Vol.16 No.4

EXPONENTIAL ATTRACTOR FOR A CLASS OF NONCLASSICAL DIFFUSION EQUATION*

Shang Yadong

( Department of Mathematics, Guangzhou University, Guangzhou 510405 China) (E-mail: ydshang@263.net)

Guo Boling

(Institute of Applied Physics and Computational Mathematics, P.O.Box 8009, Beijing 100088, China)(E-mail: gbl@mail.iapem.ac.cn) (Received Jan. 14, 2002; revised Mar. 25, 2003)

Abstract In this paper, we consider the asymptotic behavior of solutions for a class of nonclassical diﬀusion equation. We show the squeezing property and the existence of exponential attractor for this equation. We also make the estimates on its fractal dimension and exponential attraction. Key Words Nonclassical;diﬀusion equation; squeezing property; exponential attractor 2000 MR Subject Classiﬁcation 35Q55. Chinese Library Classiﬁcation O175.29.

1. Introduction

The nonclassical diﬀusion equations

n

ut − ν∆ut −

i=1

[σ(uxi )]xi + g(u) = f (x, t)

(1.1)

arise in many diﬀerent areas of mathematics and physics. They have been used, for instance, to model thermodynamics processes[1], [2], ﬂuid ﬂow inﬁssured rock [3],consolidation of clay [4], and shear in second order ﬂuids [5-7]. For the physical interpretation of ”ν∆ut ”, we refer to [1-3]. The equations of (1.1) with a one time derivative appearing in the highest order term are called pseudo-parabolic or Sobolev-Galpern equations [8-12]. Aifantis [13] proposed a general frame for establishing the equations. The existence and uniqueness, andregularity of solutions for the nonclassical diﬀusion equations have been investigated by many authors, such as Showalter [14], Davis [15],

*This work was supported by the National Natural Science Foundation of China (Grant No.10271034)

290

Shang Yadong and Guo Boling

Vol.16

Quarteroni [16], Karch [17], Shi et al [18], Liu and Wang [19], Liu,Wan and Lu [20], Li et.al [21] and theirreferences therein. In this paper,we consider the following initial boundary value problem of the nonclassical diﬀusion equation: ut − ν∆ut − λ∆u + g(u) = f (x), (x, t) ∈ Ω × R+ , u(x, 0) = u0 (x), u(x, t) = 0, x ∈ Ω, (x, t) ∈ ∂Ω × R .

+

(1.2) (1.3) (1.4)

where λ is a positive constant, g : R → R is a smooth function, and f (x) ∈ L2 (Ω), Ω ⊂ Rn is a smooth bounded open set, ∂Ω is theboundary of Ω. The existence of the 1 compact global attractor in H0 (Ω) for the equation (1.2) has been established by Li et al [21]. Our aim of this paper is to show the existence of ﬁnite dimensional exponential attractor for this equation. The outline of this paper is as follows: in Section 2, we state some basic results on the existence of exponential attractors and recall some known resultsconcerning the existence and uniqueness of solutions. Section 3 contains our main results; we ﬁrst establish the Lipschitz continuity of the dynamical system S(t) associated with Eq.(1.2), then we prove that the semigroup S(t) satisﬁes the squeezing property and deduce the existence of the exponential attractor. Throughout this paper, we denote by · the norm of H = L2 (Ω) with the usual inner product (·,·). We also use · p for the norm of Lp (Ω) for 1 ≤ p ≤ ∞( · 2 = · ). Generally, · X denotes the norm of Banach space X. For convenience, we put ν ≡ 1 in (1.2). In the sequel, we always assume that g satisﬁes conditions: g(s) ≥ µ; (G1 )g ∈ C 1 (R), ∃µ ∈ R, such that lim |s|→∞ s (G2 )∃c, γ ≥ 0, such that g (s) ≤ c(1 + |s|γ ), where 0 ≤ γ < ∞( as n = 1, 2), γ ≤ 2n (as n ≥ 3). n−2

2. PreliminariesLet D(A), V be two Hilbert spaces, D(A) be dense in V and compactly imbedded into V. We study du + Au + g(u) = f (x), dt u(0) = u0 , x ∈ Ω u |∂Ω = 0. t > 0, x ∈ Ω. (2.1) (2.2) (2.3)

No.4

Exponential attractor for a class of nonclassical diﬀusion equation

291

where Ω is a bounded open set in Rn , ∂Ω is smooth. A is a positive self adjoint operator with a compact inverse. Let...