Abstract
In this chapter we study the existence and characterisation of pullback attractors for a non-autonomous version of the Chafee–Infante equation on the domain (0, π),
when there exist 0 < b 0 < B 0 such that
Theorem 12.1 guarantees the local existence and uniqueness of solutions for an initial u(s) ∈ H 0 1(0, π), and Proposition 12.8 ensures that these solutions are globally defined. Monotonicity properties of the solutions of this equation (from Corollary 12.6) will play an essential role in our analysis, and we will be able to prove the existence of maximal and minimal bounded global solutions, ξ m ( ⋅) and ξ M ( ⋅), which provide ‘bounds’ on the asymptotic dynamics of the system, i.e. any bounded global solution ψ( ⋅) satisfies
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Angenent S (1988) The zero set of a solution of a parabolic equation. J Reine Angew Math 390:79-96
Burton TA, Hutson V (1991) Permanence for nonautonomous predator-prey systems. Differential Integral Equations 4:1269–1280
Butler G, Freedman H, Waltman P (1986) Uniformly persistent dynamical systems. Proc Amer Math Soc 96:425–430
Cantrell RS, Cosner C (1996) Practical persistence in ecological models via comparison methods. Proc Roy Soc Edinburgh Sect A 126:247–272
Cantrell RS, Cosner C, Hutson V (2003) Spatial ecology via reaction-diffusion equations. Wiley series in mathematical and computational biology. Wiley, Chichester
Chafee N, Infante EF (1974) A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl Anal 4:17–37
Chepyzhov VV, Vishik MI (2002) Attractors for equations of mathematical physics. Colloquium Publications 49, American Mathematical Society, Providence, RI
Hale JK (1988) Asymptotic behavior of dissipative systems. Mathematical surveys and monographs, American Mathematival Society, Providence, RI
Henry D (1981a) Geometric theory of semilinear parabolic equations. Lecture notes in mathematics, vol 840. Springer, Berlin Heidelberg New York
Kostin IN (1995) Lower semicontinuity of a non-hyperbolic attractor. J Lond Math Soc 52:568–582
Langa JA, Robinson JC, Rodríguez-Bernal A, Suárez A (2009) Permanence and asymptotically stable complete trajectories for non-autonomous Lotka–Volterra models with diffusion. SIAM J Math Anal 40:2179–2216
Matano H (1982) Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J Fac Sci Univ Tokyo Sect IA Math 29:401–441
Pilyugin SY (1999) Shadowing in dynamical systems. Lecture notes in mathematics, vol 1706. Springer, Berlin Heidelberg New York
Raquepas JB, Dockery JD (1999) Dynamics of a reaction-diffusion equation with nonlocal inhibition. Phys D 134:94–110
Robinson JC (2001) Infinite-dimensional dynamical systems. Cambridge University Press, Cambridge
Rosa R (2003) Finite dimensional feedback control via inertial manifold theory with application to the Chafee–Infante equation. J Dynam Differential Equations 15:61–86
Sell GR, You Y (2002) Dynamics of evolutionary equations. Applied mathematical sciences, vol 143. Springer, Berlin Heidelberg New York
Wang B (2011) Almost periodic dynamics of perturbed infinite-dimensional dynamical systems. Nonlinear Anal 74:7252–7260
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Carvalho, A.N., Langa, J.A., Robinson, J.C. (2013). A non-autonomous Chafee–Infante equation. In: Attractors for infinite-dimensional non-autonomous dynamical systems. Applied Mathematical Sciences, vol 182. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4581-4_13
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4581-4_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4580-7
Online ISBN: 978-1-4614-4581-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)