Random dynamics of non-autonomous fractional stochastic p -Laplacian equations on $${\mathbb {R}}^N$$
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00107-5 ORIGINAL PAPER
Random dynamics of non‑autonomous fractional stochastic p‑Laplacian equations on ℝN Renhai Wang1 · Bixiang Wang2 Received: 28 May 2020 / Accepted: 21 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract This article is concerned with the random dynamics of a wide class of nonautonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise on the entire space ℝN . We first establish the wellposedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders p, q ≥ 2 . We then prove that the equation has a unique bi-spatial pullback random attractor that is measurable, compact in L2 (ℝN ) ∩ Lp (ℝN ) ∩ Lq (ℝN ) and attracts all random subsets of L2 (ℝN ) under the topology of L2 (ℝN ) ∩ Lp (ℝN ) ∩ Lq (ℝN ) . In addition, we establish the upper semicontinuity of these attractors in L2 (ℝN ) ∩ Lp (ℝN ) ∩ Lq (ℝN ) when the density of noise shrinks to zero. The idea of uniform tail estimates and the method of asymptotic a priori estimates are applied to prove the pullback asymptotic compactness of the solutions in L2 (ℝN ) ∩ Lp (ℝN ) ∩ Lq (ℝN ) to overcome the non-compactness of Sobolev embeddings on ℝN as well as the almost sure nondifferentiability of the sample paths of the Wiener process. Keywords Fractional p-Laplacian · Stochastic p-Laplacian equation · Pullback random attractor · Upper semi-continuity · Unbounded domain Mathematics Subject Classification 35B40 · 35B41 · 37L30
Communicated by Ti-Jun Xiao. * Renhai Wang rwang‑[email protected] Bixiang Wang [email protected] 1
Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China
2
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA
Vol.:(0123456789)
R. Wang and B. Wang
1 Introduction This paper is concerned with the dynamics of the following non-autonomous, nonlocal, fractional, stochastic p-Laplacian equation driven by multiplicative white noise defined on the entire space ℝN :
ut + 𝜆u + (−𝛥)sp u = f1 (t, x, u) + f2 (t, x, u) + g(t, x) + 𝛼u◦
dW , x ∈ ℝN , t > 𝜏, dt
(1)
with initial condition:
u(𝜏, x) = u𝜏 (x), x ∈ ℝN ,
(2)
∞ where 𝜏 ∈ ℝ , 𝜆, 𝛼 > 0 , g ∈ Lloc (ℝ, L2 (ℝN )) , (−𝛥)sp with 0 < s < 1 and 2 ≤ p < ∞ is the non-linear, non-local, fractional p-Laplace operator, and W is a two-sided real-valued Wiener process on a probability space. The symbol ◦ indicates that the stochastic equation is interpreted in view of Stratonovich integration. The timedependent non-linear drift terms f1 , f2 ∶ ℝ × ℝN × ℝ → ℝ have polynomial growth of orders p ≥ 2 and q ≥ 2 , respectively. Fractional differential equations have wide applications in physics, chemistry, finance, biology and other fields of science; see, e.g., [21, 69] and the reference therein. The existence, regularity and upper semi-continuity of random attractors of stochastic differential equations with linear fractional Laplacia
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