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Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains. (English) Zbl 1058.35102
The authors investigate conditions for a reaction diffusion equation to be dissipative in an unbounded domain. They show how the combination of the reaction mechanism with the diffusion allows to obtain a global compact attractor. Several generalizations of the previous results to almost monotonic nonlinearities and other cases of initial data are obtained.

MSC:
35K57 Reaction-diffusion equations
35B41 Attractors
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[1] Abergel, F., Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. differential equations, 83, 85-108, (1990) · Zbl 0706.35058
[2] Amann, H., Global existence for semilinear parabolic systems, J. reine angew. math., 360, 47-83, (1985) · Zbl 0564.35060
[3] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Leipzig, 1993, pp. 9-126. · Zbl 0810.35037
[4] H. Amann, Linear and Quasilinear Parabolic Problems, Abstract Linear Theory, Vol. 1, Birkhäuser, Basel, 1995. · Zbl 0819.35001
[5] Amann, H.; Hieber, M.; Simonett, G., Bounded H∞-calculus for elliptic operators, Differential integral equations, 3, 613-653, (1994) · Zbl 0799.35060
[6] Arendt, W.; Batty, C.J.K., Exponential stability of a diffusion equation with absorption, Differential integral equations, 6, 1009-1024, (1993) · Zbl 0817.35007
[7] Arendt, W.; Batty, C.J.K., Absorption semigroups and Dirichlet boundary conditions, Math. ann., 295, 427-448, (1993) · Zbl 0788.47031
[8] Arrieta, J.M.; Carvalho, A.N.; Rodriguez-Bernal, A., Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. differential equations, 156, 376-406, (1999) · Zbl 0938.35077
[9] Arrieta, J.M.; Carvalho, A.N., Abstract parabolic problems with critical nonlinearities and applications to navier – stokes and heat equations, Trans. amer. math. soc., 352, 285-310, (2000) · Zbl 0940.35119
[10] Arrieta, J.M.; Carvalho, A.N.; Rodriguez-Bernal, A., Attractors for parabolic problems with nonlinear boundary conditionsuniform bounds, Comm. partial differential equations, 25, 1-37, (2000) · Zbl 0953.35021
[11] Babin, A.V.; Vishik, M.I., Attractors of partial differential evolution equations in unbounded domain, Proc. roy. soc. Edinburgh, 116A, 221-243, (1990) · Zbl 0721.35029
[12] Babin, A.V.; Vishik, M.I., Attractors of evolution equations, (1991), North-Holland Amsterdam · Zbl 0765.35023
[13] Ball, J.M., Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. am. math. soc., 63, 370-373, (1977) · Zbl 0353.47017
[14] Baras, P.; Goldstein, J., The heat equation with a singular potential, Trans. AMS, 284, 121-139, (1984) · Zbl 0556.35063
[15] Brezis, H.; Cazenáve, T., A nonlinear heat equation with singular initial data, J. anal. math., 68, 277-304, (1996) · Zbl 0868.35058
[16] Brezis, H.; Kato, T., Remarks on the Schrödinger operators with singular complex potentials, J. math. pures appl., 58, 137-151, (1979) · Zbl 0408.35025
[17] Browder, F.E., Estimates and existence theorems for elliptic boundary value problems, Proc. NAS, 45, 365-375, (1959) · Zbl 0093.29402
[18] Cabré, X.; Martel, Y., Existence versus explosion instantaneé pour des équations de la chaleur lineaires avec potential singulier, C. R. acad. sci., 329, 11, 973-978, (1999) · Zbl 0940.35105
[19] J.W. Cholewa T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Notes Series, Vol. 278, Cambridge University Press, Cambridge, 2000. · Zbl 0954.35002
[20] Daners, D.; Merino, S., Gradient-like parabolic flow in \(BUC(R\^{}\{N\})\), Proc. roy. soc. Edinburgh, 128A, 1281-1291, (1998) · Zbl 0920.35072
[21] Davies, E.B., Heat kernels and spectral theory, (1989), Cambridge University Press Cambridge · Zbl 0699.35006
[22] Efendiev, M.A.; Zelik, S.V., The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Commun. pure appl. math., 54, 625-688, (2001) · Zbl 1041.35016
[23] Escher, J.; Scarpellini, B., On the asymptotics of solutions of parabolic equations on unbounded domains, Nonlinear anal., 33, 483-507, (1998) · Zbl 0933.35100
[24] Feireisl, E.; Laurencot, Ph.; Simondon, F.; Toure, H., Compact attractors for reaction-diffusion equations in Rn, C. R. acad. sci. Paris ser. I, 319, 147-151, (1994) · Zbl 0806.35075
[25] Feireisl, E.; Laurencot, Ph.; Simondon, F., Global attractors for degenerate parabolic equations on unbounded domains, J. differential equations, 129, 239-261, (1996) · Zbl 0862.35058
[26] Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana univ. math. J., 34, 425-447, (1985) · Zbl 0576.35068
[27] Fujita, H., On the blowing up of solutions of the Cauchy problem for ut = δ u+ u1+α, J. fac. sci. univ. Tokyo sect. A math., 16, 109-124, (1966) · Zbl 0163.34002
[28] Hale, J.K., Asymptotic behavior of dissipative systems, (1988), AMS Providence, RI · Zbl 0642.58013
[29] Hempel, R.; Voigt, J., The spectrum of a Schrödinger operator in \(Lp(R\^{}\{N\})\) is p-independent, Commun. math. phys., 104, 243-250, (1986) · Zbl 0593.35033
[30] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer Berlin · Zbl 0456.35001
[31] Hieber, M.; Koch Medina, P.; Merino, S., Linear and semilinear parabolic equation on \(BUC(R\^{}\{N\})\), Math. narch., 179, 107-118, (1996) · Zbl 0863.35044
[32] Hieber, M.; Koch Medina, P.; Merino, S., Diffusive logistic growth on \(R\^{}\{N\}\), Nonlinear anal. TMA, 27, 879-894, (1996) · Zbl 0858.35055
[33] Koch Medina, P.; Schätti, G., Long-time behavior for reaction-diffusion equations on \(R\^{}\{N\}\), Nonlinear anal. TMA, 25, 831-870, (1995) · Zbl 0847.35065
[34] Ladyženskaya, O.A., Attractors for semigroups and evolution equations, (1991), Cambridge University Press Cambridge
[35] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Basel · Zbl 0816.35001
[36] Marion, M., Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. math. anal., 20, 816-844, (1989) · Zbl 0684.35055
[37] Merino, S., On the existence of the compact global attractor for semilinear reaction diffusion systems on RN, J. differential equations, 132, 87-106, (1996) · Zbl 0867.35045
[38] Mielke, A., The complex ginzburg – landau equation on large and unbounded domainssharper bounds and attractors, Nonlinearity, 10, 199-222, (1997) · Zbl 0905.35043
[39] Mielke, A.; Schneider, G., Attractors for modulation equations on unbounded domains—existence and comparison, Nonlinearity, 8, 743-768, (1995) · Zbl 0833.35016
[40] Mizoguchi, N.; Yanagida, E., Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation, Math. ann., 307, 663-675, (1997) · Zbl 0872.35046
[41] Mora, X., Semilinear parabolic problems define semiflows on Ck spaces, Trans. AMS, 278, 21-55, (1983) · Zbl 0525.35044
[42] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1999), Springer Berlin · Zbl 0153.13602
[43] Rodriguez-Bernal, A.; Wang, B., Attractors for partly dissipative reaction diffusion systems in \(R\^{}\{N\}\), J. math. anal. appl., 252, 790-803, (2000) · Zbl 0977.35028
[44] Rodriguez-Bernal, A.; Zuazua, E., Parabolic singular limit of a wave equation with localized boundary damping, Discr. cont. dyn. systems, 1, 303-346, (1995) · Zbl 0891.35075
[45] Simon, B., Schrödinger semigroups, Bull. AMS., 7, 447-526, (1982) · Zbl 0524.35002
[46] Souplet, P., Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Comm. partial differential equations, 24, 951-973, (1999) · Zbl 0926.35064
[47] Souplet, P., Decay of heat semigroups in L∞ and applications to nonlinear parabolic problems in unbounded domains, J. funct. anal., 173, 343-360, (2000) · Zbl 0954.35032
[48] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, 2nd edition, (1997), Springer Berlin · Zbl 0871.35001
[49] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), Veb Deutscher Berlin · Zbl 0387.46032
[50] Vázquez, J.L.; Zuazua, E., The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. funct. anal., 173, 103-153, (2000) · Zbl 0953.35053
[51] Wang, B., Attractors for reaction diffusion equations in unbounded domains, Physica D, 128, 41-52, (1999) · Zbl 0953.35022
[52] Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations in Lp, Indiana univ. math. J., 29, 79-102, (1980) · Zbl 0443.35034
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