Uniform Global Attractors for Nonautonomous Evolution Inclusions

Abstract

In this note, we prove the existence and provide basic structure properties of compact (in the natural phase space) uniform global attractor for all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions. The obtained results allow to reduce the problem of the complete qualitative investigation of various nonlinear systems into the “small” (compact) part of the natural phase space.

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1 Introduction and Setting of the Problem

For evolution triple ($V_i;H;V_i^*$)¹ and multivalued map $A_i:\mathbb {R}_+\times V\rightrightarrows V^*$, $i=1,2,\ldots , N$, $N=1,2,\ldots ,$ we consider a problem of longtime behavior (in the natural phase space H) of all globally defined weak solutions for nonautonomous evolution inclusion

$$\begin{aligned} y'(t)+\sum _{i=1}^NA_i(t,y(t))\ni \bar{0}, \end{aligned}$$

(3.1)

as $t\rightarrow +\infty$. Let $\langle \cdot ,\cdot \rangle _{V_i}:{V_i}^*\times {V_i}\rightarrow \mathbb {R}$ be the pairing in ${V_i}^*\times {V_i}$ that coincides on $H\times {V_i}$ with the inner product $(\cdot ,\cdot )$ in the Hilbert space H.

To introduce the assumptions on parameters of Problem (3.1) let us introduce additional constructions. A function $\varphi \in L_\gamma ^\mathrm{\mathrm loc}(\mathbb {R}_+)$, $\gamma >1$, is called translation bounded in $L_\gamma ^\mathrm{\mathrm loc}(\mathbb {R}_+)$, if

$$\begin{aligned} \sup _{t\ge 0}\int _{t}^{t+1}|\varphi (s)|^\gamma ds<+\infty ; \end{aligned}$$

Chepyzhov and Vishik [2, p. 105]. A function $\varphi \in L_1^\mathrm{\mathrm loc}(\mathbb {R}_+)$ is called translation uniform integrable (t.u.i.) in $L_1^\mathrm{\mathrm loc}(\mathbb {R}_+)$, if

$$ \lim _{K\rightarrow +\infty }\sup _{t\ge 0}\int _{t}^{t+1}|\varphi (s)|\mathbf{I}{\{|\varphi (s)|\ge K\}}ds=0. $$

Note that Dunford–Pettis compactness criterion provides that a function $\varphi \in L_1^\mathrm{\mathrm loc}(\mathbb {R}_+)$ is t.u.i. in $L_1^\mathrm{\mathrm loc}(\mathbb {R}_+)$ if and only if for every sequence of elements $\{\tau _n\}_{n\ge 1}\subset \mathbb {R}_+$ the sequence $\{\varphi (\,\cdot \,+\tau _n)\}_{n\ge 1}$ contains a subsequence which converges weakly in $L_1^\mathrm{\mathrm loc}(\mathbb {R}_+)$. Note that for any $\gamma >1$ every translation bounded in $L_{\gamma }^\mathrm{\mathrm loc}(\mathbb {R}_+)$ function is t.u.i. in $L_1^\mathrm{\mathrm loc}(\mathbb {R}_+)$; Gorban et al. [3].

Throughout this paper, we suppose that the listed below assumptions hold:

Assumption 1

Let $p_i\ge 2$, $q_i>1$ are such that $\frac{1}{p_i}+\frac{1}{q_i}=1$, for each for $i=1,2,\ldots , N$, and the embedding ${V_i}\subset H$ is compact one, for some for $i=1,2,\ldots , N$.

Assumption 2

(Growth Condition) There exist a t.u.i. in $L_1^\mathrm{loc}(\mathbb {R}_+)$ function $c_1:\mathbb {R}_+\rightarrow \mathbb {R}_+$ and a constant $c_2>0$ such that

$$ \max _{i=1}^N\Vert d_i\Vert _{{V_i}^*}^{q}\le c_1(t)+c_2\sum _{i=1}^N\Vert u\Vert _{V_i}^{p} $$

for any $u\in {V_i}$, $d_i\in A_i(t,u)$, $i=1,2,\ldots ,N$, and a.e. $t> 0$.

Assumption 3

(Signed Assumption) There exists a constant $\alpha >0$ and a t.u.i. in $L_1^\mathrm{loc}(\mathbb {R}_+)$ function $\beta :\mathbb {R}_+\rightarrow \mathbb {R}_+$ such that

$$ \sum _{i=1}^N\langle d_i,u\rangle _{V_i}\ge \alpha \sum _{i=1}^N \Vert u\Vert _{V_i}^p-\beta (t) $$

for any $u\in {V_i}$, $d_i\in A_i(t,u)$, $i=1,2,\ldots ,N$, and a.e. $t> 0$.

Assumption 4

(Strong Measurability) If $C\subseteq {V_i}^*$ is a closed set, then the set $\{(t,u)\in (0,+\infty )\times {V_i}\,:\, A_i(t,u)\cap C\ne \emptyset \}$ is a Borel subset in $(0,+\infty )\times {V_i}$.

Assumption 5

(Pointwise Pseudomonotonicity) Let for each $i=1,2,\ldots ,N$ and a.e. $t> 0$, two assumptions hold:

• (a) for every $u\in {V_i}$ the set $A_i(t,u)$ is nonempty, convex, and weakly compact one in ${V_i}^*$;

• (b) if a sequence $\{u_n\}_{n\ge 1}$ converges weakly in ${V_i}$ toward $u\in {V_i}$ as $n\rightarrow +\infty$, $d_n\in A_i(t,u_n)$ for any $n\ge 1$, and $\mathop {\text {lim}\,\text {sup}}\limits _{n\rightarrow +\infty }\langle d_n,u_n-u\rangle _{V_i}\le 0$, then for any $\omega \in {V_i}$ there exists $d(\omega )\in A_i(t,u)$ such that

  $$\mathop {\text {lim}\,\text {inf}}\limits _{n\rightarrow +\infty } \langle d_n, u_n-\omega \rangle _{V_i}\ge \langle d(\omega ), u-\omega \rangle _{V_i}.$$

Let $0\le \tau<T<+\infty$. As a weak solution of evolution inclusion (3.1) on the interval $[\tau ,T]$, we consider an element $u(\cdot )$ of the space $\cap _{i=1}^NL_{p_i}(\tau ,T;{V_i})$ such that for some $d_i(\cdot )\in L_{q_i}(\tau ,T;{V_i}^*)$, $i=1,2,\ldots ,N$, it is fulfilled:

$$\begin{aligned} -\int \limits _\tau ^T(\xi '(t),y(t))dt+ \sum _{i=1}^N\int \limits _\tau ^T\langle d_i(t),\xi (t)\rangle _{V_i} dt= 0\quad \forall \xi \in C_0^\infty ([\tau ,T];{V_i}), \end{aligned}$$

(3.2)

and $d_i(t)\in A_i(t,y(t))$ for each $i=1,2,\ldots ,N$ and a.e. $t\in (\tau ,T)$.

2 Preliminary Properties of Weak Solutions

Zgurovsky and Kasyanov [4, p. 225] provide the existence of a weak solution of Cauchy problem (3.1) with initial data $y(\tau )=y^{(\tau )}$ on the interval $[\tau ,T]$, for any $y^{(\tau )}\in H$. For fixed $\tau$ and T, such that $0\le \tau<T<+\infty$, we denote

$$ \mathscr {D}_{\tau ,T}(y^{(\tau )})=\{y(\cdot )\ | \ y\,\text {is a weak solution of}\, (3.1)\,\text {on}\, [\tau ,T], \ y(\tau )=y^{(\tau )}\},\quad y^{(\tau )}\in H. $$

We remark that $\mathscr {D}_{\tau ,T}(y^{(\tau )})\ne \emptyset$, if $0\le \tau<T<+\infty$ and $y^{(\tau )}\in H$. Moreover, the concatenation of Problem (3.1) weak solutions is a weak solutions too, i.e., if $0\le \tau<t<T$, $y^{(\tau )}\in H$, $y(\cdot )\in \mathscr {D}_{\tau ,t}(y^{(\tau )})$, and $v(\cdot )\in \mathscr {D}_{t,T}(y(t))$, then

$$z(s)=\left\{ \begin{array}{ll} y(s),&{}s\in [\tau ,t],\\ v(s),&{}s\in [t,T], \end{array}\right. $$

belongs to $\mathscr {D}_{\tau ,T}(y^{(\tau )})$; cf. Zgurovsky et al. [5, pp. 55–56].

Gronwall lemma provides that for any finite time interval $[\tau ,T]\subset \mathbb {R}_+$ each weak solution y of Problem (3.1) on $[\tau ,T]$ satisfies estimates

$$\begin{aligned} \Vert y(t)\Vert _H^2 -2 \int _0^t \beta (\xi )d\xi + 2\alpha \sum _{i=1}^N\int _s^t\Vert y(\xi )\Vert _{V_i}^p d\xi \le \Vert y(s)\Vert _H^2- 2 \int _0^s \beta (\xi )d\xi , \end{aligned}$$

(3.3)

$$\begin{aligned} \Vert y(t)\Vert _H^2\le \Vert y(s)\Vert _H^2 e^{-2\alpha \gamma (t-s)}+2 \int _s^t (\beta (\xi )+\alpha \gamma )e^{-2\alpha \gamma (t-\xi )}d\xi , \end{aligned}$$

(3.4)

where $t,s\in [\tau ,T]$, $t\ge s$; $\gamma$ is a constant that does not depend on y, s, and t; see Zgurovsky and Kasyanov [4, p. 225]. Therefore, any weak solution y of Problem (3.1) on a finite time interval $[\tau ,T]\subset \mathbb {R}_+$ can be extended to a global one, defined on $[\tau ,+\infty )$.

For each $\tau \ge 0$ and $y^{(\tau )}\in H$ let $\mathscr {D}_\tau (y^{(\tau )})$ be the set of all weak solutions (defined on $[\tau ,+\infty )$) of Problem (3.1) with initial data $y(\tau )=y^{(\tau )}$. Let us consider the family $\mathscr {K}_\tau ^+=\cup _{y^{(\tau )}\in H}\mathscr {D}_\tau (y^{(\tau )})$ of all weak solutions of Problem (3.1) defined on the semi-infinite time interval $[\tau ,+\infty )$.

Consider the Fréchet space $C^\mathrm{loc}(\mathbb {R}_+;H)$. We remark that the sequence $\{f_n\}_{n\ge 1}$ converges in $C^\mathrm{loc}(\mathbb {R}_+;H)$ toward $f\in C^\mathrm{loc}(\mathbb {R}_+;H)$ as $n\rightarrow +\infty$ iff the sequence $\{\varPi _{t_1,t_2}f_n\}_{n\ge 1}$ converges in $C([t_1,t_2];H)$ toward $\varPi _{t_1,t_2}f$ as $n\rightarrow +\infty$ for any finite interval $[t_1,t_2]\subset \mathbb {R}_+$, where $\varPi _{t_1,t_2}$ is the restriction operator to the interval $[t_1,t_2]$; Chepyzhov and Vishik [6, p. 918]. We denote $T(h)y(\cdot )=y_h(\cdot )$, where $y_h(t)=y(t+h)$ for any $y\in C^\mathrm{loc}(\mathbb {R}_+;H)$ and $t,h \ge 0$.

Let us consider united trajectory space that includes all globally defined on any $[\tau ,+\infty )\subseteq \mathbb {R}_+$ weak solutions of Problem (3.1) shifted to $\tau =0$:

$$ \mathscr {K}^+=\mathrm{cl}_{C^\mathrm{loc}(\mathbb {R}_+;H)}\left[ \bigcup \limits _{\tau \ge 0}\left\{ y(\,\cdot +\,\tau )\,:\, y\in \mathscr {K}_{\tau }^+\right\} \right] , $$

where $\mathrm{cl}_{C^\mathrm{loc}(\mathbb {R}_+;H)}[\,\cdot \,]$ is the closure in $C^\mathrm{loc}(\mathbb {R}_+;H)$. Note that $T(h)\{y(\,\cdot +\tau )\,:\, y\in \mathscr {K}_{\tau }^+\}\subseteq \{y(\,\cdot \,+\tau +h)\,:\, y\in \mathscr {K}_{\tau +h}^+\}$ for any $\tau ,h\ge 0$. Moreover,

$$ T(h)\mathscr {K}^+\subseteq \mathscr {K}^+ \text{ for } \text{ any } h\ge 0, $$

because

$$ \rho _{C^\mathrm{loc}(\mathbb {R}_+;H)}(T(h)u,T(h)v)\le \rho _{C^\mathrm{loc}(\mathbb {R}_+;H)}(u,v) \text{ for } \text{ any } u,v\in C^\mathrm{loc}(\mathbb {R}_+;H), $$

where $\rho _{C^\mathrm{loc}(\mathbb {R}_+;H)}$ is a standard metric on Fréchet space $C^\mathrm{loc}(\mathbb {R}_+;H)$; Zgurovsky and Kasyanov [4, p. 226].

The following Lemma 3.1 and Theorem 3.1 are keynote for the existence of compact (in the natural phase space H) uniform global attractor for all weak solutions of Problem (3.1).

Lemma 3.1

(Zgurovsky and Kasyanov [4]) Let Assumptions (1)–(5) hold. Then, there exist positive constants $c_3$ and $c_4$ such that the following inequalities hold:

$$ \Vert y(t)\Vert _H^2\le \Vert y(s)\Vert _H^2 e^{-c_3(t-s)}+c_4, $$

for each $y\in \mathscr {K}^+,$ $t\ge s\ge 0$.

Theorem 3.1

(Zgurovsky and Kasyanov [4]) Let Assumptions (1)–(5) hold. Let $\{y_n\}_{n\ge 1}\subset \mathscr {K}^+$ be a bounded in $L_\infty (\mathbb {R}_+;H)$ sequence. Then, there exist a subsequence $\{y_{n_k}\}_{k\ge 1}\subset \{y_{n}\}_{n\ge 1}$ and an element $y\in \mathscr {K}^+$ such that

$$ \max \limits _{t\in [\tau ,T]}\Vert y_{n_k}(t)-y(t)\Vert _H\rightarrow 0,\quad k\rightarrow +\infty , $$

for any finite time interval $[\tau ,T]\subset (0,+\infty )$.

3 Uniform Global Attractor for all Weak Solutions of Problem (3.1)

Let us define the multivalued semi-flow (m-semi-flow) $G:\mathbb {R}_+\times H\rightarrow 2^{H}$:

$$\begin{aligned} G(t,y_0):=\{y(t)\,:\, y(\cdot )\in \mathscr {K}^+ \text{ and } y(0)=y_0 \}, \quad t\ge 0,\,y_0\in H. \end{aligned}$$

(3.5)

For each $t\ge 0$ and $y_0\in H$, the set $G(t,y_0)$ is nonempty. Moreover, the following two conditions hold:

• (i) $G\left( 0,\cdot \right) =I$ is the identity map;

• (ii) $G\left( t_1+t_2,y_0\right) \subseteq G\left( t_1,G\left( t_2,y_0\right) \right) ,\,\,\forall t_1,t_2\in \mathbb {R}_+,$ $\forall y_0\in H,$

where $G\left( t,D\right) =\underset{y\in D}{\cup }G\left( t,y\right) ,\,\,D\subseteq H$.

We denote by ${\text {dist}}_H(C,D)=\sup _{c\in C}\inf _{d\in D}\rho (c,d)$ the Hausdorff semi-distance between nonempty subsets C and D of the Polish space H. Recall that the set ${\mathscr {R}} \subset H$ is a global attractor of the m-semi-flow G if it satisfies the following conditions:

• (i) ${\mathscr {R}}$ attracts each bounded subset $B\subset H$, i.e.,

  $$\begin{aligned} {\text {dist}}_H(G(t,B),{\mathscr {R}})\rightarrow 0,\quad t\rightarrow +\infty ; \end{aligned}$$

  (3.6)

• (ii) ${\mathscr {R}}$ is negatively semi-invariant set, i.e., ${\mathscr {R}} \subseteq G\left( t,{\mathscr {R}} \right)$ for each $t\ge 0$;

• (iii) ${\mathscr {R}}$ is the minimal set among all nonempty closed subsets $C\subseteq H$ that satisfy (3.6).

The main result of this paper has the following form.

Theorem 3.2

Let Assumptions (1)–(5) hold. Then, the m-semi-flow G, defined in (3.5), has a compact global attractor ${\mathscr {R}}$ in the phase space H.

4 Proof of Theorem 3.2

Lemma 3.1 and Theorem 3.1 imply the following properties for the m-semiflow G, defined in (3.5):

• (a) for each $t\ge 0$, the mapping $G(t,\,\cdot \,):H\rightarrow 2^{H}\setminus \{\emptyset \}$ has a closed graph;

• (b) for each $t\ge 0$ and $y_0\in H$, the set $G(t,y_0)$ is compact in H;

• (c) the set $G(1,\tilde{C})$, where $\tilde{C}:=\{z\in H\,:\, \Vert z\Vert _H^2< c_4+1 \}$, is precompact and attracts each bounded subset $C\subset H$.

Indeed, property (a) follows from Theorem 3.1; property (b) directly follows from (a) and Theorem 3.1; property (c) holds, because of Lemma 3.1 and since the set $G(1,\tilde{C})$ is precompact in H (Theorem 3.1).

According to properties (a)–(c), Mel’nik and Valero [7, Theorems 1, 2, Remark 2, Proposition 1] yields that the m-semi-flow G has a compact global attractor ${\mathscr {R}}$ in the phase space H.

5 Conclusions

For the class of nonautonomous differential-operator inclusions with pointwise pseudomonotone operators, the dynamics (as $t\rightarrow +\infty$) of all global weak solutions defined on $[0,+\infty )$ is examined. The existence of a compact global attractor in the natural phase space H is proved. The results obtained allow one to study the dynamics of solutions for new classes of evolution inclusions related to nonlinear mathematical models of geophysical and socioeconomic processes and for fields with interaction functions of pseudomonotone type satisfying the power growth and sign conditions. For applications, one can consider new classes of problems with degeneracy, feedback control problems, problems on manifolds, problems with delay, stochastic partial differential equations, etc. (see Balibrea et al. [8]; Hu and Papageorgiou [9]; Gasinski and Papageorgiou [10]; Kasyanov [11]; Kasyanov, Toscano, and Zadoianchuk [12]; Mel’nik and Valero [13]; Denkowski, Migórski, and Papageorgiou [14]; Gasinski and Papageorgiou [10]; Zgurovsky et al. [5]; etc., see, also, [16–31]) involving differential operators of pseudomonotone type and the corresponding choice of the phase spaces. This note is a continuation of Zgurovsky and Kasyanov [4, 15].