Long-time dynamics of the parabolic $p$-Laplacian equation

In this paper, we study the long-time behaviour of solutions of Cauchy problem for the parabolic $p$-Laplacian equation with variable coefficients. Under mild conditions on the coefficient of the principal part and without upper growth restriction on the source function, we prove that this problem possesses a compact and invariant global attractor in $L^{2}(R^{n})$.


Introduction
The main goal of this paper is to discuss the long-time behaviour (in the terms of attractors) of the solutions for the following equation with the initial data u(0, x) = u 0 (x), (1.2) where p ≥ 2, g ∈ L 2 (R n ), u 0 ∈ L 2 (R n ), n ≥ 2. Here, the functions σ, β and f satisfy the following assumptions: (1.5) The understanding of the long-time behaviour of dynamical systems is one of the most important problems of modern mathematics. One way of approaching to this problem is to analyse the existence of the global attractor. The existence of the global attractors for the parabolic equations has extensively been studied by many authors. We refer to [1][2][3][4][5][6][7] and the references therein for the reaction-diffusion equations and to [3,[8][9][10][11][12][13][14][15] for the evolution p-Laplacian equations. When σ(x) ≡ 1, β(x) ≡ λ, the existence of the global attractor for equation (1.1) was studied in [3,[8][9][10][11] for bounded domains and in [12][13][14][15][16] for unbounded domains.
In this paper we deal with the equation (1.1) which contains the variable coefficients σ(·) and β(·). This type of equations have recently taken an interest by several authors. In [17], for the case β(x) ≡ 0, the authors have shown the existence of the global attractor for equation (1.1) in a bounded domain. In that paper the diffusion coefficient σ(·) is assumed to be like |x| α for α ∈ (0, p) and due to the studying in a bounded domain the authors prove the asymptotic compactness property of the solutions by using the compact embeddings of Sobolev Spaces. The existence of the global attractor for equation (1.1), under the assumption σ(x) ∼ |x| α + |x| γ , α ∈ (0, p), γ > p + n 2 (p − 2), has been shown in [18]. Although the authors in [18] have studied the problem in an arbitrary domain, the compact embeddings could also be used to obtain the asymptotic compactness of solutions because of the conditions imposed on σ(·).
The main novelty in our paper is the following: (i) we weaken the conditions on the function σ(·) which are given in [17] and [18], so that the embedding of the space with the norm ∇u L p σ (Ω) + u L 2 (Ω) into the space L 2 (Ω) is not compact, for each subdomain Ω ⊂ R n ; (ii) we remove the upper growth condition on the source term.
The absence of the upper growth condition on f and the lack of the compact embedding cause some difficulties for the existence of the solutions and the asymptotic compactness of the solution operator in L 2 (R n ). We prove the existence of the solutions by Galerkin's method and to overcome the difficulties related to the limit transition in the source term f, we apply the weak compactness theorem in the Orlicz spaces. To prove the asymptotic compactness of the solutions, we first establish the validity of the energy equalities by using the approximation of the weak solutions by the bounded functions and then apply the approach of [19] by using the weak compactness argument.
Our main result is as follows: The present paper is organized as follows. In the next section, we give some definitions and lemmas which will be used in the following sections. In section 3, the well-posedness of problem (1.1)-(1.2) is proved. In section 4, we show the existence of the absorbing set and present the proof of the asymptotic compactness to establish our main result.

Preliminaries
This section is devoted to give some definitions and lemmas which will be used in the next sections. In order to study problem (1.1)-(1.2), let us begin with the introduction of the spaces W and W b .
, we define the spaces W and W b as the closure of C ∞ 0 (R n ) in the following norms respectively, One can show that W is a separable, reflexive Banach space and W b is a separable Banach space. Now, before giving the definition of the weak solution of problem (1.1)-(1.2), let us define the operator A : W → W * as Aϕ = −div(σ(x) |∇ϕ| p−2 ∇ϕ) + β(x)ϕ, where W * is the dual of W . It is easy to show that the operator A : W → W * is bounded, monotone and hemicontinuous.
, is called the weak solution to problem (1.1)-(1.2), where ·, · is the dual form between W and W * .
Remark 2.1. To give a meaning to the third term on the left hand side of the equality given in the definition, it is enough to see that f (u) ∈ L 1 (0, T ; L 1 (R n )) + L 2 (0, T ; L 2 (R n )). Let χ Ω1 and χ Ω2 be the characteristic functions of the sets Taking into account that u ∈ L ∞ (0; T ; L 2 (R n )), we get f (u)χ Ω2 ∈ L 2 (0, T ; L 2 (R n )). Since .
is satisfied for all u ∈ C ∞ 0 (R n ) and r > 0, where B(0, r) = {x : x ∈ R n , |x| < r} and the positive constant C depends on r and n.
By adding u 2 L 2 (R n \B(0,r)) to the both sides of the above inequality we get the claim of the lemma.
Proof. The Holder inequality yields , for every r > 0. From the assumption (1.3) it follows that On the other hand by condition (1.4), we have Taking into account the previous lemma and (2.4) -(2.5), we obtain the result.
Proof. By the definition of W, for any u ∈ W there exists a sequence which according to the Lemma 2.2 yields Now, let us show that lim for every measurable E ⊂ R n . By the last equality, we find Then by Egorov's theorem for any δ > 0 there exists a measurable E δ ⊂ B(0, r) such that mes(E δ ) < δ and By (2.10) and (2.12), we get By the same way, one can show that every subsequence of {u m } ∞ m=1 has a subsequence satisfying the above equality. So, we have By (2.9), (2.11) and (2.13), we obtain (2.8). Now, let us show that for any By (1.4) and (2.15), we obtain lim sup applying Lebesgue's convergence theorem, we get which together with (2.16) yields (2.14). By (2.8) and (2.14), for every u ∈ W and k ∈ N there Now, using the argument done in the proof of (2 Remark 2.2. By (2.8) and (2.14), for every u ∈ W and k ∈ N there exists a sequence On the other hand, by the definition of

Well-posedness
We prove the existence of the weak solution to problem (1.1)-(1.2) by Galerkin's method. Proof. Let us consider the approximate solutions is a basis of the space W b and the functions {c mk (t)} m k=1 are the solutions of the following problem : Since by the last equality where F (u) = u 0 f (s)ds. Integrating the last inequality over (0, T ) with respect to the variable s and taking into account (3.4), we get for every ε ∈ (0, T ). By the estimates (3.3)-(3.5) and the boundedness of the operator A : as m → ∞, for some χ ∈ L Since by (2.4) and (3.4), the sequence {u m } is bounded in L p (0, T ; W 1, 2n n+2 loc (R n )), using (3.5) and Aubin type compact embedding theorem (see [21,Corollary 4]), we have the compactness of {u m } in L 1 (ε, T ; L 1 loc (R n )) for every ε ∈ (0, T ). Hence there exists subsequences {u and ε k ց 0 such that u (k) mn → u a.e. on (ε k , T ) × B(0, k), as n → ∞. Now, applying the diagonalization procedure, we obtain (up to a subsequence {u as m → ∞. Now, because the sign of the function f (u) is the same as the sign of u, together with (3.4), it follows that where u + m = max{u m , 0} and u − m = min{u m , 0}. Since the function f (·) is continuous, increasing, positive for x > 0 and f (0) = 0, we can define an N -function (see [22] for definition) which has a complementary N -function G as follows: By definition of G(·) and (3.9) 1 , we get and consequently we obtain for every k ∈ N, where L * G ((0, T ) × B(0, k)) is the Orlicz space (see [22] for definition). On the other hand, defining g(s) = − f −1 (−s) for s > 0, we can construct a new N -function Φ such as Φ(y) = |y| 0 g(ξ)dξ.
Choosing y = − f (u − m ) and taking into account (3.9) 2 , we get and consequently for every k ∈ N. By using (3.8), continuity of f (·) and the functions max{s, 0} and min{s, 0}, it can be inferred that for every k ∈ N, where Ψ is the complementary N -function to Φ and E F , E Ψ are the closures of the set of bounded functions in the spaces L * F ((0, T ) × B(0, k)) and L * Ψ ((0, T ) × B(0, k)), respectively. The last two approximations together with (3.6) 1 yield that Taking into account (3.8) and applying Fatou's lemma to (3.17), we obtain T 0 R n f (u(t, x))u(t, x))dxdt ≤ c 2 .
As it was mentioned in the Remark 2.1, the last inequality gives us that f (u) ∈ L 1 (0, T ; L 1 (R n ) + L 2 (R n )).

Now, since the operator
Since u m (0) → u 0 in L 2 (R n ), taking into account (3.6) and (3.8), and applying Fatou's lemma, we find lim sup On the other hand, by Remark 2.1 and Lemma 2.3, we can test (3.18) by B k (u) on (ε, T ) × R n , which gives us . Taking into account Lemma 2.3 and passing to the limit as k → ∞, in the last equality, we obtain Since, the sequence {f (u(t, x))B k (u)(t, x)} ∞ k=1 is non-decreasing and B k (u) → u in C([0, T ]; L 2 (R n )), by monotone convergence theorem, we have Since u ∈ C([0, T ]; L 2 (R n )), passing to the limit in the last equality as ε → 0, we get . Taking into account the last equality in (3.20), we obtain χ = Au, which completes the proof of the existence of the solution.  1)-(1.2), with initial data u 0 and v 0 , respectively. Then where c is the same constant in (1.5).
Proof. Denoting w = u − v, we have Testing (3.23) 1 by B k (w) on (ε, T ) × R n and taking into account the monotonicity of the function f , we get By the definition of B k (·) and monotonicity of the function s p−1 for s ≥ 0, we have By the last two inequalities, we find Passing to the limit as k → ∞ and ε → 0 in the above inequality and taking into account (3.23) 2 , we obtain which by Gronwall's lemma yields (3.22).

Existence of the global attractor
We begin with the existence of the absorbing set for the semigroup {S(t)} t≥0 .
Theorem 4.1. Assume that the conditions (1.3)-(1.5) are satisfied. Then the semigroup {S(t)} t≥0 has a bounded absorbing set in L 2 (R n ), that is, there is a bounded set B 0 in L 2 (R n ) such that for any bounded subset B of L 2 (R n ), there exists a T 0 = T 0 (B) > 0 such that S(t)B ⊂ B 0 for every t ≥ T 0 .
Proof. Multiplying the equation (3.1) j by the function c mj (t), for each j, adding these relations for j = 1, ..., m, we get the following equality : by taking into account Lemma 2.2 in (4.1), we obtain for some c 1 > 0 and c 2 > 0. By (3.6) 1 and (3.6) 2 , we have u m → u weakly in C([ε, T ]; L 2 (R n )), ∀T > ε > 0, which yields u m (t) → u(t) weakly in L 2 (R n ), ∀t > 0. On the other hand, since u m (0) → u 0 strongly in L 2 (R n ), passing to the limit in the last inequality we find that is an absorbing set for {S(t)} t≥0 . Now, let's prove the asymptotic compactness property of the semigroup {S(t)} t≥0 .
These estimates can be justified by using Galerkin's approximation as it was done in the previous section. So, repeating the argument done in the proof of Theorem 3.1, for the subsequence of u k , without changing the name of it, we have (4.4) where χ ∈ L p p−1 (0, T ; W * ), w ∈ L ∞ (0, T ; L 2 (R n )) ∩ L p (0, T ; W ) ∩ W 1,2 (ε, T ; L 2 (R n )) and T 0 R n f (w(t, x))w(t, x)dxdt < ∞. Now, putting u k instead of u in (1.1) and passing to the limit, we find w t + χ + f (w) = g. Taking into account Remark 2.1 and Lemma 2.3, and testing the above equation by B k (w) on (s, T ) × R n , we obtain , ∀T ≥ s > 0. Again repeating the argument done in the proof of Theorem 3.1, passing to the limit as k → ∞ and integrating the obtained equality from ε to T with respect to s, we get Now, putting u k instead of u in (1.1), testing this equation by B m (u k ) on (s, T )×R n , integrating the obtained equality from ε to T with respect to s and passing to the limit as m → ∞, we obtain for every subsequence {m k } ∞ k=1 . Now, using the argument done at the end of the proof of [19,Lemma 3.4], let us show that the sequence {S(t m )ϕ m } ∞ m=1 is relatively compact in L 2 (R n ). If not, then there exists ε 0 > 0 such that the set {S(t m )ϕ m } ∞ m=1 has no finite ε 0 -net in L 2 (R n ). This means that there exists a subsequence {m k } ∞ k=1 , such that S(t mi )ϕ mi − S(t m k )ϕ m k L 2 (R n ) ≥ ε 0 , i = k.