans. Am. Math. Soc. 337 (1993), 549–590.CrossRefGoogle Scholar