Attractors for parabolic problems in weighted spaces

The purpose of this paper is to investigate the asymptotic behavior of the solutions of parabolic equations with singular initial data in weighted spaces Lrd(x)(W) where d(x) is the distance to the boundary. We first establish the existence of the attractor for that equation in Lrd(x)(W) and then show the existence of the exponential attractor in L2d(x)(W). In contrast to our previous results, we get the existence of attractors in weak topology spaces.

Anahtar Kelimeler:

Attractor, parabolic equation, singular initial data, weighted space

Attractors for parabolic problems in weighted spaces

Keywords:

Attractor, parabolic equation, singular initial data, weighted space,

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Arrieta, J.M., Carvalho, A.N.: Abstract parabolic problems with critical nonlinearities and applications to NavierStokes and heat equations. Trans. Amer. Math. Soc. 352, 285–310 (1999).
Arrieta, J.M., Carvalho, A.N., Rodr´ıguez-Bernal, A.: Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds. Commun. Partial Differential Equations 25, 1–37 (2000).
Babin, A., Nicolaenko, B.: Exponential attractors of reaction-diffusion systems in unbounded domains. J. Dynam. Differential Equations 7, 567–590 (1995).
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Amsterdam. North-Holland 1992.
Babin, A.V., Vishik, M.I.: Attractors of partial differential evolution equations in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A 116, 221–243 (1990).
Br´ ezis, H., Cazenave, T.: A nonlinear heat equation with singular initial data. J. Anal. Math. 68, 277–304 (1996). Eden, A., Foias, C., Kalantarov, V.: A remark on two constructions of exponential attractors for α -contractions. J. Dynam. Differential Equations 10, 37–45 (1998).
Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Ensembles inertiels pour des ´ equations d’´ evolution dissipatives. C. R. Acad. Sci. Paris, 310, S´ erie I, 559–562 (1990).
Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential attractors for dissipative evolution equations. Res. Appl. Math., Vol. 37. Paris. Masson - John Wiley 1994.
Fila, M., Souplet, P., Weissler, F.B.: Linear and nonlinear heat equations in L q δ spaces and universal bounds for global solutions. Math. Ann. 320, 87–113 (2001).
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Providence. American Mathematical Society 1988.
Li, X.: Non-autonomous parabolic problems with singular initial data in weighted spaces. Rocky Mountain J. Math. 42, 1215–1245 (2012).
Li, X. Ruan, S.: Attractors for non-autonomous parabolic problems with singular initial data. J. Differential Equations 251, 728–757 (2011).
Loayza, M.: The heat equation with singular nonlinearity and singular initial data. J. Differential Equations 229, 509–528 (2006).
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd ed. New York. Springer 1997. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam. North-Holland 1978.
Wang, B.: Attractors for reaction-diffusion equations in unbounded domains. Phys. D 128, 41–52 (1999).
Yosida, K.: Functional Analysis. New York. Springer 1965.