Parabolic problems in R n with spatially variable exponents

Abstract

We study the asymptotic behavior of parabolic $p (x)$ -Laplacian problems of the form $\frac{\partial u_{λ}}{\partial t} - div (D^{λ} | \nabla u_{λ} |^{p (x) - 2} \nabla u_{λ}) + a {| u_{λ} |}^{p (x) - 2} u_{λ} = B (u_{λ})$ in $L^{2} (R^{n})$ , where $n ⩾ 1$ , $p \in L^{\infty} (R^{n})$ such that $2 < p_{-} : = ess inf p (x) ⩽ p (x) ⩽ p_{+} : = ess sup p (x)$ , $D^{λ} \in L^{\infty} (R^{n})$ , $\infty > M ⩾ D^{λ} (x) ⩾ σ > 0$ a.e. in $R^{n}$ , $λ \in [0, \infty)$ , $B : L^{2} (R^{n}) \to L^{2} (R^{n})$ is a globally Lipschitz map and $a : R^{n} \to R$ is a non-negative continuous function such that there exists $R_{1} > 0$ with ${x \in R^{n}; a (x) = 0} \subset B_{R_{1}} (0)$ , $inf_{x \in R^{n} ∖ B_{R_{1}} (0)} a (x) > 0$ , and $\int_{R^{n} ∖ B_{R_{1}} (0)} \frac{1}{a {(x)}^{2 / (p (x) - 2)}} d x < + \infty .$ We also study the sensitivity of the problem according to the variation of the diffusion coefficients.

Keywords

attractors unbounded domains

1. Introduction

Let us consider the problem $\{\begin{matrix} \frac{\partial u_{λ}}{\partial t} (t) - div (D^{λ} {| \nabla u_{λ} (t) |}^{p (x) - 2} \nabla u_{λ} (t)) + a {| u_{λ} (t) |}^{p (x) - 2} u_{λ} (t) = B (u_{λ} (t)), & t > 0, \\ u_{λ} (0) = u_{0, λ}, \end{matrix}$ (1) where $p \in L^{\infty} (R^{n})$ such that $2 < p_{-} : = ess inf p (x) ⩽ p (x) ⩽ p_{+} : = ess sup p (x)$ , $u_{0, λ} \in H : = L^{2} (R^{n})$ , $n ⩾ 1$ , $B : H \to H$ is a globally Lipschitz map with Lipschitz constant $L ⩾ 0$ , $D^{λ} \in L^{\infty} (R^{n})$ , $\infty > M ⩾ D^{λ} (x) ⩾ σ > 0$ a.e. in $R^{n}$ , $λ \in [0, \infty)$ , $D^{λ} \to D^{λ_{1}}$ in $L^{\infty} (R^{n})$ as $λ \to λ_{1}$ and $a : R^{n} \to R$ is a non-negative continuous function satisfying the following conditions: there exists $R_{1} > 0$ such that ${x \in R^{n}; a (x) = 0} \subset B_{R_{1}} (0), inf_{x \in R^{n} ∖ B_{R_{1}} (0)} a (x) > 0$ and $\int_{R^{n} ∖ B_{R_{1}} (0)} \frac{1}{a {(x)}^{2 / (p (x) - 2)}} d x < + \infty,$ (C) where $B_{R_{1}} (0)$ denotes the closed ball with center at zero and radius $R_{1}$ .

The problem (1) was considered in [10] for a constant exponent p. The variable exponent case requires the use of function spaces with spatially dependent exponents and in the most of the proofs it is necessary to breach the estimates of the operator $- div (D^{λ} | \nabla u_{λ} (t) |^{p (x) - 2} \nabla u_{λ} (t)) + a {| u_{λ} (t) |}^{p (x) - 2} u_{λ} (t)$ and for the solutions into four cases (see [12]).

In this work we prove the existence of a global attractor in $L^{2} (R^{n})$ for each positive finite diffusion coefficient and we show that the family of attractors is upper semicontinuous with respect to positive finite diffusion parameters.

The paper is organized as follows. In Section 2 we provide some preliminary results about a specific weighted space and some properties of the operator. In Section 3 we prove that the semigroup associated with problem (1) has a maximal compact invariant global attractor. Section 4 is devoted to obtain uniform estimates for the solutions and to prove continuity of the solutions with respect to the initial values and upper semicontinuity of attractors.

2. Preliminaries

In this work, we consider the weighted space $E : = {u \in W^{1, p (x)} (R^{n}); \int_{R^{n}} a (x) {| u (x) |}^{p (x)} d x < + \infty}$ with the norm given by $∥ u ∥_{E} : = ∥ \nabla u ∥_{p (x)} + ∥ u_{a} ∥_{p (x)}$ where $∥ \cdot ∥_{p (x)}$ is the norm in $L^{p (x)} (R^{n})$ and $u_{a} (x) : = a {(x)}^{1 / p (x)} u (x)$ . We refer the reader to [4,5] and references therein to see properties of the Lebesgue and Sobolev spaces with variable exponents. In particular, with $L^{p (x)} (R^{n}) : = {u : R^{n} \to R : u is measurable, \int_{R^{n}} {| u (x) |}^{p (x)} d x < \infty}$ and $L_{+}^{\infty} (R^{n}) : = {q \in L^{\infty} (R^{n}) : ess inf q ⩾ 1}$ , define $ρ (u) : = \int_{R^{n}} {| u (x) |}^{p (x)} d x, ∥ u ∥_{p (x)} : = inf {λ > 0 : ρ (\frac{u}{λ}) ⩽ 1}$ for $u \in L^{p (x)} (R^{n})$ and $p \in L_{+}^{\infty} (R^{n})$ . Furthermore, $W^{1, p (x)} (R^{n}) : = {u \in L^{p (x)} (R^{n}); | \nabla u | \in L^{p (x)} (R^{n})},$ which is a Banach space with the norm $∥ u ∥_{W^{1, p (x)} (R^{n})} : = ∥ \nabla u ∥_{p (x)} + ∥ u ∥_{p (x)} .$ Since ${x \in R^{n}; a (x) = 0} \subset B_{R_{1}} (0)$ and $inf_{x \in R^{n} ∖ B_{R_{1}} (0)} a (x) > 0$ , there follows the continuous inclusion $E ↪ W^{1, p (x)} (R^{n})$ . With some computation we can show that $(E, ∥ \cdot ∥_{E})$ is a reflexive Banach space. Moreover, we have the following lemma.

Lemma 1.
$E ↪ L^{2} (R^{n})$ and the inclusion is compact.
Proof.
Let $θ (x) : = \frac{p (x)}{2}$ and $θ^{'} (x)$ be the conjugate exponent of $θ (x)$ , i.e., $\frac{1}{θ (x)} + \frac{1}{θ^{'} (x)} = 1$ . Let $u \in E$ and denote $b : = ∥ u_{a} ∥_{p (x)}$ . Then, we have $\begin{array}{rcl} \int_{R^{n} ∖ B_{R_{1}} (0)} \frac{| u (x) |^{2}}{b^{2}} d x & = & \int_{R^{n} ∖ B_{R_{1}} (0)} \frac{a {(x)}^{1 / θ (x)}}{a {(x)}^{1 / θ (x)}} \frac{| u (x) |^{2}}{b^{2}} d x \\ ⩽ & \int_{R^{n} ∖ B_{R_{1}} (0)} \frac{1}{θ^{'} (x)} \frac{1}{a {(x)}^{θ^{'} (x) / θ (x)}} d x \\ + \int_{R^{n} ∖ B_{R_{1}} (0)} \frac{1}{θ (x)} a (x) {(\frac{| u (x) |}{b})}^{p (x)} d x \\ ⩽ & \int_{R^{n} ∖ B_{R_{1}} (0)} \frac{1}{a {(x)}^{2 / p (x) - 2}} d x \\ + \int_{R^{n} ∖ B_{R_{1}} (0)} {(\frac{a {(x)}^{1 / p (x)} | u (x) |}{b})}^{p (x)} d x \\ ⩽ & C + 1 . \end{array}$ Thus $\int_{R^{n} ∖ B_{R_{1}} (0)} | u (x) |^{2} d x ⩽ (C + 1) ∥ u_{a} ∥_{p (x)}^{2}$ for all $u \in E$ . Thus $∥ u ∥_{L^{2} (R^{n} ∖ B_{R_{1}} (0))} ⩽ C ∥ u ∥_{E}$ . Since $∥ u ∥_{L^{2} (B_{R_{1}} (0))} ⩽ C ∥ u ∥_{W^{1, p (x)} (B_{R_{1}} (0))} ⩽ C ∥ u ∥_{W^{1, p (x)} (R^{n})} ⩽ C ∥ u ∥_{E},$ it follows that $E ↪ L^{2} (R^{n})$ .

Now we show that $E \subset \subset L^{2} (R^{n})$ . By condition (C), given $ε > 0$ , there is $R_{2} > 0$ large enough, such that $\int_{R^{n} ∖ B_{R_{2} (0)}} \frac{1}{a {(x)}^{2 / (p (x) - 2)}} d x < ε .$ Let ${u_{n}}$ be a bounded sequence in E, then there is $w \in E$ such that $u_{n} ⇀ w$ in E (up to a subsequence). Applying the above estimate, taking $R_{3} : = max {R_{1}, R_{2}}$ and using the Hölder inequality we obtain $\int_{R^{n} ∖ B_{R_{3}} (0)} {| (u_{n} - w) (x) |}^{2} d x = \int_{R^{n} ∖ B_{R_{3}} (0)} \frac{1}{a {(x)}^{1 / θ (x)}} a {(x)}^{1 / θ (x)} {| (u_{n} - w) (x) |}^{2} d x < C ε .$ In the bounded set $B_{R_{3}} (0)$ we have compact embeddings, and so, it is easy to conclude that $u_{n} \to w$ in $L^{2} (B_{R_{3}} (0))$ . □
Remark 1.
Since $C_{0}^{\infty} (R^{n}) \subset E$ , the inclusion $E ↪ L^{2} (R^{n})$ is dense.
Lemma 2.
$E ↪ L^{q (x)} (R^{n})$ for $p : R^{n} \to R$ Lipschitz continuous, $1 < p_{-} ⩽ p_{+} < \infty$ , and $q \in L^{\infty} (R^{n})$ satisfying condition $p (x) ⩽ q (x) ⩽ p^{} (x)$ for a.e.* $x \in R^{n}$ , where $p^{} (x) : = \{\begin{matrix} \frac{n p (x)}{n - p (x)}, & p (x) < n, \\ \infty, & p (x) ⩾ n . \end{matrix}$
Proof.
By [4,5] we have $W^{1, p (x)} (R^{n}) ↪ L^{q (x)} (R^{n})$ for $p (x) ⩽ q (x) ⩽ p^{} (x)$ for a.e. $x \in R^{n}$ . Since $E ↪ W^{1, p (x)} (R^{n})$ the result follows. □

Now, we consider, for each $λ \in [0, \infty)$ , the operator $A_{1}^{D^{λ}} : E \to E^{}$ , $\begin{array}{rcl} {⟨ A_{1}^{D^{λ}} u, v ⟩}_{E^{}, E} & : = & \int_{R^{n}} D^{λ} (x) {| \nabla u (x) |}^{p (x) - 2} \nabla u (x) \nabla v (x) d x \\ + \int_{R^{n}} a (x) {| u (x) |}^{p (x) - 2} u (x) v (x) d x \end{array}$ for each $u \in E$ . We have the following estimates and properties of the operator (see [12]).
Lemma 3.

If $∥ u ∥_{E} ⩽ 1$ , then ${⟨ A_{1}^{D^{λ}} u, u ⟩}_{E^{}, E} ⩾ \frac{min {1, σ}}{2^{p_{+} - 1}} ∥ u ∥_{E}^{p_{+}}$ .

If $∥ u ∥_{E} > 1$ , then* ${⟨ A_{1}^{D^{λ}} u, u ⟩}_{E^{}, E} ⩾ \{\begin{matrix} \frac{min {1, σ}}{2^{p_{+} - 1}} ∥ u ∥_{E}^{p_{+}}, & ∥ \nabla u ∥_{p (x)} ⩽ 1 and ∥ u_{a} ∥_{p (x)} ⩽ 1, \\ \frac{min {1, σ}}{2^{p_{-} - 1}} ∥ u ∥_{E}^{p_{-}}, & ∥ \nabla u ∥_{p (x)} ⩾ 1 and ∥ u_{a} ∥_{p (x)} ⩾ 1, \\ σ ∥ \nabla u ∥_{p (x)}^{p_{+}} + ∥ u_{a} ∥_{p (x)}^{p_{-}}, & ∥ \nabla u ∥_{p (x)} ⩽ 1 and ∥ u_{a} ∥_{p (x)} ⩾ 1, \\ σ ∥ \nabla u ∥_{p (x)}^{p_{-}} + ∥ u_{a} ∥_{p (x)}^{p_{+}}, & ∥ \nabla u ∥_{p (x)} ⩾ 1 and ∥ u_{a} ∥_{p (x)} ⩽ 1 . \end{matrix}$

If $u \in E$ , then* ${∥ A_{1}^{D^{λ}} u ∥}_{E^{}} ⩽ \{\begin{matrix} ∥ \nabla u ∥_{p (x)}^{p_{+}} + ∥ u_{a} ∥_{p (x)}^{p_{+}} + 1, & ∥ \nabla u ∥_{p (x)} ⩽ 1 and ∥ u_{a} ∥_{p (x)} ⩽ 1, \\ ∥ \nabla u ∥_{p (x)}^{p_{-}} + ∥ u_{a} ∥_{p (x)}^{p_{-}} + 1, & ∥ \nabla u ∥_{p (x)} ⩾ 1 and ∥ u_{a} ∥_{p (x)} ⩾ 1, \\ ∥ \nabla u ∥_{p (x)}^{p_{+}} + ∥ u_{a} ∥_{p (x)}^{p_{-}} + 1, & ∥ \nabla u ∥_{p (x)} ⩽ 1 and ∥ u_{a} ∥_{p (x)} ⩾ 1, \\ ∥ \nabla u ∥_{p (x)}^{p_{-}} + ∥ u_{a} ∥_{p (x)}^{p_{+}} + 1, & ∥ \nabla u ∥_{p (x)} ⩾ 1 and ∥ u_{a} ∥_{p (x)} ⩽ 1 . \end{matrix}$

Lemma 4.
The operator* $A_{1}^{D^{λ}} : E \to E^{}$ is monotone, coercive and hemicontinuous.*

Then, using Theorem 2.4 in [1] we conclude that the operator $A_{1}^{D^{λ}} : E \to E^{}$ is maximal monotone and $A_{1}^{D^{λ}} (E) = E^{}$ . According to Example 2.3.7 in [2], the operator $A^{D^{λ}}$ , the realization of $A_{1}^{D^{λ}}$ at $H : = L^{2} (R^{n})$ , given by $A^{D^{λ}} (u) = A_{1}^{D^{λ}} u$ , if $u \in D (A^{D^{λ}}) : = {u \in E; A_{1}^{D^{λ}} u \in H}$ , is a maximal monotone operator in H. From now on, we write $- div (D^{λ} | \nabla u |^{p (x) - 2} \nabla u) + a {| u |}^{p (x) - 2} u$ to mean $A^{D^{λ}} (u)$ as just defined.

We still observe that $A^{D^{λ}}$ is the subdifferential of the convex, proper and lower semicontinuous non-negative map $φ^{D^{λ}}$ , where $φ^{D^{λ}} : L^{2} (R^{n}) \to R \cup {+ \infty}$ is defined by $φ^{D^{λ}} (u) : = \{\begin{matrix} \int_{R^{n}} \frac{D^{λ} (x)}{p (x)} {| \nabla u (x) |}^{p (x)} d x + \int_{R^{n}} \frac{a (x)}{p (x)} {| u (x) |}^{p (x)} d x, & u \in E, \\ + \infty, & otherwise . \end{matrix}$

By [1] we know that the domain of $A^{D^{λ}}$ is a dense subset of $D (φ^{D^{λ}}) = E$ . As $E \subset H$ , and the imbedding is continuous and compact, we get $\bar{D (A^{D^{λ}})} = H = L^{2} (R^{n})$ . Moreover, the operator $A^{D^{λ}} : D (A^{D^{λ}}) \subset H \to H$ generates a nonlinear semigroup $S^{D^{λ}}$ .
Proposition 1.
The semigroup $S^{D^{λ}}$ is compact.
Proof.
By [6,8] we have that the semigroup $S^{D^{λ}}$ is compact if the level sets $M_{k} : = {u \in D (φ^{D^{λ}}); ∥ u ∥_{H} ⩽ k, φ^{D^{λ}} (u) ⩽ k}$ are compact in H for any $k > 0$ . We verify this condition. Since $E \subset \subset H$ and $M_{k} = \bar{M_{k}}$ it is sufficient to show that for each $k > 0$ , $M_{k}$ is a bounded set in E. Let $k > 0$ fixed. For $u \in M_{k}$ we have ${⟨ A_{1}^{D^{λ}} u, u ⟩}_{E^{}, E} ⩽ k p_{+}$ . Using Lemma 3, we obtain $\frac{{⟨ A_{1}^{D^{λ}} u, u ⟩}_{E^{}, E}}{∥ u ∥_{E}} ⩾ \{\begin{matrix} \frac{min {1, σ}}{2^{p_{+} - 1}} ∥ u ∥_{E}^{p_{+} - 1}, & ∥ \nabla u ∥_{p (x)} ⩽ 1 and ∥ u_{a} ∥_{p (x)} ⩽ 1, \\ \frac{min {1, σ}}{2^{p_{-} - 1}} ∥ u ∥_{E}^{p_{-} - 1}, & ∥ \nabla u ∥_{p (x)} ⩾ 1 and ∥ u_{a} ∥_{p (x)} ⩾ 1, \\ \frac{1}{2} ∥ u_{a} ∥_{p (x)}^{p_{-} - 1}, & ∥ \nabla u ∥_{p (x)} ⩽ 1 and ∥ u_{a} ∥_{p (x)} ⩾ 1, \\ \frac{σ}{2} ∥ \nabla u ∥_{p (x)}^{p_{-} - 1}, & ∥ \nabla u ∥_{p (x)} ⩾ 1 and ∥ u_{a} ∥_{p (x)} ⩽ 1 \end{matrix}$ and the result follows. □
Definition 1 ([3]).

A function $u_{λ} \in C ([0, T]; H)$ is a strong solution to (1) if $u_{λ}$ is absolutely continuous in any compact subinterval of $(0, T)$ , $u_{λ} (t) \in D (A^{D^{λ}})$ for a.e. $t \in (0, T)$ , and $\frac{d u_{λ}}{d t} (t) + A^{D^{λ}} (u_{λ} (t)) = B (u_{λ} (t)) for a.e. t \in (0, T) .$

A function $u_{λ} \in C ([0, T]; H)$ is called a weak solution to (1) if there is a sequence ${u_{λ}^{n}}_{n \in N}$ of strong solutions convergent to $u_{λ}$ in $C ([0, T]; H)$ .

By Proposition 1 in [3], problem (1) determines a continuous semigroup of nonlinear operators $T_{λ} (t) : H \to H$ , where $T_{λ} (t) u_{0, λ}$ is the global weak solution of (1). This semigroup is such that $R^{+} \times H ∋ (t, u_{0, λ}) \mapsto T_{λ} (t) u_{0, λ} \in H$ is a continuous map. Additionally, if $u_{0 λ} \in D (A^{D^{λ}})$ , then $u_{λ} (\cdot) = T_{λ} (\cdot) u_{0, λ}$ is a Lipschitz continuous strong solution of (1). As the operator is a subdifferential of a convex, proper and lower semicontinuous map φ in H with $min {φ (u) : u \in H} = 0$ and $f (\cdot) : = B (u (\cdot)) \in L^{2} (0, T; H)$ , it follows from [2, Theorem 3.6] that the weak solution is in fact a strong solution of (1). So, in this work we will just call solution of (1).

Now, we recall here two useful lemmas from [13] which we will use in the sequence.

Lemma 5 (The Uniform Gronwall lemma).

Let g, h, y, be three positive locally integrable functions on $] t_{0}, + \infty [$ such that $y^{'}$ is locally integrable on $] t_{0}, + \infty [$ , and which satisfy $\begin{array}{rcl} \frac{d y}{d t} ⩽ g y + h for t ⩾ t_{0}, \\ \int_{t}^{t + r} g (s) d s ⩽ a_{1}, \int_{t}^{t + r} h (s) d s ⩽ a_{2}, \int_{t}^{t + r} y (s) d s ⩽ a_{3} for t ⩾ t_{0}, \end{array}$ where r, $a_{1}$ , $a_{2}$ , $a_{3}$ are positive constants. Then $y (t + r) ⩽ (\frac{a_{3}}{r} + a_{2}) exp (a_{1}) \forall t ⩾ t_{0} .$

Lemma 6.
Let y be a positive absolutely continuous function on $(0, \infty)$ which satisfies $y^{'} + γ y^{p} ⩽ δ,$ with $p > 1$ , $γ > 0$ , $δ ⩾ 0$ . Then, for $t ⩾ 0$ , $y (t) ⩽ {(δ / γ)}^{1 / p} + {[γ (p - 1) t]}^{- 1 / (p - 1)} .$

In this work, we prove that the semigroup associated with problem (1) has a maximal compact invariant global attractor $A_{λ}$ in H and that ${A_{λ}}_{λ \in [0, \infty)}$ is upper semicontinuous at $λ_{1} \in [0, \infty)$ , that means, $sup_{a_{λ} \in A_{λ}} {dist}_{H} (a_{λ}, A_{λ_{1}}) \to 0 as λ \to λ_{1} .$
3. Global attractors

In this section, we prove the existence of the minimal closed global B-attractor, which is compact and invariant, for problem (1). To do this, we need to prove that the semigroup determined by (1) is of class K and is bounded dissipative in H, see Theorems 3 and 2.

Following the ideas in [9], in the next theorem we prove some estimates needed for our goal in this section.

Theorem 1.
Let $u_{0 λ} \in D (A^{D^{λ}})$ and $u_{λ} (\cdot) = T_{λ} (\cdot) u_{0 λ}$ be the global solution of ( 1). For all $T > 0$ we have:
$\int_{0}^{T} ∥ u_{λ} (s) ∥_{E}^{p_{-}} d s ⩽ C_{1} (∥ u_{0 λ} ∥_{H}, T)$ ;

$∥ u_{λ} ∥_{L^{\infty} (0, T; H)} ⩽ C_{2} (∥ u_{0 λ} ∥_{H}, T)$ ; where $C_{1}$ and $C_{2}$ are locally bounded functions.

Proof.
Let be $u_{0 λ} \in D (A^{D^{λ}})$ and $T > 0$ . We have that $u_{λ} (\cdot) : = T_{λ} (\cdot) u_{0 λ}$ is a global solution of problem (1). So, $u_{λ} (\cdot) \in C ([0, T]; H)$ and $u_{λ} (t) \in D (A^{D^{λ}})$ for a.e. $t \in I : = (0, T)$ . Consider the measurable sets $I_{1} : = {t \in I; ∥ \nabla u_{λ} (t) ∥_{p (x)} < 1 and ∥ {(u_{λ} (t))}_{a} ∥_{p (x)} < 1}$ , $I_{2} : = {t \in I; ∥ \nabla u_{λ} (t) ∥_{p (x)} ⩾ 1 and ∥ {(u_{λ} (t))}_{a} ∥_{p (x)} ⩾ 1}$ , $I_{3} : = {t \in I; ∥ \nabla u_{λ} (t) ∥_{p (x)} < 1 and ∥ {(u_{λ} (t))}_{a} ∥_{p (x)} ⩾ 1}$ and $I_{4} : = {t \in I; ∥ \nabla u_{λ} (t) ∥_{p (x)} ⩾ 1 and ∥ {(u_{λ} (t))}_{a} ∥_{p (x)} < 1}$ . So $I = I_{1} \cup I_{2} \cup I_{3} \cup I_{4}$ .

Multiplying the equation on (1) by $u_{λ}$ and using Lemma 3, we have that $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} & = & - {⟨ A^{D^{λ}} (u_{λ} (t)), u_{λ} (t) ⟩}_{E^{}, E} + ⟨ B (u_{λ} (t)), u_{λ} (t) ⟩ \\ ⩽ & \{\begin{matrix} - \frac{min {1, σ}}{2^{p_{+} - 1}} {∥ u_{λ} (t) ∥}_{E}^{p_{+}} + {∥ B (u_{λ} (t)) ∥}_{H} {∥ u_{λ} (t) ∥}_{H}, & t \in I_{1}, \\ - \frac{min {1, σ}}{2^{p_{-} - 1}} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} + {∥ B (u_{λ} (t)) ∥}_{H} {∥ u_{λ} (t) ∥}_{H}, & t \in I_{2}, \\ - σ {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{+}} - {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} + {∥ B (u_{λ} (t)) ∥}_{H} {∥ u_{λ} (t) ∥}_{H}, & t \in I_{3}, \\ - σ {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} - {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{+}} + {∥ B (u_{λ} (t)) ∥}_{H} {∥ u_{λ} (t) ∥}_{H}, & t \in I_{4} \end{matrix} \\ ⩽ & \{\begin{matrix} - C_{1} {∥ u_{λ} (t) ∥}_{E}^{p_{+}} + C_{2} {∥ u_{λ} (t) ∥}_{E}^{2} + C_{3} {∥ u_{λ} (t) ∥}_{E}, & t \in I_{1}, \\ - \tilde{C_{1}} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} + C_{2} {∥ u_{λ} (t) ∥}_{E}^{2} + C_{3} {∥ u_{λ} (t) ∥}_{E}, & t \in I_{2}, \\ - σ {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{+}} - {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} + C_{2} {∥ u_{λ} (t) ∥}_{E}^{2} + C_{3} {∥ u_{λ} (t) ∥}_{E}, & t \in I_{3}, \\ - σ {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} - {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{+}} + C_{2} {∥ u_{λ} (t) ∥}_{E}^{2} + C_{3} {∥ u_{λ} (t) ∥}_{E}, & t \in I_{4}, \end{matrix} \end{array}$ where $C_{1} = \frac{min {1, σ}}{2^{p_{+} - 1}}$ , $\tilde{C_{1}} = \frac{min {1, σ}}{2^{p_{-} - 1}}$ , $C_{2} > 0$ and $C_{3} ⩾ 0$ are constants which do not depend on the parameter λ. Thus, $\begin{array}{rclr} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} & ⩽ & \{\begin{matrix} - \frac{C_{1}}{2} {∥ u_{λ} (t) ∥}_{E}^{p_{+}} + C_{4}, & t \in I_{1}, \\ - \frac{\tilde{C_{1}}}{2} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} + C_{5}, & t \in I_{2}, \\ - \frac{σ}{2} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{+}} - \frac{1}{2} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} + C_{6}, & t \in I_{3}, \\ - \frac{σ}{2} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} - \frac{1}{2} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{+}} + C_{7}, & t \in I_{4} \end{matrix} & (2) \end{array}$ for suitable positive constants $C_{4}$ , $C_{5}$ , $C_{6}$ and $C_{7}$ which do not depend on the parameter λ.

So, $\begin{array}{rcl} \int_{0}^{T} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} d t & = & \int_{I_{1}} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} d t + \int_{I_{2}} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} d t \\ + \int_{I_{3}} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} d t + \int_{I_{4}} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} d t \\ ⩽ & - \frac{C_{1}}{2} \int_{I_{1}} {∥ u_{λ} (t) ∥}_{E}^{p_{+}} d t + C_{4} T - \frac{\tilde{C_{1}}}{2} \int_{I_{2}} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} d t + C_{5} T \\ - \frac{σ}{2} \int_{I_{3}} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{+}} d t - \frac{1}{2} \int_{I_{3}} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} d t + C_{6} T \\ - \frac{σ}{2} \int_{I_{4}} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} d t - \frac{1}{2} \int_{I_{4}} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{+}} d t + C_{7} T . \end{array}$

Then, $\begin{array}{rcl} {∥ u_{λ} (T) ∥}_{H}^{2} + C_{1} \int_{I_{1}} {∥ u_{λ} (t) ∥}_{E}^{p_{+}} d t + \tilde{C_{1}} \int_{I_{2}} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} d t \\ + σ \int_{I_{3}} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{+}} d t + \int_{I_{3}} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} d t + σ \int_{I_{4}} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} d t \\ + \int_{I_{4}} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{+}} d t ⩽ ∥ u_{0 λ} ∥_{H}^{2} + C_{8} . \end{array}$ In particular, ${∥ u_{λ} (T) ∥}_{H}^{2} + \tilde{C_{1}} \int_{I_{2}} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} d t ⩽ ∥ u_{0 λ} ∥_{H}^{2} + C_{8}$ and $u_{λ} \in L^{p_{-}} (I_{2}, E)$ . Also, with a direct proof, we have $u_{λ} \in L^{p_{-}} (I_{1}, E)$ . Then, ${∥ u_{λ} (T) ∥}_{H}^{2} + \tilde{C_{1}} \int_{I_{1} \cup I_{2}} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} d t ⩽ ∥ u_{0 λ} ∥_{H}^{2} + C_{9} .$ Similarly, ${∥ u_{λ} (T) ∥}_{H}^{2} + σ \int_{I_{3} \cup I_{4}} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} d t ⩽ ∥ u_{0 λ} ∥_{H}^{2} + C_{10}$ and ${∥ u_{λ} (T) ∥}_{H}^{2} + \int_{I_{3} \cup I_{4}} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} d t ⩽ ∥ u_{0 λ} ∥_{H}^{2} + C_{11} .$ Therefore, ${∥ u_{λ} (T) ∥}_{H}^{2} + C_{12} \int_{0}^{T} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} d t ⩽ ∥ u_{0 λ} ∥_{H}^{2} + C_{13} .$ (3) This shows that (i) is satisfied.

Also, with the same arguments as above, we obtain $\begin{array}{rcl} {∥ u_{λ} (t) ∥}_{H}^{2} & ⩽ & {∥ u_{λ} (t) ∥}_{H}^{2} + C_{12} \int_{0}^{t} {∥ u_{λ} (s) ∥}_{E}^{p_{-}} d s \\ ⩽ & ∥ u_{0 λ} ∥_{H}^{2} + C_{14}, \forall t \in I . \end{array}$ This shows that (ii) is satisfied. □
Theorem 2.
Let* ${T_{λ} (t)}$ be the semigroup associated with the problem ( 1) on H. Then, $T_{λ} (t)$ is bounded dissipative in H.
Proof.
It is sufficient to consider initial data $u_{0 λ} \in D (A^{D^{λ}})$ . Consider the embedding constant $C_{E H} > 0$ from $E ↪ H$ and take $t_{0} > 0$ . Let $t > t_{0}$ and $u_{λ} (\cdot) = T_{λ} (\cdot) u_{0 λ}$ .

If $∥ \nabla u_{λ} (t) ∥_{p (x)} < 1$ and $∥ {(u_{λ} (t))}_{a} ∥_{p (x)} < 1$ , then ${∥ u_{λ} (t) ∥}_{H} ⩽ C_{E H} {∥ u_{λ} (t) ∥}_{E} ⩽ 2 C_{E H} : = D_{1} .$

If $∥ \nabla u_{λ} (t) ∥_{p (x)} ⩾ 1$ and $∥ {(u_{λ} (t))}_{a} ∥_{p (x)} ⩾ 1$ , then from (2) $\frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} ⩽ - \frac{\tilde{C_{1}}}{2} {∥ u_{λ} (t) ∥}_{E}^{p_{-}} + C_{5} ⩽ - \frac{\tilde{C_{1}}}{2} C_{E H}^{- p_{-}} {∥ u_{λ} (t) ∥}_{H}^{p_{-}} + C_{5} .$ Hence the function $y (t) : = ∥ u_{λ} (t) ∥_{H}^{2}$ satisfies the differential inequality in $I_{2}$ $y^{'} (t) ⩽ - \tilde{C_{1}} C_{E H}^{- p_{-}} y {(t)}^{\frac{p_{-}}{2}} + 2 C_{5} .$ Therefore, from Lemma 6, $y (t) = {∥ u_{λ} (t) ∥}_{H}^{2} ⩽ {(2 C_{7} C_{E H}^{- p_{-}} C_{5})}^{\frac{2}{p_{-}}} + {(C_{7} C_{E H}^{- p_{-}} (\frac{p_{-}}{2} - 1))}^{\frac{- 2}{p_{-} - 2}} : = D_{2}$ for $t \in I_{2}$ , $t > t_{0}$ .

If $∥ \nabla u_{λ} (t) ∥_{p (x)} < 1$ and $∥ {(u_{λ} (t))}_{a} ∥_{p (x)} ⩾ 1$ , then from (2) $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} & ⩽ & - \frac{σ}{2} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{+}} - \frac{1}{2} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{-}} + C_{6} \\ ⩽ & - \frac{σ {(2 C_{E H})}^{- p_{+}}}{2} {∥ u_{λ} (t) ∥}_{H}^{p_{+}} + C_{6} . \end{array}$

If $∥ \nabla u_{λ} (t) ∥_{p (x)} ⩾ 1$ and $∥ {(u_{λ} (t))}_{a} ∥_{p (x)} < 1$ , then from (2) $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} & ⩽ & - \frac{σ}{2} {∥ \nabla u_{λ} (t) ∥}_{p (x)}^{p_{-}} - \frac{1}{2} {∥ {(u_{λ} (t))}_{a} ∥}_{p (x)}^{p_{+}} + C_{7} \\ ⩽ & - \frac{{(2 C_{E H})}^{- p_{+}}}{2} {∥ u_{λ} (t) ∥}_{H}^{p_{+}} + C_{7} . \end{array}$ Therefore, again using Lemma 6, we obtain $\begin{array}{rcl} {∥ u_{λ} (t) ∥}_{H}^{2} ⩽ D_{3} for t \in I_{3}, t > t_{0}, \\ {∥ u_{λ} (t) ∥}_{H}^{2} ⩽ D_{4} for t \in I_{4}, t > t_{0} . \end{array}$

So, taking $D > max {D_{1}, D_{2}^{1 / 2}, D_{3}^{1 / 2}, D_{4}^{1 / 2}}$ , we have ${∥ u_{λ} (t) ∥}_{H} ⩽ D$ (4) for all $t > t_{0}$ . Note that D does not depend on the initial data. So, the set ${ω_{0} \in H : ∥ ω_{0} ∥_{H} ⩽ D}$ attracts bounded sets of H in the H-norm. The proof is completed. □

Since $D (A^{D^{λ}})$ is dense in H and $E \subset H$ compactly, following the arguments as in the proof of Theorem 3 in [9], we obtain the following result.
Theorem 3.
Let ${T_{λ} (t)}$ be the semigroup associated with the problem ( 1) on H. Then $T_{λ} (t) : H \to H$ is of class K, i.e., $T_{λ} (t)$ is compact for each $t > 0$ .

As a consequence of Theorems 2 and 3 (see [7]) we obtain the following result. Theorem 4.
The semigroup ${T_{λ} (t)}$ associated with problem ( 1) has a minimal closed global B-attractor $A^{λ}$ in H, which is compact and invariant.

4. Continuity with respect to the initial values and upper semicontinuity of attractors

Using (4) and (3) we obtain the following estimates in H for the solutions $u_{λ}$ of the problem (1), uniformly on $λ \in [0, \infty)$ .

Lemma 7.

Let $u_{λ}$ is a solution of ( 1 ) in $[0, \infty)$ . Given $t_{0} > 0$ , there exists a positive number $r_{0}$ such that $∥ u_{λ} (t) ∥_{H} ⩽ r_{0}$ , for each $t ⩾ t_{0}$ and $λ \in [0, \infty)$ .

For each fixed $λ \in [0, \infty)$ , there exists a positive constant ${\tilde{r}}_{0} = {\tilde{r}}_{0} (u_{0, λ}, t_{0})$ such that $∥ u_{λ} (t) ∥_{H} < {\tilde{r}}_{0}$ , for all $t \in [0, t_{0}]$ and, for initial conditions in bounded subsets of H, we have ${\tilde{r}}_{0}$ constant on $λ \in [0, \infty)$ .

Corollary 1.
There is a bounded set $B_{0}$ in H such that $A_{λ} \subset B_{0}$ , for all $λ \in [0, \infty)$ .
Lemma 8.
If $u_{λ}$ is a solution of ( 1 ) in $[0, \infty)$ , then there exist positive constants $r_{1} > 0$ and $t_{1} > t_{0}$ such that $∥ u_{λ} (t) ∥_{E} ⩽ r_{1}$ , for each $t ⩾ t_{1}$ and $λ \in [0, \infty)$ , with $t_{0}$ as in the Lemma 7 .
Proof.
Take $t_{1} > t_{0}$ . If $u_{λ}$ is solution of (1), we have $\begin{array}{rclr} \frac{d}{d t} φ^{D^{λ}} (u_{λ} (t)) & ⩽ & \frac{1}{2} {∥ B (u_{λ} (t)) ∥}_{H}^{2} & (5) \\ ⩽ & \frac{1}{2} {(L r_{0} + {∥ B (0) ∥}_{H})}^{2} = \frac{1}{2} R^{2} & (6) \end{array}$ for all $t ⩾ t_{0}$ , $λ \in [0, \infty)$ , where $R = R (L, r_{0}) > 0$ is a constant ( $r_{0} = r_{0} (t_{0})$ ). By definition of subdifferential, $\begin{array}{rclr} \frac{1}{2} \frac{d}{d t} {∥ u_{λ} (t) ∥}_{H}^{2} + φ^{D^{λ}} (u_{λ} (t)) & ⩽ & ⟨ \frac{\partial u_{λ} (t)}{\partial t} (t), u_{λ} (t) ⟩ + ⟨ \partial φ^{D^{λ}} (u_{λ} (t)), u_{λ} (t) ⟩ \\ ⩽ & {∥ B (u_{λ} (t)) ∥}_{H} {∥ u_{λ} (t) ∥}_{H} ⩽ R r_{0} & (7) \end{array}$ for all $t ⩾ t_{0}$ and $λ \in [0, \infty)$ . Let $t ⩾ t_{0}$ and $r = t_{1} - t_{0}$ . Integrating (7) from t to $t + r$ we obtain $\int_{t}^{t + r} φ^{D^{λ}} (u_{λ} (τ)) d τ ⩽ \frac{1}{2} {∥ u_{λ} (t) ∥}_{H}^{2} + R r_{0} r ⩽ A$ (8) for all $λ \in [0, \infty)$ , where A is a positive constant. From (5), (8) and Lemma 5, $φ^{D^{λ}} (u_{λ} (t + r)) ⩽ \frac{A}{r} + \frac{1}{2} R^{2} r = : {\tilde{r}}_{1}$ for all $t ⩾ t_{0}$ and $λ \in [0, \infty)$ . So $\frac{σ}{p_{+}} \int_{R^{n}} {| \nabla u_{λ} (t + r, x) |}^{p (x)} d x + \frac{1}{p_{+}} \int_{R^{n}} {| {(u_{λ} (t + r))}_{a} (x) |}^{p (x)} d x ⩽ {\tilde{r}}_{1}$ for all $t ⩾ t_{0}$ and $λ \in [0, \infty)$ . This implies that $∥ u_{λ} (t) ∥_{E} ⩽ r_{1}$ for all $t > t_{1}$ and $λ \in [0, \infty)$ . □
Remark 2.
If $u_{λ}$ is a solution of (1) with initial conditions in bounded subsets of E, then there is a constant ${\tilde{r}}_{3} > 0$ such that $∥ u_{λ} (t) ∥_{E} < {\tilde{r}}_{3}$ , for all $λ \in [0, \infty)$ and for all $t \in [0, t_{1}]$ .

As an important consequence of Lemma 8, it follows that $⋃_{λ \in [0, \infty)} A_{λ}$ is a bounded subset of E and once $E \subset \subset H$ , we can conclude the following corollary.
Corollary 2.
$A : = \bar{⋃_{λ \in [0, \infty)} A_{λ}}$ is a compact subset of H.

The next two theorems are about the continuity of the solutions with respect to the initial values and the upper semicontinuity of attractors.
Theorem 5.
For each $λ \in [0, \infty)$ , let $u_{λ}$ be the solution of ( 1 ) with $u_{λ} (0) = u_{0, λ}$ . If ${u_{0, λ}; λ \in [0, \infty)}$ is a bounded set in E and $u_{0, λ} \to u_{0, λ_{1}}$ in H as $λ \to λ_{1}$ , then for each $T > 0$ , $u_{λ} \to u_{λ_{1}}$ in $C ([0, T]; H)$ as $λ \to λ_{1}$ .
Proof.
Let $T > 0$ be fixed and suppose that ${u_{0, λ}; λ \in [0, \infty)}$ is a bounded set in E and $u_{0, λ} \to u_{0, λ_{1}}$ in H as $λ \to λ_{1}$ . Subtracting two equations in (1) and making the inner product with $u_{λ} - u_{λ_{1}}$ , we derive $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} ∥ u_{λ} - u_{λ_{1}} ∥_{H}^{2} \\ ⩽ L ∥ u_{λ} - u_{λ_{1}} ∥_{H}^{2} \\ + {∥ D^{λ} - D^{λ_{1}} ∥}_{L^{\infty} (R^{n})} [\int_{R^{n}} {| \nabla u_{λ_{1}} |}^{p (x) - 1} | \nabla u_{λ} | d x + \int_{R^{n}} {| \nabla u_{λ_{1}} |}^{p (x)} d x] \\ ⩽ L ∥ u_{λ} - u_{λ_{1}} ∥_{H}^{2} \\ + {∥ D^{λ} - D^{λ_{1}} ∥}_{L^{\infty} (R^{n})} \\ \times (\int_{R^{n}} \frac{1}{p (x)} {| \nabla u_{λ} |}^{p (x)} d x + \int_{R^{n}} \frac{1}{q (x)} {| \nabla u_{λ_{1}} |}^{p (x)} d x + \int_{R^{n}} {| \nabla u_{λ_{1}} |}^{p (x)} d x), \end{array}$ a.e. in $(0, T)$ . Using Lemma 8 and Remark 2, we have that there exists a constant $C > 0$ such that $(\int_{R^{n}} \frac{1}{p (x)} {| \nabla u_{λ} |}^{p (x)} d x + \int_{R^{n}} \frac{1}{q (x)} {| \nabla u_{λ_{1}} |}^{p (x)} d x + \int_{R^{n}} {| \nabla u_{λ_{1}} |}^{p (x)} d x) ⩽ C$ for all $t ⩾ 0$ and $λ \in [0, \infty)$ . So, $\frac{1}{2} \frac{d}{d t} ∥ u_{λ} - u_{λ_{1}} ∥_{H}^{2} ⩽ L ∥ u_{λ} - u_{λ_{1}} ∥_{H}^{2} + {∥ D^{λ} - D^{λ_{1}} ∥}_{L^{\infty} (R^{n})} \cdot C .$

Integrating this last inequality from 0 to t, $t ⩽ T$ , and using Gronwall–Bellman’s lemma we obtain ${∥ u_{λ} (t) - u_{λ_{1}} (t) ∥}_{H}^{2} ⩽ (∥ u_{0, λ} - u_{0, λ_{1}} ∥_{H}^{2} + 2 C T {∥ D^{λ} - D^{λ_{1}} ∥}_{L^{\infty} (R^{n})}) e^{2 L T}$ for all $t \in [0, T]$ .

Therefore, since $D^{λ} \to D^{λ_{1}}$ in $L^{\infty} (R^{n})$ as $λ \to λ_{1}$ , we conclude that $u_{λ} \to u_{λ_{1}}$ in $C ([0, T]; H)$ as $λ \to λ_{1}$ , whenever $u_{0, λ} \to u_{0, λ_{1}}$ in H as $λ \to λ_{1}$ . □

Repeating the same arguments used at the proof of Theorem 2 in [10] for p-Laplacian problems we obtain the following theorem. Theorem 6.
The family of global attractors ${A_{λ}; λ \in [0, \infty)}$ of the problem ( 1 ) is upper semicontinuous at $λ_{1}$ .

5. Final remarks

If we consider the problem $\begin{array}{rclr} \{\begin{matrix} \frac{\partial u_{λ}}{\partial t} (t) + A^{D^{λ}} (u_{λ} (t)) = B (u_{λ} (t)) - T (u_{λ} (t)), & t > 0, \\ u_{λ} (0) = u_{0, λ} \in H, \end{matrix} & (9) \end{array}$ with $B : H \to H$ as in the problem (1), and $T : H \to H$ is such that:

$⟨ T u, u ⟩ ⩾ 0$ for all $u \in H$ ;

$⟨ T u - T v, u - v ⟩ ⩾ 0$ for all $u, v \in H$ ;

T is demicontinuous; that is, $T u_{n} ⇀ T u$ as $u_{n} \to u$ .

Then in this case

A^{D^{λ}} + T

is a maximal monotone operator in H by Corollary 2.6 in [1]. So by Proposition 1 in [3], problem (9) determines a continuous semigroup of nonlinear operators

T_{λ} (t) : H \to H

, where

T_{λ} (t) u_{0, λ}

is the global weak solution of (9). We can repeat the arguments in Section 3 and we obtain that the semigroup

{T_{λ} (t)}

associated with problem (9) has a minimal closed global B-attractor

A^{λ}

in H, which is compact and invariant. It is worth noting that the uniform estimates remain valid on H as in Lemma 7, but in order to obtain the uniform estimates on E as in Lemma 8 and Remark 2 it is necessary additionally suppose that T is locally Lipschitz. Then we can also conclude that the family of global attractors

{A_{λ}; λ \in [0, \infty)}

of the problem (9) is upper semicontinuous at

λ_{1}

The authors in [11] studied the continuity of the flows and upper semicontinuity of global attractors varying the exponents for bounded domains. It would also be interesting to study the unbounded domain case for the problem (1).

Footnotes

Acknowledgements

The authors would like to thank the referee for his/her suggestions and comments.

This work was partially supported by the Brazilian research agency FAPEMIG grant CEX-APQ-04098-10 and Science without Borders – CAPES – PVE – Process 88881.0303888/2013-01. C.O. Alves was partially supported by CNPq – Grant 304036/2013-7.

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