The attractor for a nonlinear reaction‐diffusion system in an unbounded domain

Abstract

In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered.

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

“…We now consider the case of general weights satisfying (5.28). We first note that the continuity (5.43) for the special weights together with estimate (5.45) and the fact that s θ 2 (s) ds < ∞ imply in a standard way the continuity of u(t) in the space L 2 θ ( ): see [10] and Proposition 2.16. Thus, (5.31) is verified for general weights as well.…”

“…This inequality is crucial for obtaining the regularity estimates in weighted spaces (see [9][10][27][28][29][30] and Section 3 below).…”

“…Since we actually need below only the case where := ‫ޒ‬ × (−1, 1) is a strip which obviously has regular boundary, in order to avoid the technicalities we do not formulate precise assumptions on the boundary ∂ (which can be found e.g. in [9] or [10]). DEFINITION 2.3.…”

“…supremum in (2.9)) in (2.8) should be naturally replaced by the integral (resp. supremum) over the boundary ∂ , see [9], [10]. REMARK 2.4.…”

“…We restrict ourselves to verify a priori estimate (3.2) only (the existence and uniqueness of a solution can be then verified in a standard way, see e.g. [9], [10]). …”