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Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

Rodríguez Bernal, Aníbal and Cholewa, Jan W. (2010) Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations. Journal of Differential Equations, 249 (3). 485-525 . ISSN 0022-0396

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Abstract

In this well-written paper, the authors consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. \par In the first part of the paper, some notions concerning dissipative systems in ordered space are recalled. Then follow results on the existence of extremal solutions and global attractors and finally on the inclusion of the global attractor in an order interval formed by the minimal and the maximal equilibria. \par In the second part of the paper, they then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, they exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in $\Bbb R^N$ with nonlinearities depending on the gradient of the solution. \par The authors consider as well systems of reaction-diffusion equations in $\Bbb R^N$ and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in $\Bbb R^N$. They further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.


Item Type:Article
Uncontrolled Keywords:Monotone semigroups; Equilibria; Asymptotic behavior of solutions; Dissipativeness; Attractors; Parabolic equations and systems; Degenerate problems; Damped wave equations; Strongly damped wave equations; Retarded functional differential equations; Dissipativity; Ordered phase space
Subjects:Sciences > Mathematics > Differential equations
ID Code:12925
Deposited On:11 Jul 2011 08:07
Last Modified:06 Feb 2014 09:36

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