Synchronization in a Quasilinear System Involving the ( p 1,p 2 ) -Laplacian

Abstract

In this article, we study the phenomenon of $ϵ$ -synchronization for a quasilinear system governed by $p$ -Laplacian, $p > 2$ . We obtain this result through the upper semicontinuity of the attractors associated with the systems and the synchronicity property of the limit system.

Keywords

upper semicontinuity of attractors synchronization

1. Introduction

In this article, we investigate a system of coupled reaction-diffusion equations of the form

\begin{aligned} (S_{n}) {\begin{cases} u_{t}^{n} = ϵ_{1}^{n} {({| u_{x}^{n} |}^{p_{1}^{n} - 2} u_{x}^{n})}_{x} + {| u^{n} |}^{q_{1}^{n} - 2} u^{n} (1 - {| u^{n} |}^{r_{1}^{n}}) - k (u^{n} - v^{n}) \\ v_{t}^{n} = ϵ_{2}^{n} {({| v_{x}^{n} |}^{p_{2}^{n} - 2} v_{x}^{n})}_{x} + {| v^{n} |}^{q_{2}^{n} - 2} v^{n} (1 - {| v^{n} |}^{r_{2}^{n}}) + k (u^{n} - v^{n}) \end{cases} \end{aligned}

defined in

W_{0}^{1, p_{1}^{n}} (0, 1) \times W_{0}^{1, p_{2}^{n}} (0, 1)

, where, for

i = 1, 2,

ϵ_{i}^{n}

and

k

are positive constants, while the exponents satisfy

\begin{aligned} q_{i}^{n} \geq 2, r_{i}^{n} > q_{i}^{n} - 2, and p_{i}^{n} > 2 . \end{aligned}

We are concerned with the phenomenon known as $ϵ$ -synchronization (Carvalho et al., 1998), that means, we prove that, when $(ϵ_{i}^{n}, p_{i}^{n}, q_{i}^{n}, r_{i}^{n}) \to (ϵ, p, q, r)$ , as $n \to \infty$ for $i = 1, 2$ , given $η > 0$ , there exists $M_{0} \in N$ such that

\begin{aligned} \underset{t \to \infty}{lim sup} ‖ u^{n} (t) - v^{n} (t) ‖_{L^{2} (0, 1)} < η, for every n, k \geq M_{0} . \end{aligned}

This adjustment in time in coupled systems is described in a large number of works, among which we mention (Carvalho et al., 1998; Hale, 1997; Rodrigues, 1996).

Synchronization is a well-known and naturally occurring phenomenon across various scientific disciplines. Since Christian Huygens’ first recorded observation in 1673 in The Horologium Oscillatorium, where he described the synchronization of two pendulum clocks mounted on a common support, similar effects have been identified in biological, electrical, mechanical, and even social systems. A comprehensive discussion of synchronization across multiple contexts is found by Strogatz (2003).

Synchronization is a strong spontaneous phenomenon observable in all areas of the human knowledge. Since the surprising first registered observation made at 1673 by Christian Huygens in The Horologium Oscillatorium on two pendulum clocks suspended on a common support till our days, the same phenomenon has been described in biological, electrical, mechanical systems, and even in social science context. According to Strogatz (2003), there are an extensive list of references confirming its occurrence in a high diversity of contexts. According to Boccaletti et al. (2018) and Ghosh et al. (2024), the analysis of synchronization phenomena in nonlinear dynamic systems began in the 20th century, with the work of Balthasar van der Pol, in electrical circuits.

Over the past decades, numerous mathematical studies have explored synchronization in coupled differential evolution systems, including finite- and infinite-dimensional cases, autonomous and non-autonomous settings, and deterministic, stochastic, and random models (see Afraimovich & Rodrigues, 1998; Caraballo & Kloeden, 2005; Carvalho & Primo, 2000; Gameiro & Rodrigues, 2001; Kloeden, 2003; Kloeden & Pavani, 2009; Rodrigues et al., 2001, and references therein).

However, despite $p$ -Laplacian problems are one of the most well-known category of nonlinear evolution systems, whose behavior has been systematically analyzed by a great number of authors, $p$ -Laplacian synchronization has been not treated up to now and, as far as we know, there is no work in the literature exhibiting the synchronization property for $p$ -Laplacian systems.

This work not only provides a further application to the synchronization theory but also it is interesting by itself. This particular system $(S_{n})$ involves equations that, when isolated seen, present a set of equilibrium states that can be qualitatively distinct for different values of $p_{i}^{n}$ , $q_{i}^{n}$ , and $ϵ_{i}$ , $i = 1, 2$ , depending on the position of the diffusion parameter $ϵ_{i}^{n}$ respectfully with bifurcation parameters, and depending if $p_{i}^{n} < q_{i}^{n}$ , $p_{i}^{n} = q_{i}^{n}$ or $p_{i}^{n} > q_{i}^{n}$ . The equilibria bifurcation scheme and stability properties as well for this kind of equation was studied and fully described by Takeuchi and Yamada (2000) and a discussion about the robustness of this problems with respect to $p_{i}^{n}, q_{i}^{n},$ and $ϵ_{i}$ can be found by Bruschi et al. (2010) and Pereira et al. (2015). When allowing initial freedom on these parameters, even if they are arbitrarily close each other, they can give rise to significantly distinct individual asymptotic tendencies which, we prove in this work, converge to a single common behavior under the coupling term.

In this article, we first exhibit a result of existence of global solution for $(S_{n})$ , then we prove that the semigroup associated with $(S_{n})$ is a gradient system and we show the existence of attractors for each value of $(ϵ_{i}^{n}, p_{i}^{n}, q_{i}^{n}, r_{i}^{n})$ . We also use an argument, based on an equivalent formulation of Ricceri’s three critical points theorem, (Bonanno, 2002, Theorem 2.3), to assure that problems $(S_{n})$ can have nontrivial attractors.

Then we show that those attractors are upper semicontinuous with respect to $n$ , and the limit system is given by the following equation:

\begin{aligned} (S L) {\begin{cases} u_{t} = ϵ {({| u_{x} |}^{p - 2} u_{x})}_{x} + {| u |}^{q - 2} u (1 - {| u |}^{r}) - k (u - v) \\ v_{t} = ϵ {({| v_{x} |}^{p - 2} v_{x})}_{x} + {| v |}^{q - 2} v (1 - {| v |}^{r}) + k (u - v) . \end{cases} \end{aligned}

The

ϵ

-synchronization of

(S_{n})

is demonstrated using the upper semicontinuity of attractors with respect to

n

and the strong synchronicity property enjoyed by

(S L)

This article is divided into seven sections. In Section 2, we prove the existence of solution and exhibit a Lyapunov functional for system $(S_{n})$ . In Section 3, we prove the existence of attractor and in Section 4, we obtain the upper semicontinuity of attractors associated to $(S_{n})$ . In Sections 5 and 7, we analyze the synchronicity of the system limit $(S L)$ and prove the $ϵ$ -synchronization for $(S_{n})$ . Nontriviality of attractors is carried in Section 6.

2. Existence and Uniqueness of Global Solutions

In this section, we establish the existence and uniqueness of a global solution in $L^{2} (0, 1) \times L^{2} (0, 1)$ for the coupled system

\begin{aligned} (S) {\begin{cases} u_{t} = ϵ_{1} {({| u_{x} |}^{p_{1} - 2} u_{x})}_{x} + {| u |}^{q_{1} - 2} u (1 - {| u |}^{r_{1}}) - k (u - v) \\ v_{t} = ϵ_{2} {({| v_{x} |}^{p_{2} - 2} v_{x})}_{x} + {| v |}^{q_{2} - 2} v (1 - {| v |}^{r_{2}}) + k (u - v), \end{cases} \end{aligned}

where

ϵ_{i}

and

k

are positive constants, while the exponents satisfy

\begin{aligned} q_{i} \geq 2, r_{i} > q_{i} - 2, and p_{i} > 2, for i = 1, 2, \end{aligned}

with initial data

(u_{0}, v_{0}) \in W_{0}^{1, p_{1}} (1, 0) \times W_{0}^{1, p_{2}} (1, 0)

In what follows we use the notation $L^{p}$ and $W_{0}^{1, p}$ to denote the spaces of functions $L^{p} (0, 1)$ and $W_{0}^{1, p} (0, 1)$ , respectively, and $D (- ϵ Δ_{p})$ to denote the domain of the operator $- ϵ Δ_{p}$ , given by the following equation:

\begin{aligned} D (- ϵ Δ_{p}) = {u \in W_{0}^{1, p} (0, 1); - ϵ Δ_{p} u \in L^{2} (0, 1)} . \end{aligned}

To prove the existence of a solution, we follow the approach by Ouardi and Hachimi (2006). The construction is carried out iteratively by considering, at each step, uncoupled problems that depend on the solutions obtained in the previous step. This process generates a sequence of solutions $(u^{n}, v^{n})$ that converges to an element $(u, v) \in C ([0, T]; L^{2}) \times C ([0, T]; L^{2})$ for each $T > 0$ . Finally, we show that $(u, v)$ satisfies system $(S)$ .

As a first step in this iterative scheme, we analyze the following problems:

\begin{aligned} (P_{1}) {\begin{cases} u_{t}^{1} = ϵ_{1} {({| u_{x}^{1} |}^{p_{1} - 2} u_{x}^{1})}_{x} + {| u^{1} |}^{q_{1} - 2} u^{1} (1 - {| u^{1} |}^{r_{1}}) - k u^{1} \\ u^{1} (0, t) = 0, u^{1} (1, t) = 0 t > 0 \\ u^{1} (0) = u_{0} \in W_{0}^{1, p_{1}} \end{cases} \end{aligned}

and

\begin{aligned} ({\tilde{P}}_{1}) {\begin{cases} v_{t}^{1} = ϵ_{2} {({| v_{x}^{1} |}^{p_{2} - 2} v_{x}^{1})}_{x} + {| v^{1} |}^{q_{2} - 2} v^{1} (1 - {| v^{1} |}^{r_{2}}) - k v^{1} \\ v^{1} (0, t) = 0, v^{1} (1, t) = 0 t > 0 \\ v^{1} (0) = v_{0} \in W_{0}^{1, p_{2}} . \end{cases} \end{aligned}

In order to apply Theorem 5.3 by Ötani (1977), we observe that the previous problems can be rewritten within an abstract framework by defining the functionals

\begin{aligned} φ_{i}^{1} (u) = \frac{ϵ_{i}}{p_{i}} \int_{0}^{1} | u_{x} (x) |^{p_{i}} d x + \int_{0}^{1} \int_{0}^{u (x)} | v |^{q_{i} + r_{i} - 2} v d v d x + k \int_{0}^{1} \int_{0}^{u (x)} v d v d x \end{aligned}

(1)

u \in W_{0}^{1, p_{i}}

and

φ_{i}^{1} (u) = + \infty

u \in L^{2} ∖ W_{0}^{1, p_{i}}

, and

\begin{aligned} φ_{i}^{2} u = \int_{0}^{1} \int_{0}^{u (x)} | v |^{q_{i} - 2} v d v d x \end{aligned}

(2)

u \in L^{q_{i}}

, and

φ_{i}^{2} u = + \infty

u \in L^{2} ∖ L^{q_{i}}

. It follows from Brézis (1973) that the subdifferentials

\partial φ_{i}^{1}

and

\partial φ_{i}^{2}

φ_{i}^{1}

and

φ_{i}^{2}

, respectively, are maximal monotone operators in

L^{2}

. Consequently, Problems

(P_{1})

and

({\tilde{P}}_{1})

can be rewritten in the following form:

\begin{aligned} (P_{1}) {\begin{cases} u_{t}^{1} + \partial φ_{1}^{1} u^{1} = \partial φ_{1}^{2} u^{1} \\ u^{1} (0) = u_{0} \in W_{0}^{1, p_{1}} \end{cases} and ({\tilde{P}}_{1}) {\begin{cases} v_{t}^{1} + \partial φ_{2}^{1} v^{1} = \partial φ_{2}^{2} v^{1} \\ v^{1} (0) = v_{0} \in W_{0}^{1, p_{2}} . \end{cases} \end{aligned}

Applying Ötani (1977, Theorem 5.3), we conclude the existence of strong solutions $u^{1} (t)$ and $v^{1} (t)$ in $[0, T)$ for $(P_{1})$ and $({\tilde{P}}_{1})$ , respectively, for each $T > 0$ .

Subsequently, for each $n \in I N$ , $n > 1$ , we consider the following problems:

\begin{aligned} (P_{n}) {\begin{cases} u_{t}^{n} + \partial φ_{1}^{1} u^{n} = \partial φ_{1}^{2} u^{n} + k v^{n - 1} \\ u^{n} (0) = u_{0} \in W_{0}^{1, p_{1}}, \end{cases} \end{aligned}

\begin{aligned} ({\tilde{P}}_{n}) {\begin{cases} v_{t}^{n} + \partial φ_{2}^{1} v^{n} = \partial φ_{2}^{2} v^{n} + k u^{n - 1} \\ v^{n} (0) = v_{0} \in W_{0}^{1, p_{2}} . \end{cases} \end{aligned}

Applying Ötani (1977, Theorem 5.3), we conclude that these problems also admit strong solutions

u^{n} (t)

and

v^{n} (t)

[0, T)

, respectively, for each

T > 0

and

n \in I N

The next step is to show that the sequence ${(u^{n}, v^{n})}$ converges to a pair $(u, v)$ which is a solution of $(S)$ . To achieve this, we will employ a compactness result established by Baras, specifically Vrabie (1995, Theorem 2.5.3).

First, observe that the sets

\begin{aligned} {f_{n} ≐ | u^{n} |^{q_{1} - 2} u^{n} - | u^{n} |^{q_{1} + r_{1} - 2} u^{n} + k v^{n - 1} - k u^{n}}_{n \in I N} \end{aligned}

(3)

and

\begin{aligned} {g_{n} ≐ | v^{n} |^{q_{1} - 2} v^{n} - | v^{n} |^{q_{1} + r_{1} - 2} v^{n} + k u^{n - 1} - k v^{n}}_{n \in I N} \end{aligned}

(4)

are uniformly integrable in

L^{1} (0, T_{0}; L^{2})

for some

T_{0}

that will be determined in what follows. This result follows directly from the lemmas presented below.

Lemma 2.1

Let $u^{n}$ and $v^{n}$ solutions of $(P_{n})$ and $({\tilde{P}}_{n})$ respectively. There is $T_{0} > 0$ , such that the sequence ${‖ u^{n} (t) ‖_{2} + ‖ v^{n} (t) ‖_{2}}$ is bounded, for all $n \in I N$ and $t \in [0, T_{0}]$ .

Proof.

For $n = 1$ , multiplying the equations $(P_{1})$ and $({\tilde{P}}_{1})$ by $u^{1}$ and $v^{1}$ , respectively, and applying Young’s inequality, we obtain

\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ u^{1} (t) ‖_{2}^{2} \leq \frac{r_{1}}{q_{1} + r_{1}} and \frac{1}{2} \frac{d}{d t} ‖ v^{1} (t) ‖_{2}^{2} \leq \frac{r_{2}}{q_{2} + r_{2}} . \end{aligned}

Fix an arbitrary $T_{0} > 0$ . Integrating from $0$ to $t$ , with $t \leq T_{0}$ , we have

\begin{aligned} max {‖ u^{1} (t) ‖_{2}^{2}, ‖ v^{1} (t) ‖_{2}^{2}} \leq M \end{aligned}

for all

t \in [0, T_{0}]

, where

M

depends on

‖ u_{0} ‖_{2}

‖ v_{0} ‖_{2}

r_{i}

q_{i}

i = 1, 2

, and

T_{0}

For each $n \in I N$ , multiplying the equations $(P_{n})$ and $({\tilde{P}}_{n})$ by $u^{n}$ and $v^{n}$ respectively, and using Young’s inequality along with previous estimates, we conclude that for all $n \in I N$ and $t \in [0, T_{0}]$

\begin{aligned} max {‖ u^{n} (t) ‖_{2}^{2}, ‖ v^{n} (t) ‖_{2}^{2}} \leq M + M (k T_{0} + k^{2} T_{0}^{2} + \dots + k^{n - 1} T_{0}^{n - 1}) . \end{aligned}

Now, choosing $T_{0}$ such that $k T_{0} < 1$ , there exists a constant $N_{1} > 0$ depending on $‖ u_{0} ‖_{2}$ , $‖ v_{0} ‖_{2}$ , $T_{0}$ , $r_{i}$ , $q_{i}$ and $k$ , such that for all $t \in [0, T_{0}]$ and $n \in I N$

\begin{aligned} ‖ u^{n} (t) ‖_{2}^{2} + ‖ v^{n} (t) ‖_{2}^{2} \leq N_{1} . \end{aligned}

In the next lemma, we establish a similar estimate for $u^{n}$ and $v^{n}$ in $W_{0}^{1, p_{1}}$ and $W_{0}^{1, p_{2}}$ , respectively.

Lemma 2.2

If $(u_{0}, v_{0}) \in W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ , there exists a constant $N_{2} > 0$ such that

\begin{aligned} ‖ u^{n} (t) ‖_{W_{0}^{1, p_{1}}} + ‖ v^{n} (t) ‖_{W_{0}^{1, p_{2}}} \leq N_{2} \end{aligned}

for all

n \in I N

and

t \in [0, T_{0}]

T_{0}

as in the previous lemma.

Proof.

In fact, since $u^{n}$ and $v^{n}$ are solutions of $(P_{n})$ and $({\tilde{P}}_{n})$ , respectively, we have

\begin{aligned} \frac{d}{d t} φ_{1}^{1} u^{n} (t) \leq \frac{d}{d t} φ_{1}^{2} u^{n} (t) + \frac{k^{2} N_{1}}{2} and \frac{d}{d t} φ_{2}^{1} v^{n} (t) \leq \frac{d}{d t} φ_{2}^{2} v^{n} (t) + \frac{k^{2} N_{1}}{2} . \end{aligned}

By integrating from $0$ to $t$ , $t \leq T_{0}$ , we obtain

\begin{aligned} ‖ u^{n} (t) ‖_{W_{0}^{1, p_{1}}} + ‖ v^{n} (t) ‖_{W_{0}^{1, p_{2}}} \leq N_{2} \end{aligned}

for all

t \in [0, T_{0}]

and

n \in I N

, where

N_{2}

depends

‖ u_{0} ‖_{W_{0}^{1, p_{1}}}

‖ v_{0} ‖_{W_{0}^{1, p_{2}}}

p_{i}

q_{i}

r_{i}

k

T_{0}

and

ϵ_{i}

Denoting by $f^{i}$ the function $f^{i} (s) = | s |^{q_{i} - 2} s - | s |^{q_{i} + r_{i} - 2} s$ , for $i = 1, 2$ , it is not difficult to see that

\begin{aligned} ‖ f^{1} (u^{n} (t)) ‖_{2}^{2} \leq N_{3} ‖ u^{n} (t) ‖_{2}^{2} and ‖ f^{2} (v^{n} (t)) ‖_{2}^{2} \leq N_{3} ‖ v^{n} (t) ‖_{2}^{2}, \end{aligned}

where

N_{3}

is independent of

(n, t, x) \in I N \times [0, T_{0}] \times [0, 1]

. Therefore, we conclude that the families

{f_{n}}_{n \in I N}

and

{g_{n}}_{n \in I N}

are uniformly integrable in

L^{1} (0, T_{0}; L^{2})

It follows from Vrabie (1995, Theorem 2.5.3) that the sets

\begin{aligned} {u^{n}; u^{n} is solution of (P_{n})}_{n \in I N} and {v^{n}; v^{n} is solution of ({\tilde{P}}_{n})}_{n \in I N} \end{aligned}

are relatively compact in

C ([0, T_{0}]; L^{2})

Then there is a subsequence still denoted by ${(u^{n}, v^{n})}$ and an element $(u, v) \in C ([0, T_{0}]; L^{2}) \times C ([0, T_{0}]; L^{2})$ such that

\begin{aligned} (u^{n}, v^{n}) \to (u, v) in C ([0, T_{0}]; L^{2}) \times C ([0, T_{0}]; L^{2}) . \end{aligned}

Since $f^{i}$ is continuous on $L^{2}$ , for $i = 1, 2$ , and $f^{1} (u (t))$ and $f^{2} (v (t))$ belong to $L^{2} (0, T_{0}, L^{2})$ , it follows from Brézis (1973, Theorem 3.6), that $(u, v)$ is a strong solution of $(S)$ . To ensure that such solution is indeed a global solution, recall that $T_{0}$ is independent of the initial data. The uniqueness of solution is easily obtained if we consider $k > N_{3}$ .

Therefore, for large values of $k$ , the system $(S)$ generates a continuous semigroup ${S (t)}$ in $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ (as stated by Ladyzhenskaya, 1991), given by the following equation:

\begin{aligned} {S (t) : W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}} \to W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}; t \geq 0}, \end{aligned}

where

\begin{aligned} S (t) (u_{0}, v_{0}) = (u (t), v (t) . \end{aligned}

Finally, we observe that the semigroup associated with the system

(S)

is indeed a gradient system, with Lyapunov functional

V : W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}} \to I R

given by the following equation:

\begin{aligned} V (ω, \bar{ω}) = φ_{1}^{1} (ω) + φ_{2}^{1} (\bar{ω}) - φ_{1}^{2} (ω) - φ_{2}^{2} (\bar{ω}) - k ⟨ ω, \bar{ω} ⟩ . \end{aligned}

Since the phase space varies with $n \in I N$ , as $A_{n} \subset W_{0}^{1, p_{1}^{n}} \times W_{0}^{1, p_{2}^{n}}$ , it becomes necessary to adopt a common phase space for all $n \in I N$ . Given that $W_{0}^{1, p_{1}^{n}} \times W_{0}^{1, p_{2}^{n}}$ is compactly embedded in $L^{2} \times L^{2}$ for all $n \in I N$ , a natural choice for the new phase space is $L^{2} \times L^{2}$ . Before proceeding, we state the following lemmas, which can be proved in the same way as Lemma 3.3 and Theorem 3.4 (Bruschi et al., 2010).

Lemma 2.3

Let ${S_{n}^{i} (t) : W_{0}^{1, p_{i}^{n}} \to W_{0}^{1, p_{i}^{n}}; t \geq 0}$ be the semigroup generated by $- ϵ_{i}^{n} Δ_{p_{i}^{n}}$ , and $(u^{n}, v^{n})$ solution of $(S_{n})$ . Then

\begin{aligned} ‖ S_{n}^{1} (h) u^{n} (t) - u^{n} (t) ‖_{2} \to 0 and ‖ S_{n}^{2} (h) v^{n} (t) - v^{n} (t) ‖_{2} \to 0 \end{aligned}

h \to 0

uniformly for

n \in I N

, for each

t > 0

. Moreover, if

T > 0

then

\begin{aligned} ‖ u^{n} (T - h) - S_{n}^{1} (h) u^{n} (T - h) ‖_{2} \to 0 and ‖ v^{n} (T - h) - S_{n}^{2} (h) v^{n} (T - h) ‖_{2} \to 0 \end{aligned}

h \to 0

, uniformly for

n \in I N

Lemma 2.4

The set $M^{n} ≐ {(u^{n}, v^{n}); (u^{n}, v^{n}) is solution of (S_{n}), n \in I N}$ is relatively compact in $C ([0, T]; L^{2}) \times C ([0, T]; L^{2})$ .

As previously mentioned, we observe that it is possible to extend the phase space to $L^{2} \times L^{2}$ by showing that, given $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ there exists a weak solution of $(S_{n})$ with initial data $(u_{0}, v_{0})$ .

In fact, consider $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ , and let ${(u_{0}^{m}, v_{0}^{m})}$ be a sequence in $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ such that $(u_{0}^{m}, v_{0}^{m}) \to (u_{0}, v_{0})$ in $L^{2} \times L^{2}$ . For each $m \in I N$ , there exists a strong solution, $(u^{m}, v^{m}) \in C ([0, T]; L^{2}) \times C ([0, T]; L^{2})$ , for system $(S)$ , with initial data $(u_{0}^{m}, v_{0}^{m})$ . By Lemma 3.1, the sequence ${(u^{m}, v^{m})}$ is bounded in $L^{2} \times L^{2}$ , consequently,

\begin{aligned} | u^{m} |^{q_{1} - 2} u^{m} - | u^{m} |^{q_{1} + r_{1} - 2} u^{m} - k u^{m} + k v^{m} \end{aligned}

and

\begin{aligned} | v^{m} |^{q_{2} - 2} v^{m} - | v^{m} |^{q_{2} + r_{2} - 2} v^{m} - k v^{m} + k u^{m} \end{aligned}

belong to

L^{1} (0, T; L^{2})

. Applying Lemmas 2.3 and 2.4, with appropriate adaptations, we obtain

(u, v) \in C ([0, T]; L^{2}) \times C ([0, T]; L^{2})

and a subsequence still denoted by

{(u^{m}, v^{m})}

, such that

\begin{aligned} (u^{m}, v^{m}) \to (u, v) in C ([0, T]; L^{2}) \times C ([0, T]; L^{2}) . \end{aligned}

Consequently,

\begin{aligned} | u^{m} |^{q_{1} - 2} u^{m} - | u^{m} |^{q_{1} + r_{1} - 2} u^{m} - k u^{m} + k v^{m} \to | u |^{q_{1} - 2} u - | u |^{q_{1} + r_{1} - 2} u - k u + k v \end{aligned}

and

\begin{aligned} | v^{m} |^{q_{2} - 2} v^{m} - | v^{m} |^{q_{2} + r_{2} - 2} v^{m} - k v^{m} + k u^{m} \to | v |^{q_{2} - 2} v - | v |^{q_{2} + r_{2} - 2} v - k v + k u \end{aligned}

L^{1} (0, T; L^{2})

. Therefore,

(u, v)

is weak solution of

(S)

, with initial data

(u_{0}, v_{0})

(see Brézis, 1973, Definition 3.1).

3. Existence of Global Attractors

In this section, we establish the existence of a global attractor in $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ for the semigroup associated with $(S)$ . It is well-known that this can be achieved by proving the asymptotic compactness and dissipativity of $S (t)$ , see Hale (1988).

The following two lemmas ensure dissipativity, first in $L^{2} \times L^{2}$ and then in $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ , respectively.

Lemma 3.1
If $(u, v)$ is solution of $(S)$ then there is a constant $N_{4} > 0$ such that
$\begin{aligned} ‖ u (t) ‖_{2}^{2} + ‖ v (t) ‖_{2}^{2} \leq N_{4} \end{aligned}$
for all $t \geq 0$ .
Proof.
In fact, we have
$\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ u (t) ‖_{2}^{2} + \frac{r_{1}}{q_{1} + r_{1}} ‖ u (t) ‖_{2}^{q_{1} + r_{1}} \leq \frac{r_{1}}{q_{1} + r_{1}} - k ‖ u (t) ‖_{2}^{2} + k ‖ u (t) ‖_{2} ‖ v (t) ‖_{2} \end{aligned}$
and
$\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ v (t) ‖_{2}^{2} + \frac{r_{2}}{q_{2} + r_{2}} ‖ v (t) ‖_{2}^{q_{2} + r_{2}} \leq \frac{r_{2}}{q_{2} + r_{2}} - k ‖ v (t) ‖_{2}^{2} + k ‖ u (t) ‖_{2} ‖ v (t) ‖_{2} . \end{aligned}$

Without loss of generality, we assume $q_{1} + r_{1} \leq q_{2} + r_{2}$ . Thus, from the Hölder Inequality, if $\bar{m} ≐ min {\frac{r_{1}}{q_{1} + r_{1}}, \frac{r_{2}}{q_{2} + r_{2}}}$ , we have
$\begin{aligned} \frac{d}{d t} (‖ u (t) ‖_{2}^{2} + ‖ v (t) ‖_{2}^{2}) + \frac{2 \bar{m}}{2^{\frac{q_{1} + r_{1} - 2}{2}}} {(‖ u (t) ‖_{2}^{2} + ‖ v (t) ‖_{2}^{2})}^{\frac{q_{1} + r_{1}}{2}} \leq 2 N_{5}, \end{aligned}$
where $N_{5}$ depends only on $q_{i}$ and $r_{i}$ . Moreover, it follows from Temam (1997, Lemma 5.1) that for all $t \geq 0$
$\begin{aligned} ‖ u (t) ‖_{2}^{2} + ‖ v (t) ‖_{2}^{2} \leq {(\frac{N_{5}}{\bar{m}} 2^{\frac{q_{1} + r_{1} - 2}{2}})}^{\frac{2}{q_{1} + r_{1}}} + {(\frac{2^{\frac{q_{1} + r_{1} - 2}{2}}}{\bar{m} (q_{1} + r_{1} - 2) t})}^{\frac{2}{q_{1} + r_{1} - 2}} . \end{aligned}$

Let $T_{1} > 0$ be such that for all $t \geq T_{1}$
$\begin{aligned} {(\frac{2^{\frac{q_{1} + r_{1} - 2}{2}}}{\bar{m} (q_{1} + r_{1} - 2) t})}^{\frac{2}{q_{1} + r_{1} - 2}} \leq 1. \end{aligned}$
If $t \in [0, T_{1}]$ , we have
$\begin{aligned} ‖ u (t) ‖_{2}^{2} + ‖ v (t) ‖_{2}^{2} \leq ‖ u_{0} ‖_{2}^{2} + ‖ v_{0} ‖_{2}^{2} + 2 N_{5} T_{1}, \end{aligned}$
and therefore,
$\begin{aligned} ‖ u (t) ‖_{2}^{2} + ‖ v (t) ‖_{2}^{2} \leq N_{4} \end{aligned}$
for $t \geq 0$ , where $N_{4}$ depends only on $‖ u_{0} ‖_{2}$ , $‖ v_{0} ‖_{2}$ , $q_{i}$ , and $r_{i}$ .
Remark 1
The constant $N_{4}$ can be chosen uniformly for initial data belonging to subsets of $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ , that are bounded in $L^{2} \times L^{2}$ , provided that $q_{i}$ and $r_{i}$ also lie within bounded subsets. Moreover, $N_{4}$ remains independent of $k$ and $p_{i}$ , for $i = 1, 2$ .
Lemma 3.2
If $(u, v)$ is solution of $(S)$ then there is a positive constant $N_{6}$ such that for all $t \geq 0$
$\begin{aligned} ‖ u (t) ‖_{W_{0}^{1, p_{1}}} + ‖ v (t) ‖_{W_{0}^{1, p_{2}}} \leq N_{6} . \end{aligned}$

Proof.
Take $T > 0$ . If $(u, v)$ is solution of $(S)$ , we have
$\begin{aligned} \frac{d}{d t} φ_{1}^{1} u (t) \leq \frac{d}{d t} φ_{1}^{2} u (t) + \frac{k^{2}}{2} ‖ v (t) ‖_{2}^{2} . \end{aligned}$

Integrating from $0$ to $t$ , with $t \leq T$ , we obtain for each $t \in [0, T]$
$\begin{aligned} ‖ u (t) ‖_{W_{0}^{1, p_{1}}} \leq N_{7}, \end{aligned}$
with $N_{7}$ depending on $p_{1}$ , $ϵ_{1}$ , $T$ , $k$ , and $N_{4}$ .

Without loss of generality we take $T = 2 T_{1}$ , $T_{1}$ as in the previous lemma. We have
$\begin{aligned} \frac{d}{d t} φ_{1}^{1} u (t) \leq (q_{1} - 1) φ_{1}^{1} u (t) + N_{8}, \end{aligned}$
(1)
where $N_{8}$ depends only on $q_{i}$ , $r_{i}$ , $k$ , and $T_{1}$ , and is independent of the initial data if $t > T_{1}$ . Moreover, as $φ_{1}^{1} (0) = 0$ , it follows that
$\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ u (t) ‖_{2}^{2} + φ_{1}^{1} u (t) \leq N_{9} + N_{10} φ_{1}^{1} u (t), \end{aligned}$
where $N_{9} = \frac{k N_{4}}{2} + \frac{r_{1}}{(q_{1} + r_{1}) η^{\frac{q_{1} + r_{1}}{r_{1}}}}$ , $N_{10} = q_{1} η^{\frac{q_{1} + r_{1}}{q_{1}}}$ , with $η$ chosen small enough so that $N_{10} < 1$ . Note that $η$ can be taken uniformly for $q_{i}$ and $r_{i}$ in bounded subsets. Thus for all $t \geq T_{1}$
$\begin{aligned} \int_{t}^{t + T_{1}} φ_{1}^{1} u (s) d s \leq \frac{N_{9} T_{1}}{1 - N_{10}} + \frac{N_{4}}{2 (1 - N_{10})} . \end{aligned}$

Then from Temam (1997, Lemma 1.1) and from (1), it follows that for any $t \geq 2 T_{1}$ ,
$\begin{aligned} ‖ u (t) ‖_{W_{0}^{1, p_{1}}} \leq {\frac{p_{1}}{ϵ_{1}} (\frac{N_{8} + 2 N_{9} T_{1}}{2 T_{1} (1 - N_{10})} + N_{8} T_{1}) e^{(q_{1} - 1) T_{1}}}^{\frac{1}{p_{1}}} \leq N_{11}, \end{aligned}$
where $N_{11}$ is taken uniformly for $p_{i}$ , $ϵ_{i}$ , $q_{i}$ , $r_{i}$ , and $k$ in bounded subsets of ${I R}_{+}^{*}$ .

Analogous estimates are obtained for $‖ v (t) ‖_{W_{0}^{1, p_{2}}}$ . Therefore, there is $N_{6} > 0$ , depending on $p_{i}$ , $q_{i}$ , $r_{i}$ , $ϵ_{i}$ , $k$ , $‖ u_{0} ‖_{W_{0}^{1, p_{1}}}$ , and $‖ v_{0} ‖_{W_{0}^{1, p_{2}}}$ , such that for all $t \geq 0$
$\begin{aligned} ‖ u (t) ‖_{W_{0}^{1, p_{1}}} + ‖ v (t) ‖_{W_{0}^{1, p_{2}}} \leq N_{6} . \end{aligned}$

Remark 2
It is easy to see that for fixed $δ > 0$ there is a constant, $N (δ) > 0$ , satisfying
$\begin{aligned} ‖ u (t) ‖_{W_{0}^{1, p_{1}}} + ‖ v (t) ‖_{W_{0}^{1, p_{2}}} \leq N (δ), \end{aligned}$
for all $t \geq δ$ and $N (δ)$ independent of the initial data.

The following result establishes the continuity of the flow in $L^{2} \times L^{2}$ , and will be used to prove the asymptotic compactness for the semigroup.
Lemma 3.3
Let $(u_{0}^{n}, v_{0}^{n}), (u_{0}, v_{0}) \in W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ be such that $(u_{0}^{n}, v_{0}^{n}) \to (u_{0}, v_{0})$ in $L^{2} \times L^{2}$ . Then, for all $t > 0$ ,
$\begin{aligned} (u_{n} (t), v_{n} (t)) \to (u (t), v (t)) \end{aligned}$
in $L^{2} \times L^{2}$ , where $(u_{n} (\cdot), v_{n} (\cdot)) = S (\cdot) (u_{0}^{n}, v_{0}^{n})$ and $(u (\cdot), v (\cdot)) = S (\cdot) (u_{0}, v_{0})$ .
Proof.
In fact,
$\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ u_{n} - u ‖_{2}^{2} + ‖ v_{n} - v ‖_{2}^{2}) \leq C (‖ u_{n} - u ‖_{2}^{2} + ‖ v_{n} - v ‖_{2}^{2}), \end{aligned}$
where $C ≐ max {C_{1}, C_{2}}$ and $C_{i} ≐ \frac{r_{i} (q_{i} - 1)^{\frac{q_{i} + r_{i} - 2}{r_{i}}}}{q_{i} + r_{i} - 2}$ , $i = 1, 2$ . Integrating from $0$ to $t$ , we obtain
$\begin{aligned} ‖ u_{n} (t) - u (t) ‖_{2}^{2} + ‖ v_{n} (t) - v (t) ‖_{2}^{2} \leq ‖ u_{0}^{n} & - u_{0} ‖_{2}^{2} + ‖ v_{0}^{n} - v_{0} ‖_{2}^{2} \\ + \int_{0}^{t} 2 C (‖ u_{n} (s) - u (s) ‖_{2}^{2} + ‖ v_{n} (s) - v (s) ‖_{2}^{2}) d s . \end{aligned}$
And the result follows from Gronwall’s inequality.

The next lemma guarantees the asymptotic compactness of the semigroup. The proof follows from the continuity of $S (t)$ and the compact embedding of $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ into $L^{2} \times L^{2}$ .
Lemma 3.4
For each $t > 0$ , the operator $S (t)$ is compact.
Proof.
Indeed, consider a bounded subset $B \subset W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ , and a sequence ${(u_{0}^{n}, v_{0}^{n})} \subset B$ . Let $N > 0$ be such that $‖ u_{0}^{n} ‖_{W_{0}^{1, p_{1}}} + ‖ v_{0}^{n} ‖_{W_{0}^{1, p_{2}}} \leq N$ , for all $n \in I N$ .

So there is a subsequence, still denoted by ${(u_{0}^{n}, v_{0}^{n})}$ and $(u_{0}, v_{0}) \in W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ such that
$\begin{aligned} (u_{0}^{n}, v_{0}^{n}) \to (u_{0}, v_{0}) in L^{2} \times L^{2} \end{aligned}$
and
$\begin{aligned} (u_{0}^{n}, v_{0}^{n}) ⇀ (u_{0}, v_{0}) in W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}} . \end{aligned}$

By Lemma 3.2 and the continuity of $S (t)$ (Lemma 3.3), we have
$\begin{aligned} (u_{n} (t), v_{n} (t)) \to (u (t), v (t)) in L^{2} \times L^{2} \end{aligned}$
and
$\begin{aligned} (u_{n} (t), v_{n} (t)) ⇀ (u (t), v (t)) in W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}} . \end{aligned}$

It follows from Tartar’s inequality that
$\begin{aligned} \tilde{ϵ} \tilde{γ} (‖ u_{n} (s) - u (s) ‖_{W_{0}^{1, p_{1}}}^{p_{1}} + ‖ v_{n} (s) - v (s) ‖_{W_{0}^{1, p_{2}}}^{p_{2}}) & \leq - \frac{1}{2} \frac{d}{d t} (‖ u_{n} (s) - u (s) ‖_{2}^{2} + ‖ v_{n} (s) - v (s) ‖_{2}^{2}) \\ + C (‖ u_{n} (s) - u (s) ‖_{2}^{2} + ‖ v_{n} (s) - v (s) ‖_{2}^{2}) \end{aligned}$
with $\tilde{ϵ} \tilde{γ} ≐ min {ϵ_{1} γ_{1}, ϵ_{2} γ_{2}}$ .

Integrating from $0$ to $t$ and using the dominated convergence theorem, we obtain
$\begin{aligned} \int_{0}^{t} (‖ u_{n} (s) - u (s) ‖_{W_{0}^{1, p_{1}}}^{p_{1}} + ‖ v_{n} (s) - v (s) ‖_{W_{0}^{1, p_{2}}}^{p_{2}}) \to 0 \end{aligned}$
as $n \to \infty$ .

Thus there exists a subsequence still denoted by $(u_{n}, v_{n})$ such that
$\begin{aligned} ‖ u_{n} (t) - u (t) ‖_{W_{0}^{1, p_{1}}} + ‖ v_{n} (t) - v (t) ‖_{W_{0}^{1, p_{2}}} \to 0 \end{aligned}$
$a e$ in $[0, T]$ .

To extend the convergence to all $t \in [0, T]$ we consider the subset $A \subset [0, T]$ such that
$\begin{aligned} (u_{n} (t), v_{n} (t)) \to (u (t), v (t)) in W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}} \end{aligned}$
for all $t \in A$ . Let be $t_{0} \in [0, T] ∖ A$ and $ψ_{1} x ≐ \frac{ϵ_{1}}{p_{1}} ‖ x ‖_{W_{0}^{1, p_{1}}}^{p_{1}}$ , for $x \in L^{2}$ , thus
$\begin{aligned} | ψ_{1} u_{n} (t_{0}) - ψ_{1} u (t_{0}) | & \leq | ψ_{1} u_{n} (t_{0}) - ψ_{1} u_{n} (θ) | + | ψ_{1} u_{n} (θ) - ψ_{1} u (θ) | + | ψ_{1} u (θ) - ψ_{1} u (t_{0}) | \\ \leq | \int_{θ}^{t_{0}} ⟨ \partial ψ_{1} u_{n} (s), \frac{d}{d t} u_{n} (s) ⟩ d s | + | ψ_{1} u_{n} (θ) - ψ_{1} u (θ) | \\ + | \int_{θ}^{t_{0}} ⟨ \partial ψ_{1} u (s), \frac{d}{d t} u (s) ⟩ d s | . \end{aligned}$
By choosing $θ$ sufficiently close to $t_{0}$ so that $ψ_{1} u_{n} (θ) \to ψ_{1} u (θ)$ , we can conclude that $ψ_{1} u_{n} (t) \to ψ_{1} u (t)$ for all $t \in [0, T]$ , since $⟨ \partial ψ_{1} u_{n} (s), \frac{d}{d t} u_{n} (s) ⟩$ is uniformly bounded. An analogous result holds for ${v_{n}}$ .

Therefore, $(u_{n} (t), v_{n} (t)) \to (u (t), v (t))$ in $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ , for all $[0, T]$ .

This completes the proof.

Once established that the semigroup ${S (t); t \in {I R}_{+}}$ is compact and dissipative, it follows from Ladyzhenskaya (1991, Theorem 2.2), the existence of a global minimal attractor which is compact and invariant in $W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ , for the semigroup ${S (t); t \in {I R}_{+}}$ .

We denote by $A_{n}$ the attractor of the semigroup ${S_{n} (t)}$ associated with the system $(S_{n})$ , with parameters $ϵ_{i}^{n}$ , $p_{i}^{n}$ , $q_{i}^{n}$ , $r_{i}^{n}$ , and $k$ , for $i = 1, 2$ , whose existence can be obtained in a similar way.
4. Upper Semicontinuity of the Global Attractors

In this section, we prove that the family of attractors ${A_{n}}$ is upper semicontinuous, that is, as $n \to + \infty$

\begin{aligned} d (A_{n}, A_{L}) \to 0, \end{aligned}

where

A_{L}

is the attractor associated with the limit system

(S L)

The next theorem ensures the continuity of the flow.

Theorem 4.1
Let $(u^{n}, v^{n})$ and $(u, v)$ solutions of $(S_{n})$ and $(S L)$ , respectively, with initial data $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ . For each $t > 0$ , if $n \to + \infty$ , then
$\begin{aligned} ‖ (u^{n} (t), v^{n} (t)) - (u (t), v (t)) ‖_{2 \times 2} \to 0. \end{aligned}$

Proof.
Take $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ . We know that there is a weak solution of $(S_{n})$ and $(S L)$ with initial data $(u_{0}, v_{0})$ for every $n \in I N$ .

Fix $δ > 0$ arbitrary, then $(u^{n} (δ), v^{n} (δ)) \in W_{0}^{1, p_{1}^{n}} \times W_{0}^{1, p_{2}^{n}}$ . Consider the new systems
$\begin{aligned} (S_{n}) {\begin{cases} u_{t}^{n} = ϵ_{1}^{n} {({| u_{x}^{n} |}^{p_{1}^{n} - 2} u_{x}^{n})}_{x} + {| u^{n} |}^{q_{1}^{n} - 2} u^{n} (1 - {| u^{n} |}^{r_{1}^{n}}) - k (u^{n} - v^{n}) \\ v_{t}^{n} = ϵ_{2}^{n} {({| v_{x}^{n} |}^{p_{2}^{n} - 2} v_{x}^{n})}_{x} + {| v^{n} |}^{q_{2}^{n} - 2} v^{n} (1 - {| v^{n} |}^{r_{2}^{n}}) + k (u^{n} - v^{n}) \end{cases} \end{aligned}$

$\begin{aligned} (S_{n}) {\begin{cases} {\bar{u}}_{t}^{n} = ϵ_{1}^{n} {({| {\bar{u}}_{x}^{n} |}^{p_{1}^{n} - 2} {\bar{u}}_{x}^{n})}_{x} + {| {\bar{u}}^{n} |}^{q_{1}^{n} - 2} {\bar{u}}^{n} (1 - {| {\bar{u}}^{n} |}^{r_{1}^{n}}) - k ({\bar{u}}^{n} - {\bar{v}}^{n}) \\ {\bar{v}}_{t}^{n} = ϵ_{2}^{n} {({| {\bar{v}}_{x}^{n} |}^{p_{2}^{n} - 2} {\bar{v}}_{x}^{n})}_{x} + {| {\bar{v}}^{n} |}^{q_{2}^{n} - 2} {\bar{v}}^{n} (1 - {| {\bar{v}}^{n} |}^{r_{2}^{n}}) + k ({\bar{u}}^{n} - {\bar{v}}^{n}) \\ ({\bar{u}}^{n} (0), {\bar{v}}^{n} (0)) = (u^{n} (δ), v^{n} (δ)) \end{cases} \end{aligned}$
and
$\begin{aligned} (S L) {\begin{cases} {\bar{u}}_{t} = ϵ {({| {\bar{u}}_{x} |}^{p - 2} {\bar{u}}_{x})}_{x} + {| \bar{u} |}^{q - 2} \bar{u} (1 - {| \bar{u} |}^{r}) - k (\bar{u} - \bar{v}) \\ {\bar{v}}_{t} = ϵ {({| {\bar{v}}_{x} |}^{p - 2} {\bar{v}}_{x})}_{x} + {| \bar{v} |}^{q - 2} \bar{v} (1 - {| \bar{v} |}^{r}) + k (\bar{u} - \bar{v}) \\ (\bar{u} (0), \bar{v} (0)) = (u (δ), v (δ)) \end{cases} \end{aligned}$
and let $({\bar{u}}^{n}, {\bar{v}}^{n})$ and $(\bar{u}, \bar{v})$ be the strong solutions of $(S_{n})$ and $(S L)$ .

From Lemma 2.4, there exists $(\tilde{u}, \tilde{v}) \in C ([0, T]; L^{2}) \times C ([0, T]; L^{2})$ and a subsequence still denoted by ${({\bar{u}}^{n}, {\bar{v}}^{n})}$ such that
$\begin{aligned} ({\bar{u}}^{n}, {\bar{v}}^{n}) \to (\tilde{u}, \tilde{v}) in C ([0, T]; L^{2}) \times C ([0, T]; L^{2}) . \end{aligned}$
We want to prove that $(\tilde{u}, \tilde{v})$ is solution of $(S L)$ with initial data $(u (δ), v (δ))$ .

Once $({\bar{u}}^{n}, {\bar{v}}^{n})$ is solution of $(S_{n})$ it follows that
$\begin{aligned} \frac{1}{2} ‖ {\bar{u}}^{n} (t) - x ‖_{2}^{2} & \leq \int_{s}^{t} ⟨ | {\bar{u}}^{n} (τ) |^{q_{1}^{n} - 2} {\bar{u}}^{n} (τ) - | {\bar{u}}^{n} (τ) |^{q_{1}^{n} + r_{1}^{n} - 2} {\bar{u}}^{n} (τ), {\bar{u}}^{n} (τ) - x ⟩ d τ \\ - \int_{s}^{t} ⟨ k ({\bar{u}}^{n} (τ) - {\bar{v}}^{n} (τ)) + y, {\bar{u}}^{n} (τ) - x ⟩ d τ + \frac{1}{2} ‖ {\bar{u}}^{n} (s) - x ‖_{2}^{2} \end{aligned}$
for all $x \in D (- ϵ_{1}^{n} Δ_{p_{1}^{n}})$ , $y = - ϵ_{1}^{n} Δ_{p_{1}^{n}} x$ and for all $0 \leq s \leq t \leq T$ . Let us prove that
$\begin{aligned} \frac{1}{2} ‖ \tilde{u} (t) - x ‖_{2}^{2} & \leq \int_{s}^{t} ⟨ | \tilde{u} (τ) |^{q - 2} \tilde{u} (τ) - | \tilde{u} (τ) |^{q + r - 2} \tilde{u} (τ), \tilde{u} (τ) - x ⟩ d τ \\ - \int_{s}^{t} ⟨ k (\tilde{u} (τ) - \tilde{v} (τ)) + y, \tilde{u} (τ) - x ⟩ d τ + \frac{1}{2} ‖ \tilde{u} (s) - x ‖_{2}^{2} \end{aligned}$
(2)
for $x \in D (- ϵ Δ_{p})$ , $y = - ϵ Δ_{p} x$ and $0 \leq s \leq t \leq T$ , and then we conclude from Brézis (1973, Proposition 3.6), that $\tilde{u}$ is a weak solution of the equation
$\begin{aligned} \frac{d w}{d t} - ϵ Δ_{p} w = | \tilde{u} |^{q - 2} \tilde{u} - | \tilde{u} |^{q + r - 2} \tilde{u} - k \tilde{u} + k \tilde{v} . \end{aligned}$

Take $x \in D (- ϵ Δ_{p}) \subset W_{0}^{1, p}$ . Then there is a sequence ${x_{l}} \subset C_{c}^{\infty} (0, 1)$ with $x_{l} \to x$ in $W^{1, p}$ , as $l \to \infty$ . Note that for each $l \in I N$ , $x_{l} \in W_{0}^{1, s}$ , whatever $s \geq 1$ .

Consider $y_{l}^{n} = - ϵ_{1}^{n} Δ_{p_{1}^{n}} x_{l}$ , for every $l \in I N$ and $n \in I N$ . We have
$\begin{aligned} \frac{1}{2} ‖ {\bar{u}}^{n} (t) - x_{l} ‖_{2}^{2} & \leq \int_{s}^{t} ⟨ | {\bar{u}}^{n} (τ) |^{q_{1}^{n} - 2} {\bar{u}}^{n} (τ) - | {\bar{u}}^{n} (τ) |^{q_{1}^{n} + r_{1}^{n} - 2} {\bar{u}}^{n} (τ), {\bar{u}}^{n} (τ) - x_{l} ⟩ d τ \\ - \int_{s}^{t} ⟨ k ({\bar{u}}^{n} (τ) - {\bar{v}}^{n} (τ)) + y_{l}^{n}, {\bar{u}}^{n} (τ) - x_{l} ⟩ d τ + \frac{1}{2} ‖ {\bar{u}}^{n} (s) - x_{l} ‖_{2}^{2} \end{aligned}$
for all $0 \leq s \leq t \leq T$ . Certainly, as $n \to \infty$ ,
$\begin{aligned} ‖ {\bar{u}}^{n} (τ) - x_{l} ‖_{2} \to ‖ \tilde{u} (τ) - x_{l} ‖_{2} \end{aligned}$
uniformly in $[0, T]$ ,
$\begin{aligned} \int_{s}^{t} ⟨ k {\bar{u}}^{n} (τ) - k {\bar{v}}^{n} (τ), {\bar{u}}^{n} (τ) - x_{l} ⟩ d τ \to \int_{s}^{t} ⟨ k \tilde{u} (τ) - k \tilde{v} (τ), \tilde{u} (τ) - x_{l} ⟩ d τ \end{aligned}$
and
$\begin{aligned} \int_{s}^{t} ⟨ | {\bar{u}}^{n} (τ) |^{q_{1}^{n} - 2} {\bar{u}}^{n} (τ) - | {\bar{u}}^{n} (τ) |^{q_{1}^{n} + r_{1}^{n} - 2} {\bar{u}}^{n} (τ), {\bar{u}}^{n} (τ) - x_{l} ⟩ d τ \\ \to \int_{s}^{t} ⟨ | \tilde{u} (τ) |^{q - 2} \tilde{u} (τ) - | \tilde{u} (τ) |^{q + r - 2} \tilde{u} (τ), \tilde{u} (τ) - x_{l} ⟩ d τ \end{aligned}$
as $({\bar{u}}^{n}, {\bar{v}}^{n}) \to (\tilde{u}, \tilde{v})$ in $C ([0, T]; L^{2}) \times C ([0, T]; L^{2})$ .

It remains to prove that $\int_{s}^{t} ⟨ y_{l}^{n}, {\bar{u}}^{n} (τ) - x_{l} ⟩ d τ \to \int_{s}^{t} ⟨ y_{l}, \tilde{u} (τ) - x_{l} ⟩ d τ$ , as $n \to \infty$ for every $l \in I N$ , where $y_{l} = - ϵ Δ_{p} x_{l}$ .

We start by proving that ${y_{l}^{n}}$ is weakly convergent $L^{2}$ . Indeed,
$\begin{aligned} ‖ y_{l}^{n} ‖_{2} = ‖ - ϵ_{1}^{n} Δ_{p_{1}^{n}} x_{l} ‖_{2} \leq C_{l} \end{aligned}$
for any $n \in I N$ , where $C_{l}$ is a positive constant which depends on $l$ . We observe that this is possible because the sequences ${ϵ_{1}^{n}}$ and ${p_{1}^{n}}$ are bounded in ${I R}_{+}^{*}$ , and $x_{l} \in C_{c}^{\infty} (0, 1)$ . Therefore, for each $l \in I N$ , the sequence ${y_{l}^{n}}_{n \in I N}$ is bounded in $L^{2}$ and then we can extract from it a subsequence still denoted by ${y_{l}^{n}}$ weakly converging to some $y_{l} \in L^{2}$ . Let us prove that $y_{l} = - ϵ Δ_{p} x_{l}$ .

We know that for every $n \in I N$ , $y_{l}^{n} = - ϵ_{1}^{n} Δ_{p_{1}^{n}} x_{l}$ , and therefore for all $ξ \in L^{2}$ ,
$\begin{aligned} \frac{ϵ_{1}^{n}}{p_{1}^{n}} ‖ ξ ‖_{W_{0}^{1, p_{1}^{n}}}^{p_{1}^{n}} \geq \frac{ϵ_{1}^{n}}{p_{1}^{n}} ‖ x_{l} ‖_{W_{0}^{1, p_{1}^{n}}}^{p_{1}^{n}} + ⟨ y_{l}^{n}, ξ - x_{l} ⟩ . \end{aligned}$

But $⟨ y_{l}^{n}, ξ - x_{l} ⟩ \to ⟨ y_{l}, ξ - x_{l} ⟩$ once $y_{l}^{n} ⇀ y_{l}$ in $L^{2}$ . And on the other hand, as $x_{l} \in C_{c}^{\infty} (0, 1)$ , it follows that $| x_{l}^{'} (x) |$ is bounded for all $x \in (0, 1)$ . From dominated convergence theorem, and remembering that $ϵ_{1}^{n} \to ϵ$ and $p_{1}^{n} \to p$ , we have
$\begin{aligned} \frac{ϵ_{1}^{n}}{p_{1}^{n}} ‖ x_{l} ‖_{W_{0}^{1, p_{1}^{n}}}^{p_{1}^{n}} \to \frac{ϵ}{p} ‖ x_{l} ‖_{W_{0}^{1, p}}^{p} \end{aligned}$
as $n \to \infty$ . Now we have to consider two cases:

The first is when $ξ \in L^{2} ∖ W_{0}^{1, p}$ . In this case, $\frac{ϵ}{p} ‖ ξ ‖_{W_{0}^{1, p}}^{p} = \infty$ , and certainly
$\begin{aligned} \frac{ϵ}{p} ‖ ξ ‖_{W_{0}^{1, p}}^{p} \geq \frac{ϵ}{p} ‖ x_{l} ‖_{W_{0}^{1, p}}^{p} + ⟨ y_{l}, ξ - x_{l} ⟩ . \end{aligned}$

In the second case, $ξ \in W_{0}^{1, p}$ , there is a sequence ${ξ_{n}} \subset C_{c}^{\infty} (0, 1)$ with $ξ_{n} \to ξ$ in $W^{1, p}$ . Thus, for every $n \in I N$ , $ξ_{n} \in W_{0}^{1, s}$ , for all $s \geq 1$ . In particular $ξ_{n} \in W_{0}^{1, p_{1}^{n}}$ , then for all $n \in I N$ ,
$\begin{aligned} \frac{ϵ_{1}^{n}}{p_{1}^{n}} ‖ ξ_{n} ‖_{W_{0}^{1, p_{1}^{n}}}^{p_{1}^{n}} \geq \frac{ϵ_{1}^{n}}{p_{1}^{n}} ‖ x_{l} ‖_{W_{0}^{1, p_{1}^{n}}}^{p_{1}^{n}} + ⟨ y_{l}^{n}, ξ_{n} - x_{l} ⟩ . \end{aligned}$
(3)

Note that $‖ ξ_{n} ‖_{W_{0}^{1, p_{1}^{n}}}^{p_{1}^{n}} \to ‖ ξ ‖_{W_{0}^{1, p}}^{p}$ and since $y_{l}^{n} ⇀ y_{l}$ and $ξ_{n} \to ξ$ in $L^{2}$ it follows that
$\begin{aligned} ⟨ y_{l}^{n}, ξ_{n} - x_{l} ⟩ \to ⟨ y_{l}, ξ - x_{l} ⟩ . \end{aligned}$
Thus, taking the limit $n \to \infty$ in (3), we obtain
$\begin{aligned} \frac{ϵ}{p} ‖ ξ ‖_{W_{0}^{1, p}}^{p} \geq \frac{ϵ}{p} ‖ x_{l} ‖_{W_{0}^{1, p}}^{p} + ⟨ y_{l}, ξ - x_{l} ⟩ \end{aligned}$
and we conclude that $y_{l} = \partial ψ (x_{l}) = - ϵ Δ_{p} x_{l}$ , where $ψ (ξ) = \frac{ϵ}{p} ‖ ξ ‖_{W_{0}^{1, p}}^{p}$ .

Therefore, $- ϵ_{1}^{n} Δ_{p_{1}^{n}} x_{l} ⇀ - ϵ Δ_{p} x_{l}$ in $L^{2}$ , which ensures for all $τ$ ,
$\begin{aligned} ⟨ y_{l}^{n}, {\bar{u}}^{n} (τ) - x_{l} ⟩ \to ⟨ y_{l}, \tilde{u} (τ) - x_{l} ⟩ . \end{aligned}$

Then, for $l \in I N$ ,
$\begin{aligned} \frac{1}{2} ‖ \tilde{u} (t) - x_{l} ‖_{2}^{2} & \leq \int_{s}^{t} ⟨ | \tilde{u} (τ) |^{q - 2} \tilde{u} (τ) - | \tilde{u} (τ) |^{q + r - 2} u (τ), \tilde{u} (τ) - x_{l} ⟩ d τ \\ - \int_{s}^{t} ⟨ k (\tilde{u} (τ) - \tilde{v} (τ)) + y_{l}, \tilde{u} (τ) - x_{l} ⟩ d τ + \frac{1}{2} ‖ \tilde{u} (s) - x_{l} ‖_{2}^{2} . \end{aligned}$
(4)

In order to prove (2), we take the limit $l \to \infty$ in (4). Since $x_{l} \to x$ in $W^{1, p}$ , we need only to prove that $y_{l} ⇀ y$ in $W^{- 1, p^{'}}$ , with $y = - ϵ Δ_{p} x$ . Note that
$\begin{aligned} ‖ y_{l} ‖_{W^{- 1, p^{'}}} = ϵ ‖ - Δ_{p} x_{l} ‖_{W^{- 1, p^{'}}} \leq ‖ x_{l} ‖_{W_{0}^{1, p}}^{p - 1} \end{aligned}$
and since ${x_{l}}$ is bounded in $W^{1, p}$ it follows that ${y_{l}}$ is bounded in $W^{- 1, p^{'}}$ . Once $W^{- 1, p^{'}}$ is reflexive there is a subsequence still denoted by ${y_{l}}$ which converges in the weak star topology for any $y \in W^{- 1, p^{'}}$ . Then
$\begin{aligned} ⟨ y_{l}, \tilde{u} (τ) - x_{l} ⟩ \to ⟨ y, \tilde{u} (τ) - x ⟩ \end{aligned}$
for all $τ$ . Now we have to prove that $y = - ϵ Δ_{p} x$ .

We know that for all $ξ \in L^{2}$ and for all $l \in I N$
$\begin{aligned} \frac{ϵ}{p} ‖ ξ ‖_{W_{0}^{1, p}}^{p} \geq \frac{ϵ}{p} ‖ x_{l} ‖_{W_{0}^{1, p}}^{p} + ⟨ y_{l}, ξ - x_{l} ⟩ . \end{aligned}$
Thus making $l \to \infty$ , we have
$\begin{aligned} \frac{ϵ}{p} ‖ ξ ‖_{W_{0}^{1, p}}^{p} \geq \frac{ϵ}{p} ‖ x ‖_{W_{0}^{1, p}}^{p} + ⟨ y, ξ - x ⟩ . \end{aligned}$
So $y = \partial ψ (x) = - ϵ Δ_{p} x$ . Since $x$ was arbitrarily taken in $D (- ϵ Δ_{p})$ , it is proved that $\tilde{u}$ satisfies (2).

By Brézis (1973, Proposition 3.6), $\tilde{u}$ is weak solution of the equation
$\begin{aligned} \begin{array}{lll} \frac{d w}{d t} - ϵ Δ_{p} w = | \tilde{u} |^{q - 2} \tilde{u} - | \tilde{u} |^{q + r - 2} \tilde{u} - k (\tilde{u} - \tilde{v}) \end{array} \end{aligned}$
(5)
and as $‖ | \tilde{u} (t) |^{q - 2} \tilde{u} (t) - | \tilde{u} (t) |^{q + r - 2} \tilde{u} (t) - k (\tilde{u} (t) - \tilde{v} (t)) ‖_{2}^{2}$ is bounded for all $t \in [0, T]$ , it follows that $| \tilde{u} |^{q - 2} \tilde{u} - | \tilde{u} |^{q + r - 2} \tilde{u} - k (\tilde{u} - \tilde{v}) \in L^{2} (0, T; L^{2})$ . Thus by Brézis (1973, Theorem 3.6), $\tilde{u}$ is strong solution of (5). Similarly, we prove that $\tilde{v}$ is strong solution of the equation
$\begin{aligned} \frac{d w}{d t} - ϵ Δ_{p} w = | \tilde{v} |^{q - 2} \tilde{v} - | \tilde{v} |^{q + r - 2} \tilde{v} + k (\tilde{u} - \tilde{v}) . \end{aligned}$

Therefore, $(\tilde{u}, \tilde{v})$ is a strong solution of the limit system $(S L)$ , with initial data $(u (δ), v (δ))$ and, by uniqueness, we have $(\tilde{u}, \tilde{v}) = (\bar{u}, \bar{v})$ .

Suppose, by contradiction, that for some $t_{0} > 0$ , $(u^{n} (t_{0}), v^{n} (t_{0}))$ does not converge to $(u (t_{0}), v (t_{0}))$ in $L^{2} \times L^{2}$ , where $(u^{n}, v^{n})$ and $(u, v)$ are solutions of $(S_{n})$ and $(S L)$ , respectively, with initial data $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ . Take $0 < δ < t_{0}$ and fix it. Consider the systems $(S_{n})$ and $(S L)$ with initial data $(u^{n} (δ), v^{n} (δ))$ and $(u (δ), v (δ))$ , respectively, and solutions $({\bar{u}}^{n}, {\bar{v}}^{n})$ and $(\bar{u}, \bar{v})$ . We know that
$\begin{aligned} ({\bar{u}}^{n}, {\bar{v}}^{n}) \to (\bar{u}, \bar{v}) in C ([0, T]; L^{2}) \times C ([0, T]; L^{2}) . \end{aligned}$
Let $s > 0$ be such that $t_{0} = δ + s$ . Thus
$\begin{aligned} (u^{n} (t_{0}), v^{n} (t_{0})) = ({\bar{u}}^{n} (s), {\bar{v}}^{n} (s)) \to (\bar{u} (s), \bar{v} (s)) = (u (t_{0}), v (t_{0})) in L^{2} \times L^{2} \end{aligned}$
and this completes the proof.

After proving the continuity of the flow for each $t > 0$ , we prove that this continuity is uniform for initial data in compact subsets of $L^{2} \times L^{2}$ . The proof is based on the Cantor’s diagonalization method.
Lemma 4.2
Let $K \subset L^{2} \times L^{2}$ be a compact set. Then for each $t > 0$
$\begin{aligned} ‖ (u^{n} (t), v^{n} (t)) - (u (t), v (t)) ‖_{2 \times 2} \to 0 \end{aligned}$
uniformly in $K$ as $n \to + \infty$ , where $(u^{n}, v^{n})$ and $(u, v)$ are solutions of $(S_{n})$ and $(S L)$ , respectively, with initial data $(u_{0}, v_{0}) \in K$ .
Proof.
Fix $t > 0$ , and define for each $n \in I N$ , the function $h^{n} : L^{2} \times L^{2} ⟶ L^{2} \times L^{2}$ given by $h^{n} (u_{0}, v_{0}) = (u^{n} (t, (u_{0}, v_{0})), v^{n} (t, (u_{0}, v_{0}))) .$ Note that fixed $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ , it follows from the continuity of the application $S_{n} (t)$ that if $(u_{0}, v_{0}) = lim_{m \to + \infty} (u_{0}^{m}, v_{0}^{m})$ in $L^{2} \times L^{2}$ then
$\begin{aligned} ‖ (u^{n} (t, (u_{0}^{m}, v_{0}^{m})), v^{n} (t, (u_{0}^{m}, v_{0}^{m}))) - (u^{n} (t, (u_{0}, v_{0})), v^{n} (t, (u_{0}, v_{0}))) ‖_{2 \times 2} \to 0 \end{aligned}$
uniformly in $n \in I N$ as $m \to + \infty$ .

Furthermore, we can also define $h : L^{2} \times L^{2} ⟶ L^{2} \times L^{2}$ given by $h (u_{0}, v_{0}) = (u (t, (u_{0}, v_{0})), v (t, (u_{0}, v_{0}))) .$ Then the family of functions ${h, h^{1}, h^{2}, h^{3}, \dots}$ is equicontinuous in $L^{2} \times L^{2}$ .

Let $K$ be a compact subset of $L^{2} \times L^{2}$ and $D = {y_{1}, y_{2}, y_{3}, \dots}$ an enumerable dense subset of $K$ , where $y_{i} = (u_{0}^{i}, v_{0}^{i}) \in K$ . We prove that there exists a subsequence of ${h^{n}}$ which converges simply in $K$ .

Consider the initial sequence ${h^{n}}$ , then ${h^{n} (y_{1})}$ is a sequence in $L^{2} \times L^{2}$ , and from Theorem 4.1, it has a subsequence ${h^{1 n} (y_{1})}$ which converges to $h (y_{1})$ .

Consider again the sequence of functions ${h^{1 n}}$ , then ${h^{1 n} (y_{2})}$ is a sequence in $L^{2} \times L^{2}$ , and from Theorem 4.1, it has a subsequence ${h^{2 n} (y_{2})}$ , which converges to $h (y_{2})$ . Repeat this reasoning for each $y_{i}$ , with $i = 1, 2, 3, \dots$ . Note that the subsequence ${h^{j n}}$ obtained at step $j$ is such that to $i = 1, 2, \dots, j$
$\begin{aligned} h^{j n} (y_{i}) \to h (y_{i}) . \end{aligned}$

Therefore, the diagonal sequence ${h^{n n}}$ is such that $h^{n n} (y_{i}) \to h (y_{i})$ for all $i \in I N$ . It follows that ${h^{n n}}$ converges uniformly on $K$ .
Remark 3
Since $\cup_{n \in I N} A_{n}$ is a bounded set in $W_{0}^{1, 2} \times W_{0}^{1, 2}$ and so relatively compact in $L^{2} \times L^{2}$ , we can take in previous lemma $K = \bar{\cup_{n \in I N} A_{n}}$ , where the closure is taken in $L^{2} \times L^{2}$ . So, we have the uniform continuity of the flow in $\cup_{n \in I N} A_{n}$ .

Now, we proceed to prove that $d (A_{n}, A_{L}) \to 0$ , as $n \to + \infty$ , where
$\begin{aligned} d (A, B) = sup_{a \in A} inf_{b \in B} {‖ a - b ‖}_{2 \times 2} \end{aligned}$
is the Hausdorff semi-distance in $L^{2} \times L^{2}$ . This result will help establish the $ϵ$ -synchronicity.
Theorem 4.3
The family of attractors ${A_{n}}_{n \in I N}$ is upper semicontinuous.
Proof.
By Lemma 3.2 there exists $T_{3} > 0$ and a positive constant $N_{6}$ such that
$\begin{aligned} ‖ u^{n} (t) ‖_{W_{0}^{1, p_{1}^{n}}} + ‖ v^{n} (t) ‖_{W_{0}^{1, p_{2}^{n}}} \leq N_{6} \end{aligned}$
for all $n \in I N$ , $t \geq T_{3}$ and any initial data $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ .

Set $B ≐ B ((0, 0); N_{6}) \subset L^{2} \times L^{2}$ . We know that $B$ absorbs bounded subsets of $L^{2} \times L^{2}$ under $S_{n} (t)$ for any $n \in I N$ in $L^{2} \times L^{2}$ . Since $A_{n}$ is bounded and invariant it follows that $\cup_{n \in I N} A_{n} \subset B$ .

But $A_{L}$ attracts bounded subsets of $L^{2} \times L^{2}$ , in particular attracts $B$ . Thus given $η > 0$ , exists $t_{0} = t (B, η) > 0$ such that
$\begin{aligned} S_{0} (t) B \subset O_{\frac{η}{2}} (A_{L}) \end{aligned}$
for all $t \geq t_{0}$ , where ${S_{0} (t)}$ is the semigroup associated with the limit system $(S L)$ . This implies that
$\begin{aligned} sup_{\binom{(u_{n}, v_{n}) \in A_{n}}{n \in I N}} d (S_{0} (t_{0}) (u_{n}, v_{n}), A_{L}) \leq \frac{η}{2} . \end{aligned}$

From the uniform continuity of the flow in compacts sets of $L^{2} \times L^{2}$ there is $n_{0} \in I N$ such that for all $n \geq n_{0}$ and $(x, y) \in \cup_{n \in I N} A_{n}$
$\begin{aligned} ‖ S_{n} (t_{0}) (x, y) - S_{0} (t_{0}) (x, y) ‖_{2 \times 2} < \frac{η}{2} . \end{aligned}$

Thus, for each $(u_{n}, v_{n}) \in A_{n}$ , with $n \geq n_{0}$
$\begin{aligned} d (S_{n} (t_{0}) (u_{n}, v_{n}), A_{L}) < η . \end{aligned}$

So ${A_{n}}_{n \in I N}$ is upper semicontinuous as $n \to \infty$ .
5. About the Limit System

We have seen in the previous section that when $n \to \infty$ the system $(S_{n})$ approximates the limit system $(S L)$ . To ensure the $ϵ$ -synchronicity, we study the attractor of the semigroup associated with the system $(S L)$ . We prove that for $k$ sufficiently large each element of this attractor is of the form $(u, u)$ , with $u \in W_{0}^{1, p}$ .

In fact, when we analyze the system $(S L)$ , it is possible to observe that if $(u, v)$ is solution of $(S L)$ with initial data $(u_{0}, v_{0})$ then $u (t)$ approximates $v (t)$ as $t \to + \infty$ , for each $k$ sufficiently large and uniformly on the initial data. This property is known as synchronization, see Carvalho et al. (1998).

Theorem 5.1
Let $(u, v)$ be solution of system $(S L)$ . For every $k > 0$ sufficiently large, as $t \to \infty$ , we have
$\begin{aligned} ‖ u (t) - v (t) ‖_{2} \to 0. \end{aligned}$

Proof.
In fact, for $\bar{C} = \frac{2 r (q - 1)^{\frac{q + r - 2}{r}}}{q + r - 2} - 4 k$ we have
$\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ u (t) - v (t) ‖_{2}^{2}) \leq \bar{C} ‖ u (t) - v (t) ‖_{2}^{2} . \end{aligned}$

Multiplying the above inequality by $e^{- \bar{C} t}$ and integrating from $0$ to $t$ we obtain, for each $k$ sufficiently large
$\begin{aligned} lim_{t \to \infty} ‖ u (t) - v (t) ‖_{2}^{2} \leq lim_{t \to \infty} e^{\bar{C} t} ‖ u_{0} - v_{0} ‖_{2}^{2} = 0. \end{aligned}$

Lemma 5.2
If $({\bar{u}}_{0}, {\bar{v}}_{0}) \in A_{L}$ , then ${\bar{u}}_{0} = {\bar{v}}_{0}$ .
Proof.
Indeed, if $({\bar{u}}_{0}, {\bar{v}}_{0}) \in A_{L}$ then for each $t > 0$ , $({\bar{u}}_{0}, {\bar{v}}_{0}) = S_{0} (t) (u_{0}^{t}, v_{0}^{t})$ with $(u_{0}^{t}, v_{0}^{t}) \in A_{L}$ . Thus given $σ > 0$ , exists $\tilde{t} > 0$ such that for all $t \geq \tilde{t}$ ,
$\begin{aligned} ‖ {\bar{u}}_{0} - {\bar{v}}_{0} ‖_{2} = ‖ u (t) - v (t) ‖_{2} < σ \end{aligned}$
with $(u (t), v (t)) = S_{0} (t) (u_{0}^{t}, v_{0}^{t})$ and $(u_{0}^{t}, v_{0}^{t}) \in A_{L}$ . Since $‖ {\bar{u}}_{0} - {\bar{v}}_{0} ‖_{2} < σ$ for all $σ > 0$ , it follows that ${\bar{u}}_{0} = {\bar{v}}_{0}$ .
6. Existence of Nontrivial Stationary Solutions

In this section, we show that the attractors $A_{n}$ are nontrivial at least in the case that $p_{i} < q_{i}$ , $i = 1, 2$ and $ϵ_{1} = ϵ_{2}$ . For this end, we use an equivalent formulation of Ricceri’s three critical points theorem, enunciated below:

Theorem 6.1
(Bonanno, 2002, Theorem 2.3) Let $X$ be a separable and reflexive real Banach space. $ϕ : X \to I R$ is a continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on $X^{*}$ ; $φ : X \to I R$ is a continuously Gateaux differentiable functional whose Gateaux derivative is compact. Assume that (i)
$lim_{‖ u ‖ \to + \infty} (ϕ (u) + λ φ (u)) = + \infty$ for each $λ > 0$ ;
(ii)
there are $R$ , $u_{0}$ and $u_{1}$ such that $ϕ (u_{0}) < R < ϕ (u_{1})$ ;
(iii)
$inf_{u \in ϕ^{- 1} (] - \infty, R])} φ (u) > \frac{((ϕ (u_{1}) - R) φ (u_{0}) + (R - ϕ (u_{0})) φ (u_{1}))}{ϕ (u_{1}) - ϕ (u_{0})}$ .
Then, there exist an open interval $Λ \subseteq [0, + \infty)$ and a positive real number $ρ$ such that, for each $λ \in Λ$ , the equation $ϕ^{'} (u) + λ φ^{'} (u) = 0$ has at least three weak solutions whose norms in $X$ are less than $ρ$ .

Taking $X = W_{0}^{1, p_{1}} \times W_{0}^{1, p_{2}}$ endowed with the norm $‖ (u, v) ‖_{X} = ‖ u_{x} ‖_{p_{1}} + ‖ v_{x} ‖_{p_{2}}$ , we define applications $ϕ : X \to I R$ and $φ : X \to I R$ by
$\begin{aligned} ϕ (u, v) = \frac{1}{p_{1}} ‖ u_{x} ‖_{p_{1}}^{p_{1}} + \frac{1}{p_{2}} ‖ v_{x} ‖_{p_{2}}^{p_{2}} \end{aligned}$
and
$\begin{aligned} φ (u, v) = - \frac{1}{q_{1}} ‖ u ‖_{q_{1}}^{q_{1}} + \frac{1}{q_{1} + r_{1}} ‖ u ‖_{q_{1} + r_{1}}^{q_{1} + r_{1}} - \frac{1}{q_{2}} ‖ v ‖_{q_{2}}^{q_{2}} + \frac{1}{q_{2} + r_{2}} ‖ v ‖_{q_{2} + r_{2}}^{q_{2} + r_{2}} + \frac{k}{2} ‖ u - v ‖_{2}^{2}, \end{aligned}$
respectively. Under the conditions imposed on parameters, the functionals $ϕ$ and $φ$ satisfy the hypotheses of Theorem 6.1. To ensure that all other hypotheses are satisfied we can choose $(u_{0}, v_{0}) = (0, 0)$ and $(u_{1}, v_{1})$ defined as follows:
$\begin{aligned} u_{1} = v_{1} = {\begin{cases} 2 x, if x \in [0, \frac{1}{2}] \\ - 2 x + 2, if x \in (\frac{1}{2}, 1] . \end{cases} \end{aligned}$
And we set
$\begin{aligned} R = \frac{1}{2} min {{[\frac{1}{ϕ (u_{1}, v_{1})} \frac{(2 q_{i} r_{i} + r_{i}^{2} + r_{i})}{p_{i}^{\frac{q_{i}}{p_{i}}} (q_{i} + 1) (q_{i} + r_{i}) (q_{i} + r_{i} + 1)}]}^{\frac{p_{i}}{q_{i} - p_{i}}}, i = 1, 2} . \end{aligned}$

Simple calculations show that $R$ , $(u_{0}, v_{0})$ and $(u_{1}, v_{1})$ defined above satisfy hypotheses $(i i)$ and $(i i i)$ of Theorem 6.1 which ensures that at least in the case that $p_{i} < q_{i}$ for $i = 1, 2$ and $ϵ_{1} = ϵ_{2}$ the family ${A_{n}}$ is not given only by trivial attractors and hence $ϵ$ -synchronization is not a trivial result.
7. Synchronization

The following theorem is the main result of this article. We prove in this theorem that the system $(S_{n})$ $ϵ$ -synchronizes.

Theorem 7.1

Consider the system

\begin{aligned} (S_{n}) {\begin{cases} u_{t}^{n} = ϵ_{1}^{n} {({| u_{x}^{n} |}^{p_{1}^{n} - 2} u_{x}^{n})}_{x} + {| u^{n} |}^{q_{1}^{n} - 2} u^{n} (1 - {| u^{n} |}^{r_{1}^{n}}) - k (u^{n} - v^{n}) \\ v_{t}^{n} = ϵ_{2}^{n} {({| v_{x}^{n} |}^{p_{2}^{n} - 2} v_{x}^{n})}_{x} + {| v^{n} |}^{q_{2}^{n} - 2} v^{n} (1 - {| v^{n} |}^{r_{2}^{n}}) + k (u^{n} - v^{n}) \end{cases} \end{aligned}

where

(u^{n} (0, t), v^{n} (0, t)) = (0, 0)

and

(u^{n} (1, t), v^{n} (1, t)) = (0, 0)

for all

t \in (0, + \infty)

and

(u^{n} (x, 0), v^{n} (x, 0)) = (u_{0} (x), v_{0} (x)) \in W_{0}^{1, p_{1}^{n}} \times W_{0}^{1, p_{2}^{n}}

ϵ_{i}^{n}

k

are positive constants,

q_{i}^{n} \geq 2

and

p_{i}^{n} > 2

and

r_{i}^{n} > q_{i}^{n} - 2

, for

i = 1, 2

and for all

n \in I N

. If

(ϵ_{i}^{n}, p_{i}^{n}, q_{i}^{n}, r_{i}^{n}) \to (ϵ, p, q, r)

n \to + \infty

then, for

k

sufficiently large the solution of

(S_{n})

ϵ

-synchronizes.

Proof.

By upper semicontinuity of attractors $A_{n}$ , given $η > 0$ exists $M_{0} \in I N$ such that if $n . k \geq M_{0}$

\begin{aligned} d (A_{n}, A_{L}) < \frac{η}{10} . \end{aligned}

Consider $(u_{0}, v_{0}) \in L^{2} \times L^{2}$ . Since $A_{n}$ is attractor, for every $n \in N$ exists $t_{n} > 0$ such that

\begin{aligned} S_{n} (t) (u_{0}, v_{0}) \in O_{\frac{η}{10}} (A_{n}), \forall t \geq t_{n} . \end{aligned}

So, for each

t \geq t_{n}

, there is

(u_{n}^{t}, v_{n}^{t}) \in A_{n}

with

\begin{aligned} ‖ (u^{n} (t), v^{n} (t)) - (u_{n}^{t}, v_{n}^{t}) ‖_{2 \times 2} < \frac{η}{10} \end{aligned}

being

(u^{n} (t), v^{n} (t)) = S_{n} (t) (u_{0}, v_{0})

, and for

n, k \geq M_{0}

there is

(u_{0}^{t}, u_{0}^{t}) \in A_{L}

such that

\begin{aligned} ‖ (u_{n}^{t}, v_{n}^{t}) - (u_{0}^{t}, u_{0}^{t}) ‖_{2 \times 2} < \frac{η}{10} . \end{aligned}

Then if $n, k \geq M_{0}$ and $t \geq t_{n}$ we have

\begin{aligned} ‖ u^{n} (t) - v^{n} (t) ‖_{2} < \frac{η}{2} \end{aligned}

which ensures

ϵ

-synchronicity.

Footnotes

ORCID iDs

Ana Claudia Pereira

Cláudia Gentile

Marcos Primo

Olimpio Miyagaki

Funding

The authors received the following financial support for the research, authorship, and/or publication of this article: Ana Claudia Pereira was partially supported by the CAPES ‐ Coordenação de Aperfeiçoamento de Pessoal de Nível Superior.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Afraimovich

V. S.

Rodrigues

H. M.

(1998). Uniform dissipativeness and synchronization of nonautonomous equations. International Conference on Differential Equations (Lisboa 1995), 3–17.

Boccaletti

Pisarchik

A. N.

Del Genio

C. I.

Amann

(2018). Synchronization: From coupled systems to complex networks. Cambridge University Press.

Bonanno

(2002). A minimax inequality and its applications to ordinary differential equations. Journal of Mathematical Analysis and Applications, 270, 210–229.

Brézis

(1973). Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. American Elsevier Publishing.

Bruschi

S. M.

Gentile

C. B.

Primo

M. R. T.

(2010). Continuity properties on

p

for

p

-Laplacian parabolic problems. Nonlinear Analysis, 72, 1580–1588.

Caraballo

Kloeden

P. E.

(2005). The persistence of synchronization under environmental noise. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2059), 2257–2267.

Carvalho

A. N.

Primo

M. R. T.

(2000). Boundary synchronization in parabolic problems with nonlinear boundary conditions. Dynamics of Continuous, Discrete and Impulsive Systems, 7(4), 541–560.

Carvalho

A. N.

Rodrigues

H. M.

Dlotko

(1998). Upper semicontinuity of attractors and synchronization. Journal of Mathematical Analysis and Applications, 220, 13–41.

Gameiro

M. F.

Rodrigues

H. M.

(2001). Applications of robust synchronization to communication systems. Applicable Analysis, 79(1–2), 21–45.

10.

Ghosh

Marwan

Small

Zhou

Heitzig

Koseska

Kiss

I. Z.

(2024). Recent achievements in nonlinear dynamics, synchronization, and networks. Chaos (Woodbury, N.Y.), 100401.

11.

Hale

J. K.

(1988). Asymptotic Behavior of Dissipative Systems. (Mathematical Surveys and Monographs, Vol. 25). American Mathematical Society.

12.

Hale

J. K.

(1997). Diffusive coupling, dissipation, and synchronization. Journal of Dynamics and Differential Equations, 9, 1–52.

13.

Kloeden

P. E.

(2003). Synchronization of nonautonomous dynamical systems. Electronic Journal of Differential Equations, 2003, 1–10.

14.

Kloeden

P. E.

Pavani

(2009). Dissipative synchronization of nonautonomous and random systems. GAMM-Mitt, 32(1), 80–92.

15.

Ladyzhenskaya

O. A.

(1991). Attractors for semigroups and evolution equations. Cambridge University Press.

16.

Ötani

(1977). On existence of strong solutions for

\frac{d u}{d t} (t) + \partial ϕ_{1} (u (t)) - \partial ϕ_{2} (u (t)) ∋ f (t)

. Journal of the Faculty of Science, University of Tokyo. Section IA, Mathematics, 24, 575–605.

17.

Ouardi

H. E.

Hachimi

A. E.

(2006). Attractors for a class of doubly nonlinear parabolic systems. Electronic Journal of Qualitative Theory of Differential Equations, 2006, 1–15.

18.

Pereira

A. C.

Moussa

C. B. G.

Miyagaki

O. H.

(2015). An example of noncontinuous attractors. Journal of Evolution Equations, 15, 979–1000.

19.

Rodrigues

H. M.

(1996). Abstract methods for synchronization and applications. Applicable Analysis, 62, 263–296.

20.

Rodrigues

H. M.

Alberto

L. F. C.

Bretas

N. G.

(2001). Uniform invariance principle and synchronization: Robustness with respect to parameter variation. Journal of Differential Equations, 169(1), 228–254.

21.

Strogatz

S. H.

(2003). Sync: The emerging science of spontaneous order. Hyperion.

22.

Takeuchi

Yamada

(2000). Asymptotic properties of a reaction-diffusion equation with degenerate

p

-Laplacian. Nonlinear Analysis, 42, 41–61.

23.

Temam

(1997). Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Applied Mathematical Sciences, Vol. 68). Springer-Verlag.

24.

Vrabie

I. I.

(1995). Compactness methods for nonlinear evolutions. John Wiley & Sons.