Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations

A global attractor in $L^2$ is shown for weakly dissipative $p$-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than $p-1$. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an $(L^2,L^r)$-attractor for $r$ belonging to a special interval, which contains the critical exponent $p$. The obtained attractor is proved to be approximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity.


1.
Introduction. This paper is concerned with the dynamical behavior of the following p-Laplace equation where p > 2, λ > 0. The source f is a continuously differential function satisfying: for all (x, s) ∈ R n × R, f (x, s)s ≥ α 1 |s| q + ψ 1 (x), |f (x, s)| ≤ α 2 |s| q−1 + ψ 2 (x), ∂f ∂s (x, s) ≥ −α 3 , (2) where q > 2, α 1 , α 2 , α 3 > 0 and ψ 2 ∈ L 2 (R n ). The force g and ψ 1 satisfy where {r} denotes the minimal integer no lesser than r. We say the equation is strongly dissipative if p ≤ q (equivalently N = 1) and weakly dissipative if p > q (equivalently N ≥ 2). Recently, Krause et al [11,12] and Li et al [14,15] have investigated the longtime behavior for the stochastic p-Laplace equation on the whole Euclid space. Khanmamedov et al [8,9,10] had earlier obtained a global attractor for the deterministic equation (i.e. (1)) on the whole space still. However, all these papers have assumed the equation was strongly dissipative. In this case, the tail estimate outside a ball can be achieved directly from the absorption of p-norm by using an interpolation. By the way, the reaction-diffusion equation (i.e. Eq. (1) with p = 2) is always strongly dissipative, its dynamical behavior in stochastic or deterministic case has been investigated by many papers, see e.g. [1,13,16,22,25,26,29,34]. The strong dissipativeness was also assumed in [19,20,21,27,32] for p-Laplace equations defined on a bounded domain with a possibly different principal part.
The weakly dissipative case was considered by Gess et al [4,5,6] and Wang [23] for stochastic p-Laplace equations but restricted on a bounded domain. In this case, it does not require the tail estimate. So the techniques are similar in both strongly and weakly dissipative cases.
In a word, it remains open whether there is an attractor for the weakly dissipative p-Laplace equation on the entire Euclid space even though the equation is nonrandom.
In this paper, we first give an affirmative answer to the above question. The main difficult arises from estimating the tail of a solution outside a ball. This estimate requires an auxiliary absorption under the p-norm. In general, the absorption may be achieved easily under the q-norm. If the equation is strongly dissipative (q ≥ p), this absorption reaches automatically the level of p-norm by an interpolation. This is the reason why the literatures mentioned above assumed the strong dissipativeness. If the equation is weakly dissipative (q < p), the interpolation lose its effectiveness. To overcome this difficulty, we define an increasing sequence q m = m(q − 2) + 2 and prove the absorption under q N -norm by an induction argument, where N is given in (3) and it is just the minimal integer such that q N ≥ p. So the absorption can reach the level of p-norm. We then use this absorption together with an half-time decomposition to derive that the p-Laplace system is asymptotically small outside a large ball and thus obtain a global attractor in the weakly dissipative case.
Our second purpose is to consider the regularity problem after proving the existence of a global attractor in L 2 . The main difficulty to consider regularity of attractors is the continuity of systems in the terminate space. Several continuity concepts were introduced in the literatures to overcome the difficulty. For instance, norm-to-weak continuity and quasi-continuity were introduced in [34] and in [17] respectively to deal with the reflexive phase space (such as L r for 1 < r < +∞ or Sobolev spaces), while quasi-weak-star-continuity (i.e. a generalization of quasicontinuity) was introduced recently in [7] to deal with non-reflexive phase space (such as L 1 or L ∞ ). The authors in a recent paper [15] gave an abstract existence result of random bi-spatial attractors without any continuity assumption in the terminate space (it depends on continuity of systems in the initial space only). The present paper will discuss the regularity of global attractors for Eq.(1) by using the abstract result given in [15] and developed by [18]. We will show by an induction method that the obtained attractor is in fact an (L 2 , L r )-attractor for any r ∈ [2, q N ] with q N = N (q − 2) + 2 in both strongly and weakly dissipative cases. Since q N is no lesser than both p and q, our result contains two critical exponents (r = p and r = q), which generalizes the main results in [8,9,28] even in the strongly dissipative case.
Our third purpose is to consider whether the obtained attractor (denoted by A ∞ ) can be approximated by the corresponding attractor (denoted by A R ) for the equation defined on a ball with the radius R. We will use the method given in [15] to prove that the attractor A ∞ is constructed as the closure of union of attractors A R over all R ∈ [R f,g , +∞), where the constant R f,g is related to both functions f (·, 0) and g(·). This means that the family A R of attractors is strictly upper semi-continuous and lower semi-continuous as the radius R tends to infinity. This continuity of attractors holds true under the r-norm with any r ∈ [2, q N ]. The result generalizes an earlier result in [2] for the lattice dynamical system with q ≥ p = 2 and generalizes the corresponding result in [15] to the weakly dissipative case. More importantly, the result presents an actual example of lower semi-continuity, even though it is extremely hard in practical examples to prove lower semi-continuity or equivalently equi-attraction of attractors as pointed out in [3, P.65].
2. Preliminaries and semigroups of solutions. We denotes the norms in L 2 and L r by · and · r respectively, where 1 ≤ r = 2. We also set u p W : = n i=1 u xi p p and thus · p + · W defines a norm in the Sobolev space W 1,p (R n ). A semilinear p-Laplace operator is defined by A : In particular, (Au, u) = u p W . We know that A is semi-continuous and monotone (see [24]).
By using the subsequent estimates in Lemma 2.1 and Lemma 3.1, one can easily establish the existence result of solutions for problem (1), precisely, for every u 0 ∈ L 2 (R n ) and T > 0, the problem (1) has a unique solution such that u(0, u 0 ) = u 0 . Continuity of solutions in L 2 is also easily established from the third hypothesis in (2). In fact, the well-posed property of (1) in the weakly dissipative case is easily obtained by a similar argument as given in [11,12] for the strongly dissipative case. The unique solution defines a continuous semigroup S(·) on L 2 by S(t)u 0 = u(t, u 0 ) for each t ≥ 0 and u 0 ∈ L 2 . The following lemma shows that the semigroup has absorption on any sufficiently large interval.
Lemma 2.1. Let (2) hold true and ψ 1 ∈ L 1 , g ∈ L 2 . Then there is a positive constant C 0 such that, for every bounded set B in L 2 , there is a large time Proof. It is elementary from condition (2) to establish the following energy inequality: where C := C( ψ 1 1 , g ) is a generic constant. Applying the classical Gronwall lemma to (7) (omitting the second terms), we have, for all t > 0, Let M = B := sup v∈B v . We choose a T large enough such that M 2 e −λT ≤ 1, then by (8), Therefore, we obtain (6) by taking T 0 = 2T .
The following Gronwall-type lemma from [30,33] will be used frequently later.
Lemma 2.2. Let y, h be nonnegative, locally integrable on [a, +∞) and y locally integrable such that where b ∈ R. Then for every t ≥ s ≥ a we have In particular, if t ≥ a + 1 we have where Proof. It is easy to prove (11) by integrating (10) twice (see [30]). Since, for all τ ∈ [t − 1, t], e −bt e bτ ≤ 1 if b ≥ 0 and e −bt e bτ ≤ e −b if b < 0, we obtain (12) from (11).
We also recall the following embedding relationship among integrable spaces, which generalizes slightly the corresponding formulation in [11,12].
The number ε may be small enough.

Absorption by induction.
This section presents the absorption of the semigroup in W 1,p and especially in L r for any r ∈ (2, q N ], where N = {(p − 2)/(q − 2)} and we define successively q m = m(q − 2) + 2, m ∈ N.
In particular, the solution u(t, u 0 ) ∈ L q N for all t > 0 and u 0 ∈ L 2 .
Proof. We first prove (14) for m = 1. Note that q 1 = q. We multiply Eq.(1) by |u| q−2 u and then integrate it on R n to find By (4), we have By the first hypothesis in (2) and the Young inequality, we have By the Young inequality again, We see from (15) Applying now Lemma 2.2 (the Gronwall-type lemma) to (19), we see from (6) that, for all t ≥ T 0 and u 0 ∈ B, By taking T 1 = 2T 0 , we see that, for all t ≥ T 1 and for some positive constant C 1 , On the other hand, integrating (19) We have proved (14) for m = 1. We then assume (14) is true for m = k − 1 < N . Since k + 1 ≤ N + 1, by (3) and Lemma 2.3 (ii), we have Multiplying Eq.(1) with |u| q k −2 u to obtain a similar formulation as (15): By the first inequality in (2), we have By the Young inequality again, The same calculation as (16) yields (Au, |u| q k −2 u) ≥ 0. Therefore, by (23) Applying Lemma 2.2 to (27), we see from the inductive hypothesis (i.e. the second term in (14) with t ≥ T k , which proves the first term in (14) for m = k. On the other hand, we integrate (27) We finish the induction proof.
The following corollary establishes the absorption in L p whether the dissipativeness is strong or weak (i.e. q ≥ p or q < p).
Therefore, the required conclusion follows from Lemma 3.1 and Lemma 2.1 immediately.
To close this section, we prove the absorption in W 1,p .
Proof. By integrating (7) on [t − 1, t] with t ≥ T 0 , we see from (6) that for all t ≥ T 0 We then multiply Eq.(1) by u t to obtain By the second inequality in condition (2), we have where q 2 = 2q − 2 is just the second number as defined in Lemma 3.1. Thus, by the Young inequality, we see from (34) and (35) that Applying Lemma 2.2 to (36), we find Therefore, the needful conclusion follows from (6), (33) and (14) with m = 1, where we use the fact that (14) holds true for m = 1 at least since N ≥ 1.

4.
Asymptotic compactness and global attractors in L 2 . This section proves the asymptotic compactness of the semigroup S(t) in L 2 (R n ). We need to estimate the integral of solutions outside a ball. The following lemma shows that the semigroup S(t) is asymptotically small outside a ball whether the dissipativeness is strong or weak. A half-time decomposition technique is used in the proof.
Proof. We take a continuously differential function ξ : We now take the inner product of Eq.
By (4), the semilinear Laplacian term can be calculated as follows By the condition (2), the nonlinear term can be estimated as follows: All above estimates imply the following differential inequality: Therefore, we obtain the following inequality by applying the classical Gronwall lemma to (43) on the interval [ t 2 , t] (rather than on [0, t], which is different from the strongly dissipative case, see, e.g. [8,9,11,12,14,15]).
We take T sufficiently large, say, T = max{T 0 , T p , T W }, where T 0 , T p , T W are defined as in Lemma 2.1, Corollary 3.2 and Lemma 3.3 respectively. Then we see from (6) that, for all t ≥ T , By (30) and (32), for each t ≥ T , the second term in the righthand side of (44) is less than Finally, since ψ 1 ∈ L 1 and g ∈ L 2 , we have Therefore, we see from (44)-(47) that there exist T := T (ε, B) ≥ max{T 0 , T p , T W } and K := K(ε) > 0 such that, for all k ≥ K, t ≥ T and u 0 ∈ B, |x|≥ √ 2k which finishes the proof of Lemma 4.1.
We obtain the following existence result of global attractors by using Lemma 2.1, Lemma 3.3, Lemma 4.1 and the classical existence theorem of global attractors (see, e.g. [3,24]). Because the following result will be included in Theorem 5.7, so we omit the proof. Remark. To guarantee existence of a global attractor in L 2 (R n ), we require only the force g ∈ L 2 in the strongly dissipative case, but we require at least g ∈ L 3 in the weakly dissipative case.

5.
Bi-spatial attractors for the p-Laplace equation. This section proves the existence of an (L 2 , L r )-attractor for the semigroup S(·) induced by Eq.(1) for every r ∈ [2, q N ]. For this result with r > 2, we need the higher regularity of ψ 1 , i.e. ψ 1 ∈ L ∞ , which is assumed in the literature investigating regularity of attractors. 5.1. Asymptotic estimates in L r . To obtain a bi-spatial attractor with the terminate space L r for each r ∈ [2, q N ], it is necessary to present some asymptotic estimates for the unbounded part of the modulus of solutions in L r . We start at the following auxiliary lemma, which presents the asymptotic estimate in L 2 .
where the subscript |u| ≥ M denotes the set {x ∈ R n ; |u(x)| ≥ M } and m(·) denotes the Lebesgue measure.
Proof. By Lemma 2.1, we know that, for all t ≥ T 0 and u 0 ∈ B, For each ε > 0, if we choose M large enough such that M ≥ C 0 /ε, then m(R n (|u(t, u 0 )| ≥ M ) < ε for all t ≥ T 0 and u 0 ∈ B, which proves (49).
To proves (50), we define a decreasing family of subsets of L 2 (when T increases) as follows: For each ε > 0, by Lemma 4.1, there are T > 0 and R > 0 such that |x|≥R w 2 (x)dx < ε 2 16 for all w ∈ K(T )(⊂ L 2 (R n )).
By Lemma 3.3 and the compact embedding W 1,p (Q R ) → L 2 (Q R ), we know the restricted set K R (T ) of K(T ) is pre-compact in L 2 (Q R ) for some larger T , where Q R = {x ∈ R n ; |x| < R} and the restricted set is defined by Therefore, K R (T ) has a finite ε/4-net in L 2 (Q R ), which together with (53) implies that K(T ) has a finite ε-net in L 2 (R n ) with the centers w i ∈ L 2 (R n ) (i = 1, 2, ..., k). We then choose a δ > 0 such that, for any set e ⊂ R n with m(e) < δ, By (49), we can choose an M such that m(|w| ≥ M )) < δ for all w ∈ K(T ) with a larger T . Given now w ∈ K(T ), there is a center w i ∈ K(T ) such that w −w i ≤ ε. Therefore, by (55), we have which proves (50).
The following lemma shows that the nonlinear function f (x, s) (given by (2)) keeps in plus or minus sign if |s| is large enough.
Therefore, by the first formulation in (2), we have for a.e. x ∈ R n and for all which implies (56) immediately.
The following lemma is a key to prove the asymptotic compactness in L r for r ∈ [2, q N ].  (57) Proof. The case r = 2 has been proved by Lemma 5.1 and thus, by Lemma 2.3, it suffices to prove (57) with r = q N . To this end, we will prove successively about m ∈ {1, 2, · · · , N } that, for each ε > 0, there exist both increasing sequences {T m } and {M m } such that, for an intrinsical constant C, where q m = m(q − 2) + 2, m ∈ {1, 2, · · · , N } and v + = max{v, 0}. We only consider here the positive part since the negative part is similar. We start at proving the case of m = 1. Since g ∈ L 2 ∩ L N +1 , by Lemma 2.3, for ε > 0, we can take δ > 0 such that, for any set e ⊂ R n with m(e) < δ, e (g 2 + |g| 3 + · · · + |g| N +1 )dx < ε.

5.2.
Abstract results on bi-spatial attractors. This subsection recalls some concepts and results about bi-spatial attractors from [15]. Let S(·) be a semigroup on a Banach space X. Assume the semigroup takes its values (for strictly positive times) in another Banach space Y in the following sense: We also assume that the pair (X, Y ) of Banach spaces has the following limitidentical property: Definition 5.4. Let S(·) be a semigroup on X taking its values in Y in the sense of (81). (83) Definition 5.5. We say that a semigroup S(·) has an absorbing set B if for every bounded set B ⊂ X there is a T := T (B) such that S(t)B ⊂ B for all t ≥ T . We say a semigroup S(·) is asymptotically compact in X (resp. in Y ) if {S(t n )u n } is pre-compact in X (resp. in Y ) whenever t n → ∞ and {u n } is bounded in X.
The following existence result of bi-spatial attractors is the weaker form of [15, Theorem 3.1] restricted on the deterministic case (the original theorem discussed random dynamical systems). Proposition 1. Let (X, Y ) be a limit-identical pair of Banach spaces. Let S(·) be a continuous semigroup on X but take its values in Y in the sense of (81). Assume further (i) S(·) has an absorbing set B, which is bounded in X; (ii) S(·) is asymptotically compact in X; (iii) S(·) is asymptotically compact in Y . Then S(·) has a unique (X, Y )-attractor A, it is just the unique global (X, X)attractor (as the same set).

5.3.
Bi-spatial attractors for p-Laplacian equations. By using the above abstract result, we obtain the following existence theorem of bi-spatial attractors for Theorem 5.6. Suppose (2) holds and ψ 1 ∈ L 1 ∩ L ∞ , g ∈ L 2 ∩ L N +1 . Then for each r ∈ [2, q N ] there is a unique (L 2 (R n ), L r (R n ))-attractor A r for the semigroup S(·) generated by Eq.(1) no matter how dissipative the equation is. Furthermore, all attractors A r ,r ∈ [2, q N ] are the same set.
Proof. We first explain that (L 2 (R n ), L r (R n )) is a limit-identical pair, although L 2 (R n ) and L r (R n ) may not be contained in any direction. Both L 2 (R n ) and L r (R n ) are continuously embedded into the distribution space D (R n ) and thus the limit-identical property follows from uniqueness of limits (see [31,Lemma 2.7]). We then see from Lemma 3.1 that S(·) takes its values in L r for all positive times. We see also from Lemma 2.1 that the ball B(0, C 0 ) is a bounded absorbing set in 1952 YANGRONG LI AND JINYAN YIN L 2 . By a standard argument, one can derive easily the asymptotic compactness in L 2 (R n ) from Lemma 3.3 and Lemma 4.1.
Next, we prove the asymptotic compactness of the semigroup in L r for each fixed r ∈ (2, q N ]. It suffices to prove that the sequence {u i } := {u(t i , u 0,i )} is pre-compact in L r whenever t i → +∞ and {u 0,i } is a bounded sequence in L 2 . To this end, we see from Lemma 5.3 that, for each ε > 0, there are I 1 ∈ N and M > 0 such that |ui|≥M |u i (x)| r dx < ε r for all i ≥ I 1 . (84) On the other hand, by the foregoing proof, the sequence {u i } has a convergent subsequence (denoted by itself) in L 2 . Hence the subsequence {u i } is a Cauchy sequence in L 2 , which implies that there is an integer I 2 ≥ I 1 such that, for all where M is the constant given in (84). According to the pair (i 1 , i 2 ) of integers, we decompose the domain R n into four parts R n = 4 j=1 D j as follows: All above estimates imply that u i1 − u i2 r ≤ 4ε if i 1 , i 2 ≥ I 2 . This means that {u i } has a Cauchy subsequence in L r (R n ) and thus the original sequence {u i } is pre-compact in L r as required. So far, we have verified all conditions in Proposition 5.6. Therefore the semigroup S(·) has a unique (L 2 (R n ), L r (R n ))-attractor A r and all these attractors A r are the same set for different r ∈ [2, q N ].
6. Approximation of attractors. This section considers the question whether the obtained attractor in Theorem 5.7 is approximated by the corresponding attractor for Eq.(1) inside a ball. It is necessary to give an additional assumption: the function f (·, 0) − g(·) has a compact support in R n , more precisely, there is a positive number R f,g such that f (x, 0) = g(x) for a.e. |x| ≥ R f,g .
In the sequel, one may regard a set K ⊂ L 2 (Q R ) as a subset of L 2 (R n ) by nullexpanding, where Q R = {x ∈ R n ; |x| < R} and the null-expansionK of K is defined byK It is obvious thatK ⊂ L r (R n ) if K ⊂ L r (Q R ) for r ≥ 2.
Theorem 6.1. Suppose (2), (91) hold and where R f,g is given in (91). The family A R is also lower semi-continuous to A ∞ under any r-norm, r ∈ [2, q N ], that is, Furthermore, we have the following construction of A ∞ : where the closure is taken in the topology of L r (R n ) and each A R is regarded as its null-expansion to the entire Euclid space.
Proof. Let S ∞ (·) be the semigroup generated by Eq.(1) in L 2 (R n ) and let S R (·) be the semigroup generated by Eq.(1) in L 2 (Q R ) (0 < R < ∞) with the Dirichelt boundary condition: We first explain that, if R ≥ R f,g , L 2 (Q R ) is a positively invariant subspace (in the null-expanding sense) under the operator S ∞ (t) : L 2 (R n ) → L 2 (R n ) for each t ≥ 0. Indeed, suppose u 0,R ∈ L 2 (Q R ) and let u R (·, u 0,R ) denote the solution of Eq.(1) defined on Q R which satisfies (96) . Then it is easy from the assumption (91) to verify that the null-expansionũ R (·, u 0,R ) of u R (·, u 0,R ) is a solution of Eq.(1) defined on R n such thatũ R (0, u 0,R ) =ũ 0,R , the null-expansion of u 0,R . Then the uniqueness of solutions implies that S ∞ (t)ũ 0,R =ũ R (t, u 0,R ) for each t ≥ 0, which proves S ∞ (t)L 2 (Q R ) ⊂ L 2 (Q R ) as required. We then prove the upper semi-continuity (93). LetÃ R be the null-expansion of A R as defined by (92). It is easy to see that both sets have the same (maximal) norms Ã R L 2 (R n ) = A R L 2 (Q R ) , which implies thatÃ R is a bounded set in L 2 (R n ) since the attractor A R is bounded in L 2 (Q R ). We will prove thatÃ R is (strictly) invariant under the operator S ∞ (t) for each t ≥ 0 and R ≥ R f,g . Indeed, ifũ 0 ∈Ã R , then there is a u 0,R ∈ A R such thatũ 0 (x) = u 0,R (x) for all x ∈ Q R . Then, by the invariance of A R under S R (t), we have w := S R (t)u 0,R ∈ A R . Letw be the nullexpansion of w. By the forgoing proof, L 2 (Q R ) is an invariant subspace under S ∞ (t), then we have S ∞ (t)ũ 0 =ũ(t, u 0,R ) =w ∈Ã R and so S ∞ (t)Ã R ⊂Ã R . Conversely, by the invariance of A R again, for each u 0,R ∈ A R , there is a v ∈ A R such that S R (t)v = u 0,R . By null-expanding, we have S ∞ (t)ṽ =ũ 0,R , which means thatũ 0,R ∈ S ∞ (t)Ã R and thusÃ R ⊂ S ∞ (t)Ã R for all t ≥ 0. We have proved that A R is indeed an invariant set under S ∞ (·) for each R ≥ R f,g . Therefore, we see from the attraction of A ∞ that, for all R ≥ R f,g , which implies that the strictly upper semi-continuity in L 2 (Q R ): In addition, we see from (98) that the strictly upper semi-continuity holds for all r-norms with r ∈ [2, q N ] (since by Theorem 5.7 A R ⊂ L r (Q R )), that is, To prove the lower semi-continuity, we need to prove A R is the same set as the restriction A ∞,R of A ∞ on L 2 (Q R ) for each R ≥ R f,g , where the restricted set is defined by By the same method as above, one can prove that each . It is obviously that A ∞,R ≤ A ∞ and so A ∞,R is a bounded set in L 2 (Q R ). Hence we see from the attraction of A R that dist(A ∞,R , A R ) = dist(S R (t)A ∞,R , A R ) → 0 as t → ∞, which implies that A ∞,R ⊂ A R . On the other hand, we have proved in (98) that the null-expansionÃ R ⊂ A ∞ . Hence, by considering their restrictions in both sides of the included relationship, we have A R ⊂ A ∞,R and thus A R = A ∞,R for all R ≥ R f,g . By this fact, we can calculate the Hausdorff semi-distance under the topology of L 2 (R n ) between A ∞ andÃ R (R ≥ R f,g ) as follows: whereũ R (x) = u(x) if |x| < R andũ R (x) = 0 if |x| ≥ R and we use the fact A R = A ∞,R in the last step. Since A ∞ is bounded and invariant in L 2 (R n ), it follows from Lemma 4.1 that for each ε > 0 there are R 0 ≥ R f,g and T > 0 such that, for all R > R 0 , Both (102) and (103) imply the lower semi-continuity in L 2 (R n ): where A R is regarded as its null-expansion. By [3, Lemma 3.2 (ii)], we see from (104) that for each u ∈ A ∞ there is a u R ∈ A R for each R ∈ [R f,g , +∞) such that u R → u in L 2 (R n ) as R → ∞. Hence A ∞ is included into the closure of the union of all A R over all R ∈ [R f,g , ∞), which together with (98) implies (95) for r = 2, i.e.
Finally, we prove that both (94) and (95) hold true under r-norms for each r ∈ (2, q N ]. To this end, let ε > 0 be arbitrarily small. Since A ∞ is bounded in L 2 (R n ) and invariant under S ∞ (t), we see from Lemma 5.3 that there is an M such that, for some T large enough, For such a number M , by (103), one can choose a R ≥ R f,g such that for all Then we obtain from (106) and (107) that for all R > R and u ∈ A ∞ , Considering the Hausdorff semi-distance of r-norm, by a similar estimate as (102), we have for all R > R Therefore A R is lower semi-continuous under r-norm for any r ∈ (2, q N ], which proves (94). By the lower semi-continuity under r-norm, we see from [3, Lemma 3.2 (ii)] that A ∞ ⊂ cl r ( R f,g ≤R<∞ A R ), which together with (98) implies that (95) holds true for any r ∈ [2, q N ] and finishes the proof.

Remark.
It remains open whether there is a random attractor even in L 2 (R n ) for stochastic p-Laplacian equations in the weakly dissipative case (q < p). In the stochastic case, one may not derive the uniform absorption as given in (30) over a large interval [t/2, t] since the sample is varied in a cocycle. The half-time decomposition technique may lose its efficacy in the stochastic case. But our method may be effective for deterministic non-autonomous systems provided the non-autonomous force satisfies some additional conditions.