Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain

Abstract

In this paper, we study the long-time behavior of solutions for the parabolic equation with nonlinear Laplacian principal part in R n . We prove the existence of a global (L 2 (R n ), L ∞ (R n ))-attractor when n p and the existence of a global (L 2 (R n ), L np/(n−p) (R n ))-attractor when n > p.  2005 Elsevier Inc. All rights reserved.

On Global Attractors for a Nonlinear Parabolic Equation of m-Laplacian Type in RN

“…Recently Khanmamedov [8] has discussed the existence of ðL 2 ; L pÃ Þ global attractor in a weak sense of the problem (1.1)-(1.2) with lu replaced by ljuj m u where p à is a certain special exponent. In these papers it is assumed that g u ðx; uÞ b ÀkðxÞ for an approprate function kðxÞ, which is essentially used to assure the uniqueness of solutions.…”

“…To ovetrcome the other one we need to prove the uniform smallness of the norm of kuðtÞk L p ðBðRÞ C Þ for large t and large R, where BðRÞ C is the complement of BðRÞ 1 fx A R N j jxj < Rg. In [8,18] this is proved for p ¼ 2 by a rather complicate argument using abstract semi-group theory. Here, we prove the fact for p b 2 by a direct cut-o¤ technique.…”

Long-time behavior of solutions of some doubly nonlinear equation

“…Note that, as a rule, the question of the existence of an attractor in articles known till now has been considered for equations with only one of the nonlinear term in the principal part of the equation. There is an extensive literature devoted to the existence of a global attractor for a parabolic equation with monotone principal part and also a porous medium-type equation (see [2,4,5,8,9,15,16,[26][27][28] and references therein). One of the latest works in this direction is [28], in which the authors have investigated the equation u t À div ru j j pÀ2 ru À Á þ gðuÞ À fðxÞ ¼ 0 under some conditions on g(s) (in particular, c 1 s j j q À k gðsÞs c 2 s j j q þk and g 0 (s) > Àl).…”

Global attractors for $p$-Laplacian differential inclusions in unbounded domains

“…Introduction. During the last ten years, many researchers have spent much effort in obtaining results on global attractors for p-Laplacian problems (see for example [1,7,12,15,16,20,24,25,37,41,42,46,48,49,50,56,57,58,59]). It is worth noting that p-laplacian equations have applications in a variety of phenomena, such as nonlinear elasticity, flows in porous media, non-Newtonian fluids and many others (see [35], [36], [38] and the references therein).…”

“…Such models appear for example when studying processes of combustion in porous media [19] or conduction of electrical impulses in nerve axons [53], [54]. On the other hand, many parabolic problems in unbounded domains have been studied over the last years [2,5,13,17,18,24,25,34,45].…”