Publication: Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains
Loading...
Full text at PDC
Publication Date
2004
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
In this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an-unbounded domain. The functional setting is based on standard Lebesgue and Sobolev-Lebesgue spaces. We show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor. In particular cases, our results allow us to improve some previous known results.
Description
UCM subjects
Unesco subjects
Keywords
Citation
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990) 85–108.
H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985) 47–83.
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.
H. Amann, Linear and Quasilinear Parabolic Problems, Abstract Linear Theory, Vol. 1, Birkhäuser, Basel, 1995.
H. Amann, M. Hieber, G. Simonett, Bounded H ∞ -calculus for elliptic operators, Differential Integral Equations 3 (1994) 613–653.
W. Arendt, C.J.K. Batty, Exponential stability of a diffusion equation with absorption, Differential Integral Equations 6 (1993) 1009–1024.
W. Arendt, C.J.K. Batty, Absorption semigroups and Dirichlet boundary conditions, Math. Ann. 295 (1993) 427–448.
J.M. Arrieta, A.N. Carvalho, A. Rodriguez–Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999) 376–406.
J.M. Arrieta, A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations, Trans. Amer. Math. Soc. 352 (2000) 285–310.
J.M. Arrieta, A.N. Carvalho, A. Rodriguez–Bernal, Attractors for parabolic problems with nonlinear boundary conditions: uniform bounds, Comm. Partial Differential Equations 25 (2000) 1–37.
A.V. Babin, M.I. Vishik, Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh 116A (1990) 221–243.
A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1991.
J.M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Am. Math. Soc. 63 (1977) 370–373.
P. Baras, J. Goldstein, The heat equation with a singular potential, Trans. AMS 284 (1984) 121–139.
H. Brezis, T. Cazenáve, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996) 277–304.
H. Brezis, T. Kato, Remarks on the Schrödinger operators with singular complex potentials, J. Math. Pures Appl. 58 (1979) 137–151.
F.E. Browder, Estimates and existence theorems for elliptic boundary value problems, Proc. NAS 45 (1959) 365–375.
X. Cabré, Y. Martel, Existence versus explosion instantaneé pour des équations de la chaleur lineaires avec potential singulier, C. R. Acad. Sci. 329 (11) (1999) 973–978.
J.W. Cholewa T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Notes Series, Vol. 278, Cambridge University Press, Cambridge, 2000.
D. Daners, S. Merino, Gradient-like parabolic flow in BUC(R N ) , Proc. Roy. Soc. Edinburgh 128A (1998) 1281–1291.
E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
M.A. Efendiev, S.V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Commun. Pure Appl. Math. 54 (2001) 625–688.
J. Escher, B. Scarpellini, On the asymptotics of solutions of parabolic equations on unbounded domains, Nonlinear Anal. 33 (1998) 483–507.
E. Feireisl, Ph. Laurencot, F. Simondon, H. Toure, Compact attractors for reaction-diffusion equations in R n , C. R. Acad. Sci. Paris Ser. I 319 (1994) 147–151.
E. Feireisl, Ph. Laurencot, F. Simondon, Global attractors for degenerate parabolic equations on unbounded domains, J. Differential Equations 129 (1996) 239–261.
A. Friedman, B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985) 425–447.
H. Fujita, On the blowing up of solutions of the Cauchy problem for u t =Δu+u 1+α , J. Fac. Sci. Univ. Tokyo Sect. A Math. 16 (1966) 109–124.
J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988.
R. Hempel, J. Voigt, The spectrum of a Schrödinger operator in L p (R N ) is p -independent, Commun. Math. Phys. 104 (1986) 243–250.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.
M. Hieber, P. Koch Medina, S. Merino, Linear and semilinear parabolic equation on BUC(R N ) , Math. Narch. 179 (1996) 107–118.
M. Hieber, P. Koch Medina, S. Merino, Diffusive logistic growth on R N , Nonlinear Anal. TMA 27 (1996) 879–894.
P. Koch Medina, G. Schätti, Long-time behavior for reaction-diffusion equations on R N , Nonlinear Anal. TMA 25 (1995) 831–870.
O.A. Ladyženskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal. 20 (1989) 816–844.
S. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on R N , J. Differential Equations 132 (1996) 87–106.
A. Mielke, The complex Ginzburg–Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity 10 (1997) 199–222.
A. Mielke, G. Schneider, Attractors for modulation equations on unbounded domains—existence and comparison, Nonlinearity 8 (1995) 743–768.
N. Mizoguchi, E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation, Math. Ann. 307 (1997) 663–675.
X. Mora, Semilinear parabolic problems define semiflows on C k spaces, Trans. AMS 278 (1983) 21–55.
M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Springer, Berlin, 1999.
A. Rodriguez-Bernal, B. Wang, Attractors for partly dissipative reaction diffusion systems in R N , J. Math. Anal. Appl. 252 (2000) 790–803.
A. Rodriguez-Bernal, E. Zuazua, Parabolic singular limit of a wave equation with localized boundary damping, Discr. Cont. Dyn. Systems 1 (1995) 303–346.
B. Simon, Schrödinger Semigroups, Bull. AMS. 7 (1982) 447–526.
P. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Comm. Partial Differential Equations 24 (1999) 951–973.
P. Souplet, Decay of heat semigroups in L ∞ and applications to nonlinear parabolic problems in unbounded domains, J. Funct. Anal. 173 (2000) 343–360.
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition, Springer, Berlin, 1997.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978.
J.L. Vázquez, E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000) 103–153.
B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D 128 (1999) 41–52.
F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p , Indiana Univ. Math. J. 29 (1980) 79–102.