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

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.

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 [5-7]. 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 [8-10], 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

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

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

(H2) , where is the set of all translation compact functions in whose definition is given in Definition 1.1 below.

(H3) and .

Let us give some comments about assumptions (H1)-(H3). The nonlinearity f is assumed to have a polynomial growth and to satisfy a standard dissipative condition. A typical example of functions satisfying conditions (H1) 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 . While (H3) is a technical condition ensuring that is embedded compactly into L²(Ω), where 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 [15-19]. 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 [8-10,20] 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, for all t > τ, u ∈ ℝ, or a weaker assumption

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 , we define the weighted space

equipped with the norm

It is easy to check that the dual space of is the space , where p' is defined by . Moreover, we define the weighted Sobolev space as the closure of in the norm

As 1 < p < ∞, is reflexive, and the dual space of will be denoted by .

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 ,

where p, q, γ, δ are related by

and δq < N, γp < N.

The inequality (1.5) implies that the embedding

This implies, by duality,

It is pointed out in [19] that

for every p, q, γ, δ satisfying with γ ≤ δ ≤ γ + 1 and δq < N, γ p < N.

From assumption (H3), it is easy to check that there exists a positive number δ such that compactly. Since the embedding is continuous, it is seen that is an evolution triplet.

We now define the following "evolution" spaces which will be useful in what follows.

endowed with the norm

The dual space of is .

Putting

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 -Δ_p,γ maps into its dual . Moreover,

(1) -Δ_p,γ is hemicontinuous, i.e., for all , the map λ ↦ 〈-Δ _p,γ(u + λv), w〉 is continuous from ℝ to ℝ.

(2) -Δ_p,γ is monotone, i.e., 〈-Δ_p,γu + Δ_p,γv,u - v〉 ≥ 0, for all .

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

(1) A function is said to be translation bounded if

(2) A function is said to be translation compact if the closure of {φ(⋅ + h)|h ∈ ℝ} is compact in .

Denote by and the sets of all translation bounded functions and of all translation compact functions in , respectively. It is well-known (see [4]) that .

Let be the closure of the set {g(· + h)|h ∈ ℝ} in . The following results were proved in [[1], Chapter 5, Proposition 3.4].

Lemma 1.3. (1) is compact.

(2) For all ;

(3) The translation group {T(h)}, which is defined by T(h)σ(s) = σ(h + s), s, h ∈ ℝ, is continuous on ;

(4) ;

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.

We denote

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

and

for all test functions φ ∈ V.

It is known (see [[1], Theorem 1.8, p. 33]) that if u ∈ V and , then u ∈ C([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

where is a basis of , which is orthonormal in L²(Ω). We get u_n from solving the problem

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

Step 2: A priori estimates. We have

By assumption (H3), we can choose δ > 0 such that , then and therefore there exists λ > 0 such that

where the last inequality follows from the Young inequality. Using (1.3) and the Cauchy inequality, we get

Hence

We show that the local solution u_n can be extended to the interval [τ, ∞). Indeed, from (2.2) we have

By the Gronwall inequality, we obtain

where we have used the facts that and

We now establish some a priori estimates for u_n. Integrating (2.2) on [τ, T], τ < t ≤ T, and using the fact that , we have

The last inequality implies that

Using hypothesis (1.2), we get

Hence, we can conclude that {f(t, u_n)} is bounded in and thus,

We have

for all , where we have used the Hölder inequality. Because of the boundedness of {u_n} in , we infer that {-Δ_p,γ u_n} is bounded in .

Step 3: Passing limits. From the above estimates, there exists a subsequence {u_μ} ⊂ {u_n} such that

up to a subsequence.

To prove that η(t) = f(t, u(t)), we argue similarly to [22,23] to deduce that

for all T > τ. In particular, we obtain from (2.5) that

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

Thus, we have

We now show that ψ = -Δ_{p, γ} u. Since -Δ_{p, γ} is monotone, we have

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

we obtain

where we have used the facts that u_n(τ) → u_τ in . On the other hand, by integrating by parts, from (2.14) we have

and therefore thanks to (2.15) and (2.16) one gets

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 , we obtain

hence

leting λ → 0 in (2.17), we conclude that

So ψ = -Δ_p,γ u. Thus,

We now show that u(τ) = u_τ. Choosing some with φ(T) = 0, observe that φ ∈ V, by the Lebesgue dominated theorem, one can check that

Doing the same in the Galerkin approximations yields

Passing to the limit as n → ∞, we have

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, +∞).

3.1. 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

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

(1) for arbitrary w ∈ E;

(2) for all w ∈ E, t₁, t₂ ∈ ℝ⁺, where G (t, B) = ∪_x∈B G (t, x), B ⊂ E.

It is called a strict multi-valued semiflow if , for all w ∈ E, t₁, t₂ ∈ ℝ⁺.

We now consider problem (1.1) with τ = 0. By Theorem 2.1, we construct a multi-valued mapping as follows

Lemma 3.1. is a strict multi-valued semiflow in the sense of Definition 3.1.

Proof. Assume that , 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, . It remains to show that . If then ξ = v(t₁), where . One can suppose that v(0) = u(t₂), where u(0) = u₀. Set

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 .

Definition 3.2. [5]A set is said to be a global attractor of the multi-valued semiflow if the following conditions hold:

• is an attracting, i.e., as t → ∞ for all bounded subsets B ⊂ E,

• is negatively semi-invariant: for arbitrary t ≥ 0,

• If ℬ is an attracting of then ,

where is the Hausdorff semi-distance.

The following theorem gives the sufficient conditions for the existence of a global attractor for the multi-valued semiflow .

Theorem 3.2. [5,7]Suppose that the strict multi-valued semiflow has the following properties:

(1) is pointwise dissipative, i.e., there exists K > 0 such that for one has ∥u(t)∥_E ≤ K if t ≥ t₀ (∥u₀∥_E);

(2) is a closed map for any t ≥ 0, i.e., if ξ_n → ξ, η_n → η, then ;

(3) is asymptotically upper semicompact, i.e., if B is a bounded set in E such that for some is bounded, any sequence with t_n → ∞ is precompact in E.

Then has a compact global attractor in E. Moreover, is invariant, i.e., for any t ≥ 0.

Lemma 3.3. 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 (H1)-(H3), where f andg are assumed to be independent of time t, the strict multi-valued semiflow 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 , we have

Noting that

we have

Therefore

Hence one can deduce that is pointwise dissipative.

We now check hypothesis (2) of Theorem 3.2. Assume that in L²(Ω). Then there exists a sequence {u_n} such that

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

• u_n → u in L²(Q_0,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 ,

• in V,

• -Δ_p,γ u_n ⇀ -Δ_p,γ u in ,

up to a subsequence. Hence, passing to the limit in the equality

we conclude that u(t) is a weak solution of (1.1) with the initial condition u(0) = η. Thus, .

For hypothesis (3), one observes that for n large enough,

where t* > 0 and B* is a bounded set in L²(Ω). Using Lemma 3.3, we conclude that, if , 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)}_n∈N in an appropriate topological space Σ. The family of such symbols is said to be the hull of σ in Σ and is denoted by , i.e.,

If the hull is a compact set in Σ, we say that σ is translation compact in Σ.

Denote ℝ_d = {(t, τ) ∈ ℝ² | τ ≤ t}. Let X be a complete metric space, and 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

Then Z is a Banach space. We say that f_n → f in the space C(ℝ; Z) if

for all t ∈ ℝ, r > 0.

Let , and

where the topology in is equipped by the local weak convergence, i.e., g_n → g in if

for all t ∈ ℝ, r > 0 and ϕ ∈ L² (Q_t,t+r). We define .

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 (H2) ensures that g is translation compact in (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 f ∈ C(ℝ; Z) is translation compact if and only if for all R > 0 one has

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

(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 , one sees that Σ is a compact set in . 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 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 x ∈ X, t, s, τ ∈ ℝ,τ ≤ s ≤ t.

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

We denote by the set of all global weak solutions (defined for all t ≥ τ) 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

Lemma 3.6. U_σ (t, τ, u_τ) is a multi-valued semiprocess. Moreover,

Proof. Given z ∈ U_σ(t, τ, u_τ)) we have to prove that z ∈ U_σ(t, s, U_σ(s, τ, u_τ)). Take 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 t ≥ s we have that z(s) = y(s) and obviously . Consequently, z(t) ∈ U_σ(t, s, U_σ(s, τ, u_τ)).

Let z ∈ U_σ(t + s, τ + s, u_τ). Then there exists such that z = u(t + s) and , so that z = v(t) ∈ u_{τ,T (s)σ} (u_τ).

Conversely, if z ∈ U_τ,T(s)σ (u_τ), then there is such that z = u(t) and so that z = v(t + s) ∈ U_σ(t + s, τ + s, u_τ).

Denote by

Definition 3.4. [6]A set is called a uniform global attractor for the family of multi-valued semiprocesses U_Σ if:

(1) it is negatively semiinvariant, i.e., for all t ≥ τ;

(2) it is uniformly attracting, i.e., , as t → ∞ , for all and τ ∈ ℝ;

(3) for any closed uniformly attracting set Y, we have (minimality).

Theorem 3.7. [[6], Theorem 2] Suppose that the family of multi-valued semiprocesses U_Σ 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, x ∈ X, we have

(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 U_Σ has a uniform global compact attractor .

The following is the key point of this subsection.

Lemma 3.8. Let conditions (H1)-(H3) hold and let {u_n}_n∈ℕ is a sequence of weak solutions of (1.1) with respect to the sequence of symbols {σ_n} ⊂ Σ such that

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 (H1) for all t ∈ ℝ and , one sees that f_n also satisfies (H1). On the other hand, noting that {u_n(τ)} is bounded in L²(Ω) and . Thus, repeating the arguments in the proof of Theorem 2.1, we obtain that

In particular, we have

up to a subsequence. Let 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

for all v ∈ V. Since in L²(τ,T; L²(Ω)), it remains to prove that in . We first show that in . Indeed,

because in Z and {u_n} is bounded in L^q(Q_τ,T). On the other hand, since is bounded in , by using Lemma 1.3 in [[21], Chapter 1] and the continuity of as in the proof of Theorem 2.1, we can conclude that weakly in . Hence, we have

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 .

Putting

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

Indeed,

and

as n → ∞ since u_n → u strongly in L²(Q_τ,t) and {g_n} is bounded in L²(Q_τ,t). In addition,

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,

So

for n ≥ n₀(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 (H1)-(H3) hold. Then the family of multi-valued semipro-cesses {U_σ (t, τ)} has a uniform global compact attractor .

Proof. We know that each symbol σ_n = (f_n, g_n) ∈ Σ satisfies the same conditions as in (H1)-(H2). Furthermore, since , we have . Hence if u_n is a weak solution of (1.1) with respect to the symbol σ_n, one has

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

that is, , for all t ≥ T₀(τ, R). Thus, {U_σ(t, τ)} fulfills condition (3) in Theorem 3.7.

We now define the set . Lemma 3.8 implies that K is compact. Moreover, since is an absorbing set, we have

for all , and t ≥ T₀(τ, B_R). It follows that any sequence {ξ_n} such that , 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 t ≥ τ. Suppose that it is not true, that is, there exist , and such that . But Lemma 3.8 implies (up to a subsequence) that , which is a contracdition. Thus, the existence of the uniform global compact attractor follows then from Theorem 3.7.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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.

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