Attractors for reaction-diffusion equations in unbounded domains

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

“…When sðv 2 Þ ¼ 1 and p ¼ 2 this fact is proved in [18] by use of an abstract theory of semi-group generated by evolution equation. Here we give a direct proof for a general case m b 0 and p b 2, though we utilize a smoothing e¤ect in L pþm .…”

“…For the case m ¼ 0 the existence of L 2 global attractor for the problem is proved by Wang in [18] under appropriate assumptions on g and f . 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.…”

“…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 FOR SEMILINEAR DEGENERATE PARABOLIC EQUATIONS ON ℝN

“…As it turns out, such a condition can be found with careful tails estimates as in [18] (see the proof of Lemma 3.2 below). More precisely, we assume that the function σ : R N → R satisfies the following assumption:…”

Upper Semicontinuity of Pullback Attractors for Non-Autonomous Generalized 2d Parabolic Equations

“…They used the technique of uniform estimates on the tails of solutions to prove the asymptotic compactness of the solution operator. This technique was developed by Wang [12] to investigate the behavior of reaction-diffusion equations in unbounded domains.…”