Publication: Dissipative mechanism of a semilinear higher order parabolic equation in R-N
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It is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems. In this paper we investigate the dissipative mechanism for some semilinear fourth-order parabolic equations in the spaces of Bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor. While for second-order reaction-diffusion equations the dissipativeness mechanism has already been satisfactorily understood (see Arrieta et al. (2004), doi:10.1142/S0218202504003234), for higher order problems in unbounded domains it has not yet been fully developed. As shown throughout the paper, one of the main differences from the case of reaction-diffusion equations stems from the lack of a maximum principle. Thus we have to rely here on suitable energy estimates for the solutions. As in the case of second-order reaction-diffusion equations, we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness. Thus, the dissipative mechanisms in second-order and fourth-order equations are similar, although the lack of a maximum principle makes the proofs more difficult and the results not as complete. Finally, we make essential use of the sharp results of Cholewa and Rodriguez-Bernal (2012), doi:10.1016/j.na.2011.08.022, on linear fourth-order equations with a very large class of linear potentials.
A.V. Babin, M.I. Vishik, Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh 116A (1990) 221–243. E. Feireisl, Ph. Laurencot, F. Simondon, H. Toure, Compact attractors for reaction–diffusion equations in Rn, C. R. Acad. Sci. Paris Ser. I 319 (1994) 147–151. S. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on RN , J. Differential Equations 132 (1996) 87–106. B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D 128 (1999) 41–52. A. Rodriguez-Bernal, B. Wang, Attractors for partly dissipative reaction diffusion systems in RN , J. Math. Anal. Appl. 252 (2000) 790–803. M.A. Efendiev, S.V. Zelik, The attractor for a nonlinear reaction–diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001) 625–688. J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal. TMA 56 (2004) 515–554. J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodriguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr. 280 (2007) 1643–1663. A. Rodriguez-Bernal, A. Vidal-López, Semistable extremal ground states for nonlinear evolution equations in unbounded domains, J. Math. Anal. Appl. 338 (2008) 675–694. J.W. Cholewa, A. Rodriguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci. 19 (2009) 1995–2037. J.W. Cholewa, A. Rodriguez-Bernal, Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations, J. Differential Equations 249 (2010) 485–525. J.W. Cholewa, A. Rodriguez-Bernal, Linear and semilinear higher order parabolic equations in RN , Nonlinear Anal. TMA 75 (2012) 194–210, doi:10.1016/j.na.2011.08.022. J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodriguez-Bernal, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci. 14 (2004) 253–294. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975) 181–205. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978. H. Amann, M. Hieber, G. Simonnett, Bounded H∞ calculus for elliptic operators, Differential Integral Equations 7 (1994) 613–653. D. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981. A.N. Carvalho, J.W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl. 310 (2005) 557–578. A.N. Carvalho, J.W. Cholewa, Strongly damped wave equations in W1,p 0 (Ω) × Lp(Ω), Discrete Contin. Dyn. Syst. (Supplement) (2007) 230–239. J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. T. Dlotko, C. Sun, Dynamics of the modified viscous Cahn–Hilliard equation in RN , Topol. Methods Nonlinear Anal. 35 (2010) 277–294. J.W. Cholewa, T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000. J.M. Arrieta, A. Rodríguez-Bernal, Non well posedness of parabolic equations with supercritical nonlinearities, Comm. Contemp. Math. 6 (2004) 733-764.