Attractors for parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities

  • Nguyen Dinh Binh1Email author and

    Affiliated with

    • Cung The Anh2

      Affiliated with

      Boundary Value Problems20122012:35

      DOI: 10.1186/1687-2770-2012-35

      Received: 21 February 2011

      Accepted: 28 March 2012

      Published: 28 March 2012

      Abstract

      Using the theory of uniform global attractors for multi-valued semiprocesses, we prove the existence of attractors for quasilinear parabolic equations related to Caffarelli-Kohn- Nirenberg inequalities, in which the conditions imposed on the nonlinearity provide the global existence of weak solutions but not uniqueness, in both autonomous and non-autonomous cases.

      Mathematics Subject Classification 2010: 35B41, 35K65, 35D30.

      Keywords

      Caffarelli-Kohn-Nirenberg inequalities non-uniqueness weak solution multivalued semiflow multi-valued semiprocess compact attractor compactness and monotonicity methods

      1. Introduction

      The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to attack the problem for a dissipative dynamical system is to consider its attractor. The existence of the attractor has been derived for a large class of PDEs (see e.g., [1, 2] and references therein) for both autonomous and non-autonomous equations. However, these researches may not be applied to a wide class of problems, in which solutions may not be unique. Good examples of such systems are differential inclusions, variational inequalities, control infinite dimensional systems and also some partial differential equations for which solutions may not be known unique as, for example, some certain semilinear wave equations with high power nonlinearities, the incompressible Navier-Stokes equation in three space dimension, the Ginzburg-Landau equation, etc. For the qualitative analysis of the above mentioned systems from the point of view of the theory of dynamical systems, it is necessary to develop a corresponding theory for multi-valued semigroups.

      In the last years, there have been some theories for which one can treat multi-valued semi-flows and their asymptotic behavior, including the generalized semiflows theory of Ball [3], theory of trajectory attractors of Chepyzhov and Vishik [4] and theories of multi-valued semiflows and semiprocesses of Melnik and Valero [57]. Thanks to these theories, several results concerning attractors in the case of equations without uniqueness have been obtained recently for differential inclusion [5, 6], parabolic equations [810], the phase-field equation [11], the wave equation [12], the three-dimensional Navier-Stokes equation [3, 13], etc. Although the existence of attractors has been derived for many classes of partial differential equations without uniqueness, to the best of our knowledge, little seems to be known for singular/degenerate equations, expecially in the quasilinear case.

      Let Ω be a bounded domain in ℝ N (N ≥ 2) containing the origin with boundary Ω. In this paper we consider the following quasilinear parabolic equation
      u t - div x - p γ u p - 2 u + f ( t , u ) = g ( x , t ) , x Ω , t > τ , u | t = τ = u τ ( x ) , x Ω , u | Ω = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ1_HTML.gif
      (1.1)

      where τ ∈ ℝ, u τ L2(Ω) are given, the nonlinearity f, the external force g, and the numbers p, γ satisfy the following conditions:

      (H1) f: ℝ × ℝ → ℝ is a continuous function satisfying
      f ( t , u ) C 1 u q - 1 + k 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ2_HTML.gif
      (1.2)
      u f ( t , u ) C 2 u q - k 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ3_HTML.gif
      (1.3)

      for some q ≥ 2, where C1, C2, k1, k2 are positive constants;

      (H2) g L c 2 ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq1_HTML.gif, where L c 2 ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq2_HTML.gif is the set of all translation compact functions in L loc 2 ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq3_HTML.gif whose definition is given in Definition 1.1 below.

      (H3) 2 N N + 2 p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq4_HTML.gif and N p - N 2 γ + 1 < N p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq5_HTML.gif.

      Let us give some comments about assumptions (H 1)-(H 3). The nonlinearity f is assumed to have a polynomial growth and to satisfy a standard dissipative condition. A typical example of functions satisfying conditions (H 1) is f (t, u) = |u|q-2u. arctan t, q ≥ 2. We refer the reader to [[1], Chapter 5, Propositions 3.3 and 3.5] for translation compact criterions in L loc 2 ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq3_HTML.gif. While (H 3) is a technical condition ensuring that D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq6_HTML.gif is embedded compactly into L2(Ω), where D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq6_HTML.gif is the natural energy space related to problem (1.1), which is defined later in this section. This is essential for proving the existence of a weak solution to problem (1.1) using the compactness method.

      Problem (1.1), which is related to some Caffarelli-Kohn-Nirenberg inequalities [14], contains some important classes of parabolic equations, such as the semilinear heat equations (when γ = 0, p = 2), semilinear singular/degenerate parabolic equations (when p = 2), the p- Laplacian equations (when γ = 0, p ≠ 2), etc. The existence and properties of solutions to problem type (1.1) have attracted interest in recent years [1519]. However, to the best of our knowledge, little seems to be known for the long-time behavior of solutions to problem (1.1).

      In this article we study the long-time behavior of solutions to problem (1.1) via the concept of uniform global attractors for multi-valued semiprocesses. Here there is no restrictions on the growth of the nonlinearity f and the conditions imposed on f provide the global existence of a weak solution to problem (1.1), but not uniqueness. Thus, when studying the long-time behavior of solutions, in order to handle nonuniqueness of solutions, we need use the theory of attractors for multi-valued semiprocesses. Following the general lines of the approach used in [810, 20] for non-degenerate parabolic equations, we prove the existence of a global compact attractor in the autonomous case, and of a uniform global compact attractor in the non-autonomous case. Noting that when the nonlinearity f does not depend on time t, the existence of an attractor for problem (1.1) in the semilinear non-degenerate case, namely when γ = 0 and p = 2, was studied in [8, 9]. Thus, our results extend some known results on the existence and long-time behavior of solutions of nondegenerate semilinear parabolic equations.

      It is worth noticing that under some additional conditions on f, for example, f u ( t , u ) - C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq7_HTML.gif for all t > τ, u ∈ ℝ, or a weaker assumption
      f ( t , u ) - f ( t , v ) ( u - v ) - C u - v 2 for all t > τ , u , v , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equa_HTML.gif

      one can prove that the weak solution of problem (1.1) is unique. Then the multivalued semiprocess turns to be a single-valued one and the uniform compact global attractor is exactly the usual uniform attractor for the family of single-valued semiprocesses [1].

      In the rest of this section, for convenience of the reader, we recall some results on function spaces related to Caffarelli-Kohn-Nirenberg inequalities and translation compact functions.

      For 1 < p < ∞ and γ < N - p p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq8_HTML.gif, we define the weighted space
      L γ p ( Ω ) = u : Ω is measurable such that x - γ u ( x ) L p ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equb_HTML.gif
      equipped with the norm
      u L γ p ( Ω ) = Ω x - p γ u ( x ) p 1 / p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equc_HTML.gif
      It is easy to check that the dual space ( L γ p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq9_HTML.gif of L γ p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq10_HTML.gif is the space L - γ p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq11_HTML.gif, where p' is defined by 1 p + 1 p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq12_HTML.gif. Moreover, we define the weighted Sobolev space D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq6_HTML.gif as the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq13_HTML.gif in the norm
      u D 0 , γ 1 , p ( Ω ) = u L γ p ( Ω ) = Ω x - p γ u ( x ) p d x 1 p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ4_HTML.gif
      (1.4)

      As 1 < p < ∞, D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq6_HTML.gif is reflexive, and the dual space of D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq6_HTML.gif will be denoted by D - γ - 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq14_HTML.gif.

      We now state some results which we will use later. The first is the Caffarelli-Kohn-Nirenberg inequality.

      Proposition 1.1. [14] Assume that 1 < p < N. Then there exists a positive constant C N,p,γ,q such that for every u C 0 ( N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq15_HTML.gif,
      N x - δ q u ( x ) q d x p / q C N , p , γ , q N x - p γ u ( x ) p d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ5_HTML.gif
      (1.5)
      where p, q, γ, δ are related by
      1 q - δ N = 1 p - γ + 1 N , γ δ γ + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ6_HTML.gif
      (1.6)

      and δq < N, γp < N.

      The inequality (1.5) implies that the embedding
      D 0 , γ 1 , p ( Ω ) L δ q ( Ω ) is continuous for p , q , γ , δ satisfying ( 1 . 6 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equd_HTML.gif
      This implies, by duality,
      L - δ q ( Ω ) D - γ - 1 , p ( Ω ) for p , q , γ , δ satisfying ( 1 . 6 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Eque_HTML.gif
      It is pointed out in [19] that
      D 0 , γ 1 , p ( Ω ) L δ q ( Ω ) compactly http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ7_HTML.gif
      (1.7)

      for every p, q, γ, δ satisfying 1 q - δ N > 1 p - γ + 1 N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq16_HTML.gif with γδγ + 1 and δq < N, γ p < N.

      From assumption (H 3), it is easy to check that there exists a positive number δ such that D 0 , γ 1 , p ( Ω ) L δ 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq17_HTML.gif compactly. Since the embedding L δ 2 ( Ω ) L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq18_HTML.gif is continuous, it is seen that D 0 , γ 1 , p ( Ω ) L 2 ( Ω ) D γ 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq19_HTML.gif is an evolution triplet.

      We now define the following "evolution" spaces which will be useful in what follows.
      L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) = u ( . , . ) : Ω × ( τ , T ) measurable: u ( . , t ) D 0 , γ 1 , p ( Ω ) for a.e . t ( τ , T ) , u ( . , t ) D 0 , γ 1 , p ( Ω ) L p ( τ , T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equf_HTML.gif
      endowed with the norm
      u L p τ , T ; D 0 , γ 1 , p ( Ω ) = τ T u ( . , t ) D 0 , γ 1 , p ( Ω ) p d t 1 / p = τ T Ω x - p γ u p d x d t 1 / p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equg_HTML.gif

      The dual space of L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq20_HTML.gif is L p ( τ , T ; D - γ - 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq21_HTML.gif.

      Putting
      Δ p , γ u = div( | x | p γ | u | p 2 u ) , u D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equh_HTML.gif

      The following proposition, which is easily proved by using similar arguments as in [[21], Chapter 2], gives some important properties of the operator -Δ p,γ .

      Proposition 1.2. The operator p,γ maps D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq6_HTML.gif into its dual D - γ - 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq14_HTML.gif. Moreover,

      (1) - Δ p,γ is hemicontinuous, i.e., for all u , v , w D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq22_HTML.gif, the map λ ↦ 〈-Δ p,γ (u + λv), wis continuous fromto ℝ.

      (2) - Δ p,γ is monotone, i.e., 〈-Δ p,γ u + Δ p,γ v,u - v〉 ≥ 0, for all u , v D 0 , γ 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq23_HTML.gif.

      Definition 1.1. Assume that ℰ is a reflexive Banach space.

      (1) A function φ L loc 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq24_HTML.gif is said to be translation bounded if
      φ L b 2 2 = φ L b 2 ( ; ) = sup t t t + 1 φ 2 d s < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equi_HTML.gif

      (2) A function φ L loc 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq24_HTML.gif is said to be translation compact if the closure of {φ(⋅ + h)|h ∈ ℝ} is compact in L loc 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq25_HTML.gif.

      Denote by L b 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq26_HTML.gif and L c 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq27_HTML.gif the sets of all translation bounded functions and of all translation compact functions in L loc 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq25_HTML.gif, respectively. It is well-known (see [4]) that L c 2 ( ; ) L b 2 ( ; ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq28_HTML.gif.

      Let ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq29_HTML.gif be the closure of the set {g(· + h)|h ∈ ℝ} in L b 2 ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq30_HTML.gif. The following results were proved in [[1], Chapter 5, Proposition 3.4].

      Lemma 1.3. (1) ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq29_HTML.gif is compact.

      (2) For all σ ( g ) , σ L b 2 2 g L b 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq31_HTML.gif;

      (3) The translation group {T(h)}, which is defined by T(h)σ(s) = σ(h + s), s, h ∈ ℝ, is continuous on ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq29_HTML.gif;

      (4) T ( h ) ( g ) = ( g ) f o r h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq32_HTML.gif;

      The rest of the article is organized as follows. In Section 2, we prove the global existence of a weak solution to problem (1.1) by using the monotonicity and compactness methods. In Section 3, the existence of global attractors for problem (1.1) is proved in both the autonomous and non-autonomous cases.

      2. Existence of a weak solution

      We denote
      Q τ , T = Ω × ( τ , T ) , V = L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) L q ( τ , T ; L q ( Ω ) ) , V = L p ( τ , T ; D - γ - 1 , p ( Ω ) ) + L q ( τ , T ; L q ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equj_HTML.gif

      where p', q' are the conjugate indexes of p, q, respectively.

      Definition 2.1. A function u(x, t) is called a weak solution of (1.1) on (τ, T) iff
      u V , d u d t V , u t = τ = u τ a . e . i n Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equk_HTML.gif
      and
      τ T u t , φ d t + τ T Ω x - p γ u p - 2 | u φ d x d t + τ T f ( t , u ) , φ d t = τ T ( g ( t ) , φ ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equl_HTML.gif

      for all test functions φV.

      It is known (see [[1], Theorem 1.8, p. 33]) that if uV and d u d t V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq33_HTML.gif, then uC([τ, T];L2(Ω)). This makes the initial condition in problem (1.1) meaningful.

      Theorem 2.1. For any τ, T ∈ ℝ, T > τ and u τ L2(Ω) given, problem (1.1) has at least one weak solution u on (τ, T). Moreover, the solution u can be extended to the whole interval (τ, +∞).

      Proof. We split the proof into three steps.

      Step 1: A Galerkin scheme. Consider the approximating solution u n (t) in the form
      u n ( t ) = k = 1 n u n k ( t ) e k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equm_HTML.gif
      where e k k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq34_HTML.gif is a basis of D 0 , γ 1 , p ( Ω ) ) L q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq35_HTML.gif, which is orthonormal in L2(Ω). We get u n from solving the problem
      d u n d t , e k + - Δ p , γ u n , e k + f ( t , u n ) , e k = ( g ( t ) , e k ) , ( u n ( τ ) , e k ) = ( u τ , e k ) , k = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equn_HTML.gif

      Using the Peano theorem in the theory of ODEs, we get the local existence of u n .

      Step 2: A priori estimates. We have
      1 2 d d t u n L 2 ( Ω ) 2 + u n D 0 , γ 1 , p ( Ω ) p + Ω f ( t , u n ) u n d x = Ω g ( t ) u n d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equo_HTML.gif
      By assumption (H 3), we can choose δ > 0 such that 1 2 - δ N > 1 p - γ + 1 N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq36_HTML.gif, then D 0 , γ 1 , p ( Ω ) L δ 2 ( Ω ) L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq37_HTML.gif and therefore there exists λ > 0 such that
      u D 0 , γ 1 , p ( Ω ) p C u L δ 2 ( Ω ) p λ ^ u L 2 ( Ω ) p λ u L 2 ( Ω ) 2 - λ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ8_HTML.gif
      (2.1)
      where the last inequality follows from the Young inequality. Using (1.3) and the Cauchy inequality, we get
      1 2 d d t u n L 2 ( Ω ) 2 + u n D 0 , γ 1 , p ( Ω ) p + C 2 u n L q ( Ω ) q - k 2 Ω 1 2 λ g ( t ) L 2 ( Ω ) 2 + λ 2 u n L 2 ( Ω ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equp_HTML.gif
      Hence
      d d t u n L 2 ( Ω ) 2 + u n D 0 , γ 1 , p ( Ω ) p + 2 C 2 u n L q ( Ω ) q 1 λ g ( t ) L 2 ( Ω ) 2 + 2 k 2 Ω + λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ9_HTML.gif
      (2.2)
      We show that the local solution u n can be extended to the interval [τ, ∞). Indeed, from (2.2) we have
      d d t u n L 2 ( Ω ) 2 + λ u n L 2 ( Ω ) 2 1 λ g ( t ) L 2 ( Ω ) 2 + 2 k 2 Ω + 2 λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equq_HTML.gif
      By the Gronwall inequality, we obtain
      u n ( t ) L 2 ( Ω ) 2 u n ( τ ) L 2 ( Ω ) 2 e - λ ( t - τ ) + 1 λ τ t e - λ ( t - s ) g ( s ) L 2 Ω 2 d s + ( 2 k 2 Ω + 2 λ ) τ t e - λ ( t - s ) d s u τ L 2 ( Ω ) 2 e - λ ( t - τ ) + 1 λ ( 1 - e - λ ) g L b 2 2 + 2 k 2 Ω λ + 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ10_HTML.gif
      (2.3)
      where we have used the facts that u n ( τ ) L 2 ( Ω ) u τ L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq38_HTML.gif and
      τ t e - λ ( t - s ) g ( s ) L 2 ( Ω ) 2 d s t - 1 t e - λ ( t - s ) g ( s ) L 2 ( Ω ) 2 d s + t - 2 t - 1 e - λ ( t - s ) g ( s ) L 2 ( Ω ) 2 d s + t - 1 t g ( s ) L 2 ( Ω ) 2 d s + e - λ t - 2 t - 1 g ( s ) L 2 ( Ω ) 2 d s + ( 1 + e - λ + e - 2 λ + ) g L b 2 2 = 1 1 - e - λ g L b 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equr_HTML.gif
      We now establish some a priori estimates for u n . Integrating (2.2) on [τ, T], τ < tT, and using the fact that u n ( τ ) L 2 ( Ω ) u τ L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq38_HTML.gif, we have
      u n ( t ) L 2 ( Ω ) 2 + τ T u n ( s ) D 0 , γ 1 , p ( Ω ) p d s + 2 C 2 τ T u n ( s ) L q ( Ω ) q d s u τ L 2 ( Ω ) 2 + 1 λ τ T g ( s ) L 2 Ω 2 d s + ( 2 k 2 Ω + 2 λ ) ( T - τ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ11_HTML.gif
      (2.4)
      The last inequality implies that
      { u n } is bounded in L ( τ , T ; L 2 ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ12_HTML.gif
      (2.5)
      { u n } is bounded in L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ13_HTML.gif
      (2.6)
      { u n } is bounded in L q ( τ , T ; L q ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ14_HTML.gif
      (2.7)
      Using hypothesis (1.2), we get
      τ T f ( t , u n ) L q ' ( Ω ) q ' d t τ T Ω ( C 1 | u n | q 1 + k 1 ) q ' d x d t τ T Ω C ( | u n | q + 1 ) d x d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equs_HTML.gif
      Hence, we can conclude that {f(t, u n )} is bounded in L q ( τ , T ; L q ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq39_HTML.gif and thus,
      f ( t , u n ) η in L q ( τ , T ; L q ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ15_HTML.gif
      (2.8)
      We have
      - Δ p , γ u n , v = - div ( x - p γ u p - 2 u ) , v = τ T Ω x - p γ u n p - 2 u n v d x d t = τ T Ω x - ( p - 1 ) γ u n p - 2 u n ( x - γ v ) d x d t u n L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) p / p v L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equt_HTML.gif

      for all v L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq40_HTML.gif, where we have used the Hölder inequality. Because of the boundedness of {u n } in L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq20_HTML.gif, we infer that {-Δ p,γ u n } is bounded in L p ( τ , T ; D - γ - 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq21_HTML.gif.

      Step 3: Passing limits. From the above estimates, there exists a subsequence {u μ } ⊂ {u n } such that
      u μ u in L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ16_HTML.gif
      (2.9)
      f ( t , u μ ) η   in   L q ( τ , T ; L q Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ17_HTML.gif
      (2.10)
      - Δ p , γ u μ ψ in L p ( τ , T ; D - γ - 1 , p ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ18_HTML.gif
      (2.11)

      up to a subsequence.

      To prove that η(t) = f(t, u(t)), we argue similarly to [22, 23] to deduce that
      lim a 0 sup μ τ T - a u μ ( t + a ) - u μ ( t ) L 2 ( Ω ) 2 d t = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ19_HTML.gif
      (2.12)
      for all T > τ. In particular, we obtain from (2.5) that
      lim a 0 sup μ τ T + a u μ ( t ) L 2 ( Ω ) 2 d t + T - a T u μ ( t ) L 2 ( Ω ) 2 d t = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ20_HTML.gif
      (2.13)
      Then, by Theorem 13.3 and Remark 13.1 in [24], we obtain that u μ u strongly in L2(τ, T; L2(Ω)), up to a subsequence. Hence, we can assume that u μ u a.e. in Q τ,T . Therefore, f(t, u μ ) → f(t, u) a.e. in Q τ,T since f is continuous. By Lemma 1.3 in [[21], Chapter 1], one has
      f ( t , u μ ) f ( t , u ) in L q ( τ , T ; L q ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equu_HTML.gif
      Thus, we have
      d u d t = ψ - f ( t , u ) + g ( t ) in V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ21_HTML.gif
      (2.14)
      We now show that ψ = -Δ p, γ u. Since -Δ p, γ is monotone, we have
      X n = τ T - Δ p , γ u n + Δ p , γ v , u n - v d t 0 , for all v V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equv_HTML.gif
      Note that {u n (T)} is bounded in L2(Ω), so by arguments as in [[21], pp. 159-160], we have that u n (T) ⇀ u(T) in L2(Ω). Because
      τ T - Δ p , γ u n , u n d t = - τ T Ω ( f ( t , u n ) u n - g ( t ) u n ) d x d t + 1 2 u n ( τ ) L 2 ( Ω ) 2 - 1 2 u n ( T ) L 2 ( Ω ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ22_HTML.gif
      (2.15)
      we obtain
      lim sup n X n τ T ( f ( t , u ) u d t + 1 2 u ( τ ) L 2 ( Ω ) 2 1 2 u ( T ) L 2 ( Ω ) 2 τ T ( ψ , v ) d t + τ T ( Δ p , γ v , u v ) d t + τ T ( g ( t ) , u ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ23_HTML.gif
      (2.16)
      where we have used the facts that u n (τ) → u τ in L 2 ( Ω ) , u ( T ) L 2 ( Ω ) 2 lim inf n u n ( T ) L 2 ( Ω ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq41_HTML.gif. On the other hand, by integrating by parts, from (2.14) we have
      - τ T ( f , u ) d t + 1 2 u ( τ ) L 2 ( Ω ) 2 - 1 2 u ( T ) L 2 ( Ω ) 2 + τ T ( g ( t ) , u ) d t = τ T ( ψ , u ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equw_HTML.gif
      and therefore thanks to (2.15) and (2.16) one gets
      τ T ( ψ + Δ p , γ v , u - v ) d t 0 , v V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equx_HTML.gif
      We now use the hemicontinuity of the operator Δ p,γ to show that ψ = -Δ p,γ u. Taking v = u - λw, where λ > 0 and w V : = L p ( τ , T ; D 0 , γ 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq42_HTML.gif, we obtain
      λ τ T ( ψ + Δ p , γ ( u - λ w ) , w ) d t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equy_HTML.gif
      hence
      τ T ( ψ + Δ p , γ ( u - λ w ) , w ) d t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ24_HTML.gif
      (2.17)
      leting λ → 0 in (2.17), we conclude that
      τ T ( ψ + Δ p , γ u , w ) d t 0 , for all w V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equz_HTML.gif
      So ψ = -Δ p,γ u. Thus,
      u = Δ p , γ u - f ( t , u ) + g ( t ) in V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equaa_HTML.gif
      We now show that u(τ) = u τ . Choosing some φ C 1 ( [ τ , T ] ; D 0 , γ 1 , p ( Ω ) L q ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq43_HTML.gif with φ(T) = 0, observe that φV, by the Lebesgue dominated theorem, one can check that
      - τ T ( u , φ ) d t + τ T Ω x - p γ u p - 2 u φ d x d t + τ T Ω f ( t , u ) φ d x d t = ( u ( τ ) , φ ( τ ) ) + τ T Ω g φ d x d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equab_HTML.gif
      Doing the same in the Galerkin approximations yields
      - τ T ( u n , φ ) d t + τ T Ω x - p γ u n p - 2 u n φ d x d t + τ T Ω f ( t , u n ) φ d x d t = ( u n ( τ ) , φ ( τ ) ) + τ T Ω g φ d x d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equac_HTML.gif
      Passing to the limit as n → ∞, we have
      - τ T ( u , φ ) d t + τ T Ω x - p γ u p - 2 u φ d x d t + τ T Ω f ( t , u ) φ d x d t = ( u τ , φ ( τ ) ) + τ T Ω g φ d x d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equad_HTML.gif

      Therefore, u(τ) = u τ and u is a weak solution of (1.1) on (τ, T).

      Finally, it is easy to check that the solution u satisfies the inequality similar to (2.3), and this implies that the solution u exists globally on the interval (τ, +∞).

      3. Existence of global attractors

      3.1. The autonomous case

      Consider the case where f and g do not depend on the time t, and let us recall the definition of multi-valued semiflows.

      Definition 3.1. [5] Let E be a Banach space. The mapping
      G : [ 0 , + ) × E 2 E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equae_HTML.gif

      is called a multi-valued semiflow if the following conditions are satisfied:

      (1) G ( 0 , w ) = w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq44_HTML.gif for arbitrary wE;

      (2) G ( t 1 + t 2 , w ) G ( t 1 , G ( t 2 , w ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq45_HTML.gif for all wE, t1, t2 ∈ ℝ+, where G (t, B) = ∪xBG (t, x), BE.

      It is called a strict multi-valued semiflow if G ( t 1 + t 2 , w ) = G ( t 1 , G ( t 2 , w ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq46_HTML.gif, for all wE, t1, t2 ∈ ℝ+.

      We now consider problem (1.1) with τ = 0. By Theorem 2.1, we construct a multi-valued mapping as follows
      G ( t , u 0 ) = { u ( t ) | u ( ) is a global weak solution of (1 .1) such that u ( 0 ) = u 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equaf_HTML.gif

      Lemma 3.1. http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif is a strict multi-valued semiflow in the sense of Definition 3.1.

      Proof. Assume that ξ G ( t 1 + t 2 , u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq48_HTML.gif, then ξ = u(t1 + t2), where u(t) is a solution of (1.1). Denoting v (t) = u(t + t2), we see that v(.) is also in the set of solutions of (1.1) with respect to initial condition v(0) = u(t2). Therefore, ξ = v ( t 1 ) G ( t 1 , u ( t 2 ) ) G ( t 2 , G ( t 2 , u 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq49_HTML.gif. It remains to show that G ( t 1 , G ( t 2 , u 0 ) ) G ( t 1 + t 2 , u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq50_HTML.gif. If ξ G ( t 1 , G ( t 2 , u 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq51_HTML.gif then ξ = v(t1), where v ( 0 ) G ( t 2 , u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq52_HTML.gif. One can suppose that v(0) = u(t2), where u(0) = u0. Set
      w ( τ ) = u ( τ ) , 0 τ < t 2 , v ( τ - t 2 ) , τ t 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equag_HTML.gif

      Since u and v are two solutions of (1.1), we obtain that w is a solution of (1.1) with w(0) = u(0) = u0. In addition, since ξ = v(t1) = w(t1 + t2), we have ξ G ( t 1 + t 2 , u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq48_HTML.gif.

      Definition 3.2. [5] A set http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif is said to be a global attractor of the multi-valued semiflow http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif if the following conditions hold:

      • http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif is an attracting, i.e., d i s t ( G ( t , B ) , A ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq54_HTML.gifas t → ∞ for all bounded subsets BE,

      • http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif is negatively semi-invariant: A G ( t , A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq55_HTML.giffor arbitrary t ≥ 0,

      • Ifis an attracting of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif then A ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq56_HTML.gif,

      where d i s t ( C , A ) = sup c C inf a A c - a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq57_HTML.gif is the Hausdorff semi-distance.

      The following theorem gives the sufficient conditions for the existence of a global attractor for the multi-valued semiflow http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif .

      Theorem 3.2. [5, 7] Suppose that the strict multi-valued semiflow http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif has the following properties:

      (1) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif is pointwise dissipative, i.e., there exists K > 0 such that for u 0 E , u ( t ) G ( t , u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq58_HTML.gif one hasu(t)∥ E K if tt0 (∥u0 E );

      (2) G ( t , . ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq59_HTML.gif is a closed map for any t ≥ 0, i.e., if ξ n ξ, η n η, ξ n G ( t , η n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq60_HTML.gif then ξ G ( t , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq61_HTML.gif;

      (3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif is asymptotically upper semicompact, i.e., if B is a bounded set in E such that for some T ( B ) , γ T ( B ) + ( B ) : = t T ( B ) G ( t , B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq62_HTML.gif is bounded, any sequence ξ n G ( t n , B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq63_HTML.gif with t n → ∞ is precompact in E.

      Then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif has a compact global attractor http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif in E. Moreover, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif is invariant, i.e., G ( t , A ) = A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq64_HTML.gif for any t ≥ 0.

      Lemma 3.3. G ( t * , . ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq65_HTML.gif is a compact mapping for each t* ∈ (0, T].

      Proof. This lemma is a direct consequence of Lemma 3.8 in Section 3.2 below.

      We now can prove the existence of a global attractor.

      Theorem 3.4. Under conditions (H 1)-(H 3), where f andg are assumed to be independent of time t, the strict multi-valued semiflow http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif generated by problem (1.1) has an invariant compact global attractor in L2(Ω).

      Proof. We will check hypotheses (1)-(3) of Theorem 3.2. First, assume u ( t ) G ( t , u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq66_HTML.gif, we have
      1 2 d d t u ( t ) L 2 ( Ω ) 2 + Ω x - p γ u p + C 2 u L q ( Ω ) q k 2 Ω + Ω u g d x k 2 Ω + ε u L 2 ( Ω ) 2 + C ε g L 2 ( Ω ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equah_HTML.gif
      Noting that
      C 2 u L q ( Ω ) q λ u L 2 ( Ω ) 2 - C , C = C q , Ω > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equai_HTML.gif
      we have
      1 2 d d t u L 2 ( Ω ) 2 + λ u L 2 ( Ω ) 2 C q , Ω , g L 2 ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ25_HTML.gif
      (3.1)
      Therefore
      u ( t ) L 2 ( Ω ) 2 u ( 0 ) L 2 ( Ω ) 2 e - 2 λ t + C q , Ω , g L 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equaj_HTML.gif

      Hence one can deduce that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq47_HTML.gif is pointwise dissipative.

      We now check hypothesis (2) of Theorem 3.2. Assume that ξ n G ( t , η n ) , ξ n ξ , η n η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq67_HTML.gif in L2(Ω). Then there exists a sequence {u n } such that
      u n ( t ) = ξ n , u n ( 0 ) = η n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equak_HTML.gif

      Using the same arguments as in the proof of Theorem 2.1, we have

      • u n u in L2(Q0,T),

      • u n (t) ⇀ u(t) in L2(Ω) for arbitrary t ∈ [0, T] (and then u(0) = η),

      • f(u n )⇀ f(u) in L q ( Q 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq68_HTML.gif,

      • d u n d t d u d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq69_HTML.gif in V,

      • p,γ u n ⇀ -Δ p,γ u in L p 0 , T ; D - γ - 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq70_HTML.gif,

      up to a subsequence. Hence, passing to the limit in the equality
      0 T u n , v + 0 T d t Ω x - p γ u n p - 2 u n v + 0 T d t Ω f ( u n ) v = 0 T d t Ω g v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equal_HTML.gif

      we conclude that u(t) is a weak solution of (1.1) with the initial condition u(0) = η. Thus, ξ G ( t , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq71_HTML.gif.

      For hypothesis (3), one observes that for n large enough,
      G ( t n , B ) = G ( t * + t n - t * , B ) G ( t * , G ( t n - t * B ) ) G ( t * , B * ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equam_HTML.gif

      where t* > 0 and B* is a bounded set in L2(Ω). Using Lemma 3.3, we conclude that, if ξ n G ( t n , B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq72_HTML.gif, then {ξ n } is precompact in L2(Ω).

      3.2. The non-autonomous case

      Let us recall some definitions and related results. The pair of functions (f(s,⋅),g(⋅,s)) = σ(s) is called a symbol of (1.1). We consider (1.1) with a family of symbols including the shifted forms σ(s + h) = (f(s + h,⋅), g (⋅, s + h)) and the limits of some sequence {σ(s + h n )}nNin an appropriate topological space Σ. The family of such symbols is said to be the hull of σ in Σ and is denoted by ( σ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq73_HTML.gif, i.e.,
      ( σ ) = c l { σ ( + h ) h } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equan_HTML.gif

      If the hull ( σ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq73_HTML.gif is a compact set in Σ, we say that σ is translation compact in Σ.

      Denote ℝ d = {(t, τ) ∈ ℝ2 | τt}. Let X be a complete metric space, P ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq74_HTML.gif and ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq75_HTML.gif be the set of all nonempty subsets and the set of all nonempty bounded subsets of the space X, respectively and let Σ be a subspace of Σ.

      Denote
      Z = { φ C ( ; ) : φ ( u ) C φ ( 1 + u q - 1 ) , for some C φ > 0 } , φ Z = sup u φ ( u ) 1 + u q - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equao_HTML.gif
      Then Z is a Banach space. We say that f n f in the space C(ℝ; Z) if
      lim n sup s [ t , t + r ] f n ( s , ) - f ( s , ) Z = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ26_HTML.gif
      (3.2)

      for all t ∈ ℝ, r > 0.

      Let f 0 C ( ; Z ) , g 0 L loc 2 , w ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq76_HTML.gif, and
      ( f 0 ) = c l C ( ; Z ) { f 0 ( + h ) h } , ( g 0 ) = c l L loc 2 , w ( ; L 2 ( Ω ) ) { g 0 ( + h ) h } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equap_HTML.gif
      where the topology in L loc 2 , w ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq77_HTML.gif is equipped by the local weak convergence, i.e., g n g in L loc 2 , w ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq77_HTML.gif if
      lim n t t + r Ω ( g n ( s , x ) - g ( s , x ) ) ϕ ( x , s ) d s d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equaq_HTML.gif

      for all t ∈ ℝ, r > 0 and ϕL2 (Qt,t+r). We define = ( f 0 ) × ( g 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq78_HTML.gif.

      In order to deal with a uniform attractor with respect to the family of symbols, one usually requires the translation compact property. Let us recall some discussions on this requirement. It is known that hypothesis (H 2) ensures that g is translation compact in L loc 2 , w ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq77_HTML.gif (see [4] for details). In addition, the following statement gives a sufficient condition for the translation compact property in C (ℝ; Z).

      Proposition 3.5. [4] The function fC(ℝ; Z) is translation compact if and only if for all R > 0 one has
      1. (1)

        |f(t, v)| ≤ C(R) for all t ∈ ℝ, v ∈ [-R, R],

         
      2. (2)

        |f(t1, v1)-f(t2, v2)| ≤ α(|t1-t2| + |v1-v2|,R), ∀t1, t2 ∈ ℝ, v1, v2 ∈ [-R, R], here C(R) > 0 and α(.,.) is a function such that α(s, R) → 0 as s → 0+.

         

      From now on, we suppose that f is translation compact. Together with the fact that g is translation compact in L loc 2 , w ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq77_HTML.gif, one sees that Σ is a compact set in L loc 2 , w ( ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq77_HTML.gif. Then it follows from [4] that T(h) : Σ → Σ is continuous and T(h)Σ ⊂ Σ for all h ∈ ℝ.

      Definition 3.3. [6] The map U : d × X P ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq79_HTML.gif is called an multi-valued semiprocess (MSP) if

      (1)U (τ, τ,.) = Id (the identity map),

      (2)U (t, τ, x) ⊂ U(t, s, U(s, τ, x)), for all xX, t, s, τ ∈ ℝ,τst.

      It is called a strict multi-valued semiprocess if U(t, τ, x) = U(t, s, U(s, τ, x)).

      We denote by D τ , σ ( u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq80_HTML.gif the set of all global weak solutions (defined for all tτ) of the problem (1.1) with data (f σ , g σ ) instead of (f, g) such that u(τ) = u τ . For each σ = (f, g) ∈ Σ, we consider the family of MSP {U σ : σ ∈ Σ} defined by
      U σ ( t , τ , u τ ) = { u = u ( t ) u ( ) D τ , σ ( u τ ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equar_HTML.gif
      Lemma 3.6. U σ (t, τ, u τ ) is a multi-valued semiprocess. Moreover,
      U σ ( t + s , τ + s , u τ ) = U T ( s ) σ ( t , τ , u τ ) f o r a l l u τ L 2 ( Ω ) , ( t , τ ) d , s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equas_HTML.gif

      Proof. Given zU σ (t, τ, u τ )) we have to prove that zU σ (t, s, U σ (s, τ, u τ )). Take y ( . ) D τ , σ ( u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq81_HTML.gif such that y(τ) = u τ and y(t) = z. Clearly, y(s) ∈ U σ (s, τ, u τ ). Then if we define z(t) = y(t) for ts we have that z(s) = y(s) and obviously z ( . ) D s , σ ( y ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq82_HTML.gif. Consequently, z(t) ∈ U σ (t, s, U σ (s, τ, u τ )).

      Let zU σ (t + s, τ + s, u τ ). Then there exists u ( ) D τ + s , σ ( u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq83_HTML.gif such that z = u(t + s) and v ( ) = u ( + s ) D τ , T ( s ) σ ( u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq84_HTML.gif, so that z = v(t) ∈ uτ,T (s)σ(u τ ).

      Conversely, if zUτ,T(s)σ(u τ ), then there is z D τ , T ( s ) σ ( u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq85_HTML.gif such that z = u(t) and v ( ) = u ( - s + ) D τ + s , σ ( u τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq86_HTML.gif so that z = v(t + s) ∈ U σ (t + s, τ + s, u τ ).

      Denote by
      U ( t , τ , x ) = σ U σ ( t , τ , x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equat_HTML.gif

      Definition 3.4. [6] A set http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif is called a uniform global attractor for the family of multi-valued semiprocesses UΣ if:

      (1) it is negatively semiinvariant, i.e., A U ( t , τ , A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq87_HTML.gif for all tτ;

      (2) it is uniformly attracting, i.e., d i s t ( U ( t , τ , B ) , A ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq88_HTML.gif, as t → ∞ , for all B ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq89_HTML.gif and τ ∈ ℝ;

      (3) for any closed uniformly attracting set Y, we have A Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq90_HTML.gif (minimality).

      Theorem 3.7. [[6], Theorem 2] Suppose that the family of multi-valued semiprocesses UΣ satisfies the following conditions:

      (1) On Σ is defined the continuous shift operator T(s)σ(t) = σ(t + s), s ∈ ℝ such that T(h)Σ ⊂ Σ, and for any (t, τ) ∈ ℝ d , σ ∈ Σ, s ∈ ℝ, xX, we have
      U σ ( t + s , τ + s , x ) = U T ( s ) σ ( t , τ , x ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equau_HTML.gif

      (2) U σ is uniformly asymtopically upper semicompact;

      (3) U σ is pointwise dissipative;

      (4) The map (x, σ) ↦ U σ (t, 0, x) has closed values and is w-upper semicontinuous.

      Then the family of multi-valued semiprocesses UΣ has a uniform global compact attractor http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif .

      The following is the key point of this subsection.

      Lemma 3.8. Let conditions (H 1)-(H 3) hold and let {u n }n∈ℕis a sequence of weak solutions of (1.1) with respect to the sequence of symbols {σ n } ⊂ Σ such that
      ( 1 ) u n ( τ ) u τ i n L 2 ( Ω ) , ( 2 ) σ n σ i n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equav_HTML.gif

      Then there exists a solution u of (1.1) with respect to the symbol σ such that u(τ) = u T and u n (t*) → u(t*) in L2(Ω) for any t* > τ, up to a subsequence.

      Proof. Let σ n = (f n , g n ). Since f satisfies (H 1) for all t ∈ ℝ and f n ( f ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq91_HTML.gif, one sees that f n also satisfies (H 1). On the other hand, noting that {u n (τ)} is bounded in L2(Ω) and g n L b 2 g L b 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq92_HTML.gif. Thus, repeating the arguments in the proof of Theorem 2.1, we obtain that
      { u n } is bounded in V = L p τ , T ; D 0 , γ 1 , p ( Ω ) L q ( τ , T ; L q ( Ω ) ) , { u n } is bounded in V = L p τ , T ; D - γ - 1 , p ( Ω ) + L q τ , T ; L q ( Ω ) , { u n } is bounded in C ( [ τ , T ] ; L 2 ( Ω ) ) , { f n ( t , u n ) } is bounded in L q ( Q τ , T ) , { - Δ p , γ u n } is bounded in L p τ , T ; D - γ - 1 , p ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equaw_HTML.gif
      In particular, we have
      u n ( t ) u ( t ) in L 2 ( Ω ) for all t [ τ , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ27_HTML.gif
      (3.3)
      up to a subsequence. Let σ n σ = ( f ¯ , g ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq93_HTML.gif in Σ, to show that u is a solution of (1.1) with respect to the symbol σ such that u(τ) = u T , we need to pass to the limits in the following relation
      τ T Ω u n v + x - p γ u n p - 2 u n v + f n ( t , u n ) v d x d t = τ T Ω g n v d x d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equax_HTML.gif
      for all vV. Since g n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq94_HTML.gif in L2(τ,T; L2(Ω)), it remains to prove that f n ( t , u n ) f ̄ ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq95_HTML.gif in L q ( Q τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq96_HTML.gif. We first show that f n ( t , u n ) f ̄ ( t , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq97_HTML.gif in L q ( Q τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq96_HTML.gif. Indeed,
      τ T Ω f n ( t , u n ) - f ̄ ( t , u n ) q d x d t = τ T Ω f n ( t , u n ) - f ̄ ( t , u n ) q ( 1 + u n q - 1 ) q ( 1 + u n q - 1 ) q d x d t sup [ τ , T ] f n - f ̄ Z q τ T Ω ( 1 + u n q ) d x d t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equay_HTML.gif
      because f n f ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq98_HTML.gif in Z and {u n } is bounded in L q (Q τ,T ). On the other hand, since { f ̄ ( t , u n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq99_HTML.gif is bounded in L q ( Q τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq96_HTML.gif, by using Lemma 1.3 in [[21], Chapter 1] and the continuity of f ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq100_HTML.gif as in the proof of Theorem 2.1, we can conclude that f ̄ ( t , u n ) f ̄ ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq101_HTML.gif weakly in L q ( Q τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq96_HTML.gif. Hence, we have
      f n ( t , u n ) - f ̄ ( t , u ) = ( f n ( t , u n ) - f ̄ ( t , u n ) ) + ( f ̄ ( t , u n ) - f ̄ ( t , u ) ) 0 weakly in L q ( Q τ , T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equaz_HTML.gif

      We now have to show that u n (t*) → u(t*) in L2(Ω) for any t* > τ. Taking into account of (3.3), we have to check that u n ( t * ) L 2 ( Ω ) u ( t * ) L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq102_HTML.gif.

      Putting
      J n ( t ) = u n ( t ) L 2 ( Ω ) 2 - 2 τ t ( g n ( s ) , u n ( s ) ) d s - ( 2 k 2 Ω + 2 λ ) ( t - τ ) , J ( t ) = u ( t ) L 2 ( Ω ) 2 - 2 τ t ( g ( s ) , u ( s ) ) d s - ( 2 k 2 Ω + 2 λ ) ( t - τ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equba_HTML.gif
      It is easy to check that the functions J n (t), J(t) are continuous and non-increasing on [τ, T]. We first show that
      J n ( t ) J ( t ) for a .e . t [ τ , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ28_HTML.gif
      (3.4)
      Indeed,
      | J n ( t ) J ( t ) | | u n ( t ) L 2 ( Ω ) 2 u ( y ) L 2 ( Ω ) 2 | + 2 | τ t [ ( g n ( s ) , u n ( s ) ) ( g ( s ) , u ( s ) ) ] d s | u n ( t ) u ( t ) L 2 ( Ω ) ( u n ( t ) L 2 ( Ω ) + u ( t ) L 2 ( Ω ) ) + 2 | τ t [ ( g n ( s ) , u n ( s ) u ( s ) ) d s | + 2 | τ t [ ( g n ( s ) g ( s ) , u ( s ) ) d s | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equbb_HTML.gif
      and
      | τ t [ ( g n ( s ) , u n ( s ) u ( s ) ) d s | g n L 2 ( Q τ , t ) u n ( t ) u ( t ) L 2 ( Ω ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equbc_HTML.gif
      as n → ∞ since u n u strongly in L2(Q τ,t ) and {g n } is bounded in L2(Q τ,t ). In addition,
      τ t ( g n ( s ) - g ( s ) , u ( s ) ) d s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equbd_HTML.gif

      as n → ∞ since g n g in L2(Q τ,t ). Then (3.4) is proved due to the fact that u n (t) → u(t) in L2(Ω) for a.e. t ∈ [τ, T].

      We choose an increasing sequence {t m } ⊂ [τ, T], t m t* such that J n (t m ) ⇀ J(t m ) as n → ∞. Then, by the continuity,
      J n ( t m ) J n ( t * ) , as m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Eqube_HTML.gif
      So
      J n ( t * ) - J ( t * ) J n ( t m ) - J ( t * ) = J n ( t m ) - J ( t m ) + J ( t m ) - J ( t * ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equbf_HTML.gif

      for nn0(ε) and any ε > 0. Hence, lim sup J n (t*) ≤ J(t*) and then lim sup ∥u n (t*)∥ ≤ ∥u(t*)∥. From the weak convergence u n (t*) ⇀ u(t*) we have then ∥u n (t*)∥ → ∥u(t*)∥, so u n (t*) → u(t*) strongly in L2(Ω) as n → ∞. This completes the proof.

      Theorem 3.9. Let conditions (H 1)-(H 3) hold. Then the family of multi-valued semipro-cesses {U σ (t, τ)} has a uniform global compact attractor http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq53_HTML.gif .

      Proof. We know that each symbol σ n = (f n , g n ) ∈ Σ satisfies the same conditions as in (H 1)-(H 2). Furthermore, since g n ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq103_HTML.gif, we have g n L b 2 g L b 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq104_HTML.gif. Hence if u n is a weak solution of (1.1) with respect to the symbol σ n , one has
      u n ( t ) L 2 ( Ω ) 2 u n ( τ ) L 2 ( Ω ) 2 e - λ ( t - τ ) + 1 λ ( 1 - e - λ ) g L b 2 2 + 2 k 2 Ω λ + 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equ29_HTML.gif
      (3.5)
      The last inequality ensures the existence of a positive number R0 such that if u n (τ) ∈ B R , the ball in L2(Ω) centered at 0 with radius R, then there exists T0 = T0(τ, R) such that
      u n ( t ) B R 0 for all t T 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equbg_HTML.gif

      that is, U ( t , τ , B R ) B R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq105_HTML.gif, for all tT0(τ, R). Thus, {U σ (t, τ)} fulfills condition (3) in Theorem 3.7.

      We now define the set K = U ( 1 , 0 , B R 0 ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq106_HTML.gif. Lemma 3.8 implies that K is compact. Moreover, since B R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq107_HTML.gif is an absorbing set, we have
      U σ n ( t , τ , B R ) = U σ n ( t , t 1 , U σ n ( t 1 , τ , B R ) = U T ( t 1 ) σ n ( 1 , 0 , U T ( τ ) σ n ( t 1 τ , 0 , B R ) ) U ( 1 , 0 , B R 0 ) K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_Equbh_HTML.gif

      for all σ n , B R ( L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq108_HTML.gif, and tT0(τ, B R ). It follows that any sequence {ξ n } such that { ξ n } U σ n ( t n , τ , B R 0 ) , σ n , t n + , B R ( L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq109_HTML.gif, is precompact in L2(Ω). It is a consequence of Lemma 3.8 that the map U σ has compact values for any σ ∈ Σ.

      Finally, let us prove that the map (σ, x) ↦ U σ (t, τ, x) is upper semicontinuous for each fixed tτ. Suppose that it is not true, that is, there exist ū L 2 ( Ω ) , t τ , σ ̄ , ε > 0 , δ n 0 , u n B δ n ( ū ) , σ n σ ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq110_HTML.gif, and ξ n U σ n ( t , τ , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq111_HTML.gif such that { ξ n } B ε ( U σ ¯ ( t , τ , u ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq112_HTML.gif. But Lemma 3.8 implies (up to a subsequence) that ξ n ξ U σ ̄ ( t , τ , ū ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-35/MediaObjects/13661_2011_Article_134_IEq113_HTML.gif, which is a contracdition. Thus, the existence of the uniform global compact attractor follows then from Theorem 3.7.

      Declarations

      Acknowledgements

      This work was supported by Vietnam's National Foundation for Science and Technology Development (NAFOSTED), Project 101.01-2010.05.

      The authors would like to thank the reviewers for valuable comments and suggestions.

      Authors’ Affiliations

      (1)
      Department of Applied Mathematics and Informatics, Hanoi University of Science and Technology
      (2)
      Department of Mathematics, Hanoi National University of Education

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      © Binh and Anh; licensee Springer. 2012

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