Abstract
Using the theory of uniform global attractors for multivalued semiprocesses, we prove the existence of attractors for quasilinear parabolic equations related to CaffarelliKohn Nirenberg inequalities, in which the conditions imposed on the nonlinearity provide the global existence of weak solutions but not uniqueness, in both autonomous and nonautonomous cases.
Mathematics Subject Classification 2010: 35B41, 35K65, 35D30.
Keywords:
CaffarelliKohnNirenberg inequalities; nonuniqueness; weak solution; multivalued semiflow; multivalued semiprocess; compact attractor; compactness and monotonicity methods1. 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 nonautonomous 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 NavierStokes equation in three space dimension, the GinzburgLandau 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 multivalued semigroups.
In the last years, there have been some theories for which one can treat multivalued semiflows and their asymptotic behavior, including the generalized semiflows theory of Ball [3], theory of trajectory attractors of Chepyzhov and Vishik [4] and theories of multivalued 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 phasefield equation [11], the wave equation [12], the threedimensional NavierStokes 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 }∈ L^{2}(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 C_{1}, C_{2}, k_{1}, k_{2 }are positive constants;
(H2) , where is the set of all translation compact functions in whose definition is given in Definition 1.1 below.
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^{q2}u. 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 L^{2}(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 CaffarelliKohnNirenberg 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 pLaplacian 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 longtime behavior of solutions to problem (1.1).
In this article we study the longtime behavior of solutions to problem (1.1) via the concept of uniform global attractors for multivalued 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 longtime behavior of solutions, in order to handle nonuniqueness of solutions, we need use the theory of attractors for multivalued semiprocesses. Following the general lines of the approach used in [810,20] for nondegenerate parabolic equations, we prove the existence of a global compact attractor in the autonomous case, and of a uniform global compact attractor in the nonautonomous 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 nondegenerate case, namely when γ = 0 and p = 2, was studied in [8,9]. Thus, our results extend some known results on the existence and longtime 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 singlevalued one and the uniform compact global attractor is exactly the usual uniform attractor for the family of singlevalued semiprocesses [1].
In the rest of this section, for convenience of the reader, we recall some results on function spaces related to CaffarelliKohnNirenberg 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 CaffarelliKohnNirenberg 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
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,γ}u + Δ_{p,γ}v,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 wellknown (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].
(3) The translation group {T(h)}, which is defined by T(h)σ(s) = σ(h + s), s, h ∈ ℝ, is continuous on ;
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 nonautonomous cases.
2. Existence of a weak solution
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];L^{2}(Ω)). This makes the initial condition in problem (1.1) meaningful.
Theorem 2.1. For any τ, T ∈ ℝ, T > τ and u_{τ }∈ L^{2}(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 L^{2}(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 L^{2}(τ, T; L^{2}(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 L^{2}(Ω), so by arguments as in [[21], pp. 159160], we have that u_{n}(T) ⇀ u(T) in L^{2}(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. 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 multivalued semiflows.
Definition 3.1. [5]Let E be a Banach space. The mapping
is called a multivalued semiflow if the following conditions are satisfied:
(2) for all w ∈ E, t_{1}, t_{2 }∈ ℝ^{+}, where G (t, B) = ∪_{x∈B }G (t, x), B ⊂ E.
It is called a strict multivalued semiflow if , for all w ∈ E, t_{1}, t_{2 }∈ ℝ^{+}.
We now consider problem (1.1) with τ = 0. By Theorem 2.1, we construct a multivalued mapping as follows
Lemma 3.1. is a strict multivalued semiflow in the sense of Definition 3.1.
Proof. Assume that , then ξ = u(t_{1 }+ t_{2}), where u(t) is a solution of (1.1). Denoting v (t) = u(t + t_{2}), we see that v(.) is also in the set of solutions of (1.1) with respect to initial condition v(0) = u(t_{2}). Therefore, . It remains to show that . If then ξ = v(t_{1}), where . One can suppose that v(0) = u(t_{2}), where u(0) = u_{0}. 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) = u_{0}. In addition, since ξ = v(t_{1}) = w(t_{1 }+ t_{2}), we have .
Definition 3.2. [5]A set is said to be a global attractor of the multivalued semiflow if the following conditions hold:
• is an attracting, i.e., as t → ∞ for all bounded subsets B ⊂ E,
• is negatively semiinvariant: for arbitrary t ≥ 0,
• If ℬ is an attracting of then ,
where is the Hausdorff semidistance.
The following theorem gives the sufficient conditions for the existence of a global attractor for the multivalued semiflow .
Theorem 3.2. [5,7]Suppose that the strict multivalued 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 ≥ t_{0 }(∥u_{0}∥_{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 multivalued semiflow generated by problem (1.1) has an invariant compact global attractor in L^{2}(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 L^{2}(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 L^{2}(Q_{0,T}),
• u_{n}(t) ⇀ u(t) in L^{2}(Ω) for arbitrary t ∈ [0, T] (and then u(0) = η),
• 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 L^{2}(Ω). Using Lemma 3.3, we conclude that, if , then {ξ_{n}} is precompact in L^{2}(Ω).
3.2. The nonautonomous 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) ∈ ℝ^{2 } 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.
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 ∈ L^{2 }(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(t_{1}, v_{1})f(t_{2}, v_{2}) ≤ α(t_{1}t_{2} + v_{1}v_{2},R), ∀t_{1}, t_{2 }∈ ℝ, v_{1}, v_{2 }∈ [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 multivalued 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 multivalued 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 multivalued 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 multivalued 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 multivalued 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 wupper semicontinuous.
Then the family of multivalued 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 L^{2}(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 L^{2}(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 L^{2}(τ,T; L^{2}(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 L^{2}(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 nonincreasing on [τ, T]. We first show that
Indeed,
and
as n → ∞ since u_{n }→ u strongly in L^{2}(Q_{τ,t}) and {g_{n}} is bounded in L^{2}(Q_{τ,t}). In addition,
as n → ∞ since g_{n }⇀ g in L^{2}(Q_{τ,t}). Then (3.4) is proved due to the fact that u_{n}(t) → u(t) in L^{2}(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 ≥ n_{0}(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 L^{2}(Ω) as n → ∞. This completes the proof.
Theorem 3.9. Let conditions (H1)(H3) hold. Then the family of multivalued semiprocesses {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 R_{0 }such that if u_{n}(τ) ∈ B_{R}, the ball in L^{2}(Ω) centered at 0 with radius R, then there exists T_{0 }= T_{0}(τ, R) such that
that is, , for all t ≥ T_{0}(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 ≥ T_{0}(τ, B_{R}). It follows that any sequence {ξ_{n}} such that , is precompact in L^{2}(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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
This work was supported by Vietnam's National Foundation for Science and Technology Development (NAFOSTED), Project 101.012010.05.
The authors would like to thank the reviewers for valuable comments and suggestions.
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