Premium
Two types of upper semi‐continuity of bi‐spatial attractors for non‐autonomous stochastic p ‐Laplacian equations on R n
Author(s) -
Yin Jinyan,
Li Yangrong
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4353
Subject(s) - mathematics , attractor , p laplacian , laplace operator , mathematical analysis , type (biology) , pure mathematics , boundary value problem , ecology , biology
We consider the long time behavior of solutions for the non‐autonomous stochastic p ‐Laplacian equation with additive noise on an unbounded domain. First, we show the existence of a unique ( L 2 , L 2 ∩ L q ) ‐pullback attractor, where q is related to the order of the nonlinearity. The main difficulty existed here is to prove the asymptotic compactness of systems in both spaces, because the Laplacian operator is nonlinear and additive noise is considered. We overcome these obstacles by applying the compactness of solutions inside a ball, a truncation method and some new techniques of estimates involving the Laplacian operator. Next, we establish the upper semi‐continuity of attractors at any intensity of noise under the topology ofL 2 ∩ L q . Finally, we prove this continuity of attractors from domains in the norm ofL 2 ∩ L q , which improves an early result by Bates et al. (2001) who studied such continuity when the deterministic lattice equations were approached by finite‐dimensional systems, and also complements Li et al. (2015) who discussed this approximation when the nonlinearity f (·,0) had a compact support. Copyright © 2017 John Wiley & Sons, Ltd.