Asymptotic Translation Uniform Integrability and Multivalued Dynamics of Solutions for Non-autonomous Reaction-Diffusion Equations

Abstract

In this note we introduce asymptotic translation uniform integrability condition for a function acting from a positive semi-axes of time-line to a Banach space. We prove that this condition is equivalent to uniform integrability condition. As a result, we obtain the corollaries for the multivalued dynamics (as time t → +∞) of solutions for non-autonomous reaction-diffusion equations.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Let $\mathbb {R}=[0,+\infty ),$ γ ≥ 1, and $\mathcal {E}$ be a real separable Banach space. As $L_{\gamma }^{\mathrm {loc}}({\mathbb R}_+; \mathcal {E})$ we consider the Fréchet space of all locally integrable functions with values in $\mathcal {E}$, that is, $\varphi \in L_\gamma ^{\mathrm {loc}}({\mathbb R}_+; \mathcal {E})$ if and only if for any finite interval $[\tau ,T]\subset {\mathbb R}_+$ the restriction of φ on [τ, T] belongs to the space $L_\gamma (\tau ,T; \mathcal {E})$. If $\mathcal {E}\subseteq L_1(\varOmega )$, then any function φ from $L_\gamma ^{\mathrm {loc}}({\mathbb R}_+; \mathcal {E})$ can be considered as a measurable mapping that acts from $\varOmega \times {\mathbb R}_+$ into ${\mathbb R}$. Further, we write φ(x, t), when we consider this mapping as a function from $\varOmega \times {\mathbb R}_+$ into ${\mathbb R}$, and φ(t), if this mapping is considered as an element from $L_\gamma ^{\mathrm {loc}}({\mathbb R}_+; \mathcal {E})$; cf. Gajewski et al. [5, Chapter III]; Temam [10]; Babin and Vishik [1]; Chepyzhov and Vishik [3]; Zgurovsky et at. [12] and references therein. A function $\varphi \in L_\gamma ^{\mathrm {loc}}({\mathbb R}_+; \mathcal {E})$ is called translation bounded in $L_\gamma ^{\mathrm {loc}}({\mathbb R}_+; \mathcal {E})$, if

$$\displaystyle \begin{aligned} \sup_{t\ge0}\int\limits_{t}^{t+1} {\|\varphi(s)\|{}_{\mathcal{E}}^{\gamma}} ds<+\infty; \end{aligned} $$

(21.1)

Chepyzhov and Vishik [4, p. 105].

Let N = 1, 2, … and $\varOmega \subset \mathbb {R}^N$ be a bounded domain. A function $\varphi \in L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ is called translation uniform integrable one (t.u.i.) in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$, if

$$\displaystyle \begin{aligned} \lim_{K\to+\infty}\sup_{t\ge0}\int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds=0; \end{aligned} $$

(21.2)

Gorban et al. [6,7,8,9]. Dunford-Pettis compactness criterion provides that a function $\varphi \in L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ is t.u.i. in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ if and only if for every sequence of elements $\{\tau _n\}_{n\ge 1}\subset {\mathbb R}_+$ the sequence {φ(⋅ + τ _n)}_n≥1 contains a subsequence which converges weakly in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$. We note that for any γ > 1 Hölder’s and Chebyshev’s inequalities imply that every translation bounded in $L_{\gamma }^{\mathrm {loc}}({\mathbb R}_+;L_{\gamma }(\varOmega ))$ function is t.u.i. in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$, because

$$\displaystyle \begin{aligned} {\int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds\le \frac{1}{K^{\gamma-1}}\sup\limits_{t\ge0}\int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|{}^\gamma dxds\to 0 \mbox{ as }K\to+\infty.} \end{aligned}$$

Let us introduce the definition of asymptotic translation uniform integrable function.

Definition 21.1

A function $\varphi \in L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ is called asymptotic translation uniform integrable one (a.t.u.i.) in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$, if

(21.3)

Remark 21.1

The limit (as K → +∞) in (21.2) ((21.3)) exists because the function

(21.4)

is nonincreasing in K > 0.

The main result of this note has the following formulation.

Theorem 21.1

Let $\varphi \in L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega )).$ Then there exists $\tilde {T}\ge 0$ such that $\varphi (\cdot \,+\tilde {T})$ is t.u.i. in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ iff φ is a.t.u.i. in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega )).$

In Sect. 21.3 we apply Theorem 21.1 to non-autonomous nonlinear reaction-diffusion system.

2 Proof of Theorem 21.1

Let us prove Theorem 21.1. The t.u.i. of $\varphi (\cdot \,+\tilde {T})$ for some $\tilde {T}\ge 0$ implies a.u.t.i. of φ(⋅) because for each sequence $\{a_n\}_{n=1,2,\ldots }\subset \overline {\mathbb {R}}$ its limit superior is no greater than its supremum, that is, (21.2) implies (21.3). Let us prove the converse statement: if φ(⋅) is a.t.u.i., then $\varphi (\cdot \,+\tilde {T})$ is t.u.i. for some $\tilde {T}\ge 0.$ We provide the proof in several steps.

Step 1

The following equalities hold:

(21.5)

Indeed, the first equality follows from a.t.u.i. of φ(⋅), the second equality holds because the mapping

is nonincreasing and for each $a:[0,+\infty )\mapsto \overline {\mathbb {R}}$ the equality

holds, and the last equality follows from the basic properties of infimum.

Step 2

We set

$$\displaystyle \begin{aligned} \delta(T):=\inf_{K>0}\sup_{t\ge T} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds, \end{aligned} $$

(21.6)

T ≥ 0, and notice that (21.5) directly implies the existence of $\tilde {T}\ge 0$ such that

(21.7)

Step 3

According to (21.6) and (21.7), for each $T\ge \tilde {T}$ there exists K _T > 0 such that

$$\displaystyle \begin{aligned} \sup_{t\ge T} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds<\delta(T)+\frac 1T<+\infty, \end{aligned} $$

(21.8)

for each K ≥ K _T.

Step 4

Since for each n = 0, 1, …

$$\displaystyle \begin{aligned} \begin{aligned} \int\limits_{{\tilde{T}}+n}^{{\tilde{T}}+n+1}\int\limits_{\varOmega}|\varphi(x,s)|dxds&= \int\limits_{{\tilde{T}}+n}^{{\tilde{T}}+n+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\le K_{\tilde{T}}\}}dxds \\ &+\int\limits_{{\tilde{T}}+n}^{{\tilde{T}}+n+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K_{\tilde{T}}\}}dxds\\ &\le K_{\tilde{T}}{\mathrm{meas}}(\varOmega)+\delta(\tilde{T})+\frac 1{\tilde{T}}<+\infty, \end{aligned} \end{aligned}$$

where the first inequality follows from (21.8), and the second inequality holds because meas(Ω) < +∞, then absolute continuity of the Lebesgue integral implies that for each $T>\tilde {T}$ and $t\in [\tilde {T},T]$ there exists $K(\tilde {T},T)>0$ such that

$$\displaystyle \begin{aligned} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds\le \int\limits_{\tilde{T}}^{T+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds <\frac 1T \end{aligned}$$

for each $K\ge K(\tilde {T},T),$ that is,

$$\displaystyle \begin{aligned} \sup_{t\in[\tilde{T},T]}\int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds\le \frac 1N, \end{aligned} $$

(21.9)

for each $T>\tilde {T}$ and $K\ge \tilde {K}_T^{\tilde {T}}:=\sup \limits _{t\in [\tilde {T},T]}\{K_T;K(\tilde {T},T)\}.$

Step 5

Inequalities (21.8) and (21.9) imply that

$$\displaystyle \begin{aligned} \sup_{t\ge \tilde{T}} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds<\delta(T)+\frac 1T, \end{aligned}$$

for each $T>\tilde {T}$ and $K\ge \tilde {K}_T^{\tilde {T}}.$ Thus, according to (21.6),

$$\displaystyle \begin{aligned} \delta(\tilde{T})=\inf_{K>0}\sup_{t\ge \tilde{T}} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds<\delta(T)+\frac 1T, \end{aligned} $$

(21.10)

for each $T>\tilde {T}.$

Step 6

Since the function

$$\displaystyle \begin{aligned} K\mapsto \sup_{t\ge \tilde{T}} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds \end{aligned}$$

is nonincreasing, we have that

$$\displaystyle \begin{aligned} \lim_{K\to+\infty}\sup_{t\ge \tilde{T}} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds=\delta(\tilde{T})<\delta(T)+\frac 1T. \end{aligned} $$

(21.11)

for each $T>\tilde {T},$ where the inequality follows from (21.10). According to (21.7), as T → +∞. Therefore, (21.11) implies that

$$\displaystyle \begin{aligned} \lim_{K\to+\infty}\sup_{t\ge \tilde{T}} \int\limits_{t}^{t+1}\int\limits_{\varOmega}|\varphi(x,s)|\chi_{\{|\varphi(x,s)|\ge K\}}dxds=0, \end{aligned}$$

that is, φ(⋅) is t.u.i.

3 Examples of Applications

Let N, M = 1, 2, …, $\varOmega \subset \mathbb {R}^{N}$ be a bounded domain with sufficiently smooth boundary ∂Ω. We consider a problem of long-time behavior of all globally defined weak solutions for the non-autonomous parabolic problem (named RD-system)

$$\displaystyle \begin{aligned} \left\{ \begin{array} [c]{l} y_{t}=a\varDelta y-f(x,t,y),\quad x\in\varOmega,\ t>0,\\ y|{}_{\partial\varOmega}=0, \end{array} \right. {} \end{aligned} $$

(21.12)

as t → +∞, where y = y(x, t) = (y ⁽¹⁾(x, t), …, y ^(M)(x, t)) is unknown vector-function, f = f(x, t, y) = (f ⁽¹⁾(x, t, y), …, f ^(M)(x, t, y)) is given function, a is real M × M matrix with positive symmetric part.

We suppose that the listed below assumptions hold.

Assumption I

Let p _i ≥ 2 and q _i > 1 are such that $\frac 1{p_i}+\frac 1{q_i}=1$, for any i = 1, 2, …, M. Moreover, there exists a positive constant d such that $\frac {1}{2}(a+a^*)\geq d I$, where I is unit M × M matrix, a ^∗ is a transposed matrix for a.

Assumption II

The interaction function $f=(f^{(1)},\ldots ,f^{(M)}):\varOmega \times {\mathbb R}_+\times {\mathbb R}^M\to {\mathbb R}^M$ satisfies the standard Carathéodory’s conditions, i.e. the mapping (x, t, u) → f(x, t, u) is continuous in $u\in {\mathbb R}^M$ for a.e. $(x,t)\in \varOmega \times {\mathbb R}_+$, and it is measurable in $(x,t)\in \varOmega \times {\mathbb R}_+$ for any $u\in {\mathbb R}^M$.

Assumption III (Growth Condition)

There exist an a.t.u.i. in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ function $c_1:\varOmega \times {\mathbb R}_+\to {\mathbb R}_+$ and a constant c ₂ > 0 such that

$$\displaystyle \begin{aligned} \sum\limits_{i=1}^M\left|f^{(i)}(x,t,u)\right|{}^{q_i}\le c_1(x,t)+c_2\sum\limits_{i=1}^M\left|u^{(i)}\right|{}^{p_i} \end{aligned}$$

for any $u=(u^{(1)},\ldots ,u^{(M)})\in {\mathbb R}^M$, and a.e. $(x,t)\in \varOmega \times {\mathbb R}_+$.

Assumption IV (Sign Condition)

There exists a constant α > 0 and an a.t.u.i. in $L_1^{\mathrm {loc}}({\mathbb R}_+; L_1(\varOmega ))$ function $\beta :\varOmega \times {\mathbb R}_+\to {\mathbb R}_+$ such that

$$\displaystyle \begin{aligned} \sum\limits_{i=1}^M f^{(i)}(x,t,u)u^{(i)}\ge \alpha \sum\limits_{i=1}^M\left|u^{(i)}\right|{}^{p_i}-\beta(x,t) \end{aligned}$$

for any $u=(u^{(1)},\ldots ,u^{(M)})\in {\mathbb R}^M$, and a.e. $(x,t)\in \varOmega \times {\mathbb R}_+$.

In further arguments we will use standard functional Hilbert spaces H = (L ₂(Ω))^M, $V=(H_0^1(\varOmega ))^M$, and V ^∗ = (H ⁻¹(Ω))^M with standard respective inner products and norms (⋅, ⋅)_H and ∥⋅∥_H, (⋅, ⋅)_V and ∥⋅∥_V, and $(\cdot ,\cdot )_{V^*}$ and $\|\cdot \|{ }_{V^*}$, vector notations p = (p ₁, p ₂, …, p _M) and q = (q ₁, q ₂, …, q _M), and the spaces

$$\displaystyle \begin{aligned} \begin{array}{l} {\mathbf L}_{\mathbf p}(\varOmega):=L_{p_1}(\varOmega)\times\ldots\times L_{p_M}(\varOmega),\quad {\mathbf L}_{\mathbf q}(\varOmega):=L_{q_1}(\varOmega)\times\ldots\times L_{q_M}(\varOmega),\\ {\mathbf L}_{\mathbf p}(\tau,T;{\mathbf L}_{\mathbf p}(\varOmega)):=L_{p_1}(\tau,T;L_{p_1}(\varOmega))\times\ldots\times L_{p_M}(\tau,T;L_{p_M}(\varOmega)),\\ {\mathbf L}_{\mathbf q}(\tau,T;{\mathbf L}_{\mathbf q}(\varOmega)):=L_{q_1}(\tau,T;L_{q_1}(\varOmega))\times\ldots\times L_{q_M}(\tau,T;L_{q_M}(\varOmega)),\ 0\le\tau<T<+\infty. \vspace{-2pt}\end{array} \end{aligned}$$

Let 0 ≤ τ < T < +∞. A function y = y(x, t) ∈L ₂(τ, T;V ) ∩L _p(τ, T;L _p(Ω)) is called a weak solution of Problem (21.12) on [τ, T], if for any function $\varphi =\varphi (x)\in (C_0^{\infty }(\varOmega ))^M$, the following identity holds

$$\displaystyle \begin{aligned} \frac{d}{dt}\int_{\varOmega}y(x,t)\cdot\varphi(x)dx+\int_{\varOmega}\{a\nabla y(x,t)\cdot \nabla\varphi(x)+f(x,t,y(x,t))\cdot\varphi(x)\}dx=0 \end{aligned} $$

(21.13)

in the sense of scalar distributions on (τ, T).

In the general case Problem (21.12) on [τ, T] with initial condition y(x, τ) = y _τ(x) in Ω has more than one weak solution with y _τ ∈ H (cf. Balibrea et al. [2] and references therein).

Assumptions I–IV and Chepyzhov and Vishik [4, pp. 283–284] (see also Zgurovsky et al. [11, Chapter 2] and references therein) provide the existence of a weak solution of Cauchy problem (21.12) with initial data y(τ) = y ^(τ) on the interval [τ, T], for any y ^(τ) ∈ H. The proof is provided by standard Faedo–Galerkin approximations and using local existence Carathéodory’s theorem instead of classical Peano results. A priori estimates are similar. Formula (21.13) and definition of the derivative for an element from $\mathcal {D}([\tau ,T];V^*+{\mathbf L}_{\mathbf q}(\varOmega ))$ yield that each weak solution y ∈ X _τ,T of Problem (21.12) on [τ, T] belongs to the space W _τ,T. Moreover, each weak solution of Problem (21.12) on [τ, T] satisfies the equality:

$$\displaystyle \begin{aligned} \int_\tau^T\int_{\varOmega}\left[\frac{\partial y(x,t)}{\partial t}\cdot\psi(x,t)+ a\nabla y(x,t)\cdot \nabla\psi(x,t)+f(x,t,y(x,t))\cdot\psi(x,t)\right]dxdt=0, \end{aligned} $$

(21.14)

for any ψ ∈ X _τ,T. For fixed τ and T, such that 0 ≤ τ < T < +∞, we denote

$$\displaystyle \begin{aligned} \mathcal{D}_{\tau,T}(y^{(\tau)})=\{y(\cdot)\ | \ y\mbox{ is a weak solution of (21.12) on } [\tau,T], \ y(\tau)=y^{(\tau)}\},\quad y^{(\tau)}\in H. \end{aligned} $$

We remark that $\mathcal {D}_{\tau ,T}(y^{(\tau )})\ne \emptyset$ and $\mathcal {D}_{\tau ,T}(y^{(\tau )})\subset W_{\tau ,T}$, if 0 ≤ τ < T < +∞ and y ^(τ) ∈ H. Moreover, the concatenation of Problem (21.12) weak solutions is a weak solutions too, i.e. if 0 ≤ τ < t < T, y ^(τ) ∈ H, $y(\cdot )\in \mathcal {D}_{\tau ,t}(y^{(\tau )})$, and $v(\cdot )\in \mathcal {D}_{t,T}(y(t))$, then

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

belongs to $\mathcal {D}_{\tau ,T}(y^{(\tau )})$; cf. Zgurovsky et al. [12, pp. 55–56].

Each weak solution y of Problem (21.12) on a finite time interval $[\tau ,T]\subset {\mathbb R}_+$ can be extended to a global one, defined on [τ, +∞). For arbitrary τ ≥ 0 and y ^(τ) ∈ H let $\mathcal {D}_\tau (y^{(\tau )})$ be the set of all weak solutions (defined on [τ, +∞)) of Problem (21.12) with initial data y(τ) = y ^(τ). Let us consider the family $\mathcal {K}_\tau ^+=\cup _{y^{(\tau )}\in H}\mathcal {D}_\tau (y^{(\tau )})$ of all weak solutions of Problem (21.12) defined on the semi-infinite time interval [τ, +∞).

Consider the Fréchet space

$$\displaystyle \begin{aligned} C^{\mathrm{loc}}(\mathbb{R}_+;H):=\{y:{\mathbb R}_+\to H\,:\, \varPi_{t_1,t_2}y\in C([t_1,t_2];H)\mbox{ for any }[t_1,t_2]\subset{\mathbb R}_+\}, \end{aligned}$$

where $\varPi _{t_1,t_2}$ is the restriction operator to the interval [t ₁, t ₂]; Chepyzhov and Vishik [3, p. 918]. We remark that the sequence {f _n}_n≥1 converges (converges weakly respectively) in $C^{\mathrm {loc}}(\mathbb {R}_+;H)$ towards $f\in C^{\mathrm {loc}}(\mathbb {R}_+;H)$ as n → +∞ if and only if the sequence $\{\varPi _{t_1,t_2}f_n\}_{n\ge 1}$ converges (converges weakly respectively) in C([t ₁, t ₂];H) towards $\varPi _{t_1,t_2}f$ as n → +∞ for any finite interval $[t_1,t_2]\subset {\mathbb R}_+$.

We denote T(h)y(⋅) = y _h(⋅), where y _h(t) = y(t + h) for any $y\in C^{\mathrm {loc}}(\mathbb {R}_+;H)$ and t, h ≥ 0.

In the non-autonomous case we notice that $T(h)\mathcal {K}_0^+\not \subseteq \mathcal {K}_{0}^+$. Therefore (see Gorban et al. [8]), we need to consider united trajectory space that includes all globally defined on any $[\tau ,+\infty )\subseteq {\mathbb R}_+$ weak solutions of Problem (21.12) shifted to τ = 0:

$$\displaystyle \begin{aligned} \mathcal{K}_\cup^+:=\bigcup\limits_{\tau\ge 0}\left\{y(\cdot\,+\tau)\in W^{\mathrm{loc}}({\mathbb R}_+)\,:\, y(\cdot)\in \mathcal{K}_{\tau}^+\right\}. \end{aligned} $$

(21.15)

Note that $T(h)\{y(\cdot \,+\tau )\,:\, y\in \mathcal {K}_{\tau }^+\}\subseteq \{y(\cdot \,+\tau +h)\,:\, y\in \mathcal {K}_{\tau +h}^+\}$ for any τ, h ≥ 0. Therefore,

$$\displaystyle \begin{aligned}T(h)\mathcal{K}_\cup^+\subseteq\mathcal{K}_\cup^+\end{aligned}$$

for any h ≥ 0. Further we consider extended united trajectory space for Problem (21.12):

$$\displaystyle \begin{aligned} \quad \mathcal{K}_{C^{\mathrm{loc}}(\mathbb{R}_+;H)}^+={\mathrm{cl}}_{C^{\mathrm{loc}}(\mathbb{R}_+;H)}\left[\mathcal{K}_\cup^+\right], \end{aligned} $$

(21.16)

where ${\mathrm {cl}}_{C^{\mathrm {loc}}(\mathbb {R}_+;H)}[\,\cdot \,]$ is the closure in $C^{\mathrm {loc}}(\mathbb {R}_+;H)$. We note that

$$\displaystyle \begin{aligned} T(h)\mathcal{K}_{C^{\mathrm{loc}}(\mathbb{R}_+;H)}^+\subseteq\mathcal{K}_{C^{\mathrm{loc}}(\mathbb{R}_+;H)}^+ \end{aligned}$$

for each h ≥ 0, because

$$\displaystyle \begin{aligned} \rho_{C^{\mathrm{loc}}(\mathbb{R}_+;H)}(T(h)u,T(h)v)\le \rho_{C^{\mathrm{loc}}(\mathbb{R}_+;H)}(u,v)\mbox{ for any }u,v\in C^{\mathrm{loc}}(\mathbb{R}_+;H), \end{aligned}$$

where $\rho _{C^{\mathrm {loc}}(\mathbb {R}_+;H)}$ is a standard metric on Fréchet space $C^{\mathrm {loc}}(\mathbb {R}_+;H)$.

Let us provide the result characterizing the compactness properties of shifted solutions of Problem (21.12) in the induced topology from $C^{\mathrm {loc}}({\mathbb R}_+;H)$.

Theorem 21.2

Let Assumptions I–IV hold. If $\{y_n\}_{n\ge 1}\subset \mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ is an arbitrary sequence, which is bounded in $L_\infty (\mathbb {R}_+;H)$ , then there exist a subsequence $\{y_{n_k}\}_{k\ge 1}\subseteq \{y_{n}\}_{n\ge 1}$ and an element $y\in \mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ such that

$$\displaystyle \begin{aligned} \|\varPi_{\tau,T}y_{n_k}-\varPi_{\tau,T}y\|{}_{C([\tau,T];H)}\to0,\quad k\to+\infty, \end{aligned} $$

(21.17)

for any finite time interval [τ, T] ⊂ (0, +∞). Moreover, for any $y\in \mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ the estimate holds

$$\displaystyle \begin{aligned} \|y(t)\|{}_H^2\le\|y(0)\|{}_H^2 e^{-c_3t}+c_4, \end{aligned} $$

(21.18)

for any t ≥ 0, where positive constants c ₃ and c ₄ do not depend on $y\in \mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ and t ≥ 0.

Proof

This statement directly follows from Gorban et al. [8, Theorem 4.1] and Theorem 21.1.

A set $\mathcal {P}\subset \mathcal {F}^{\mathrm {loc}}(\mathbb {R}_+)\cap L_{\infty }(\mathbb {R}_+;H)$ is said to be a uniformly attracting set (cf. Chepyzhov and Vishik [3, p. 921]) for the extended united trajectory space $\mathcal {K}_{\mathcal {F}^{\mathrm {loc}}(\mathbb {R}_+)}^+$ of Problem (21.12) in the topology of $\mathcal {F}^{\mathrm {loc}}(\mathbb {R}_+)$, if for any bounded in $L_{\infty }(\mathbb {R}_+;H)$ set $\mathcal {B}\subseteq \mathcal {K}_{\mathcal {F}^{\mathrm {loc}}(\mathbb {R}_+)}^+$ and any segment $[t_1,t_2]\subset {\mathbb R}_+$ the following relation holds:

$$\displaystyle \begin{aligned} \mbox{dist}_{\mathcal{F}_{t_1,t_2}}(\varPi_{t_1,t_2}T(t)\mathcal{B},\varPi_{t_1,t_2}\mathcal{P})\to 0, \quad t\to+\infty, \end{aligned} $$

(21.19)

where $\mbox{dist}_{\mathcal {F}_{t_1,t_2}}$ is the Hausdorff semi-metric.

A set $\mathcal {U}\subset \mathcal {K}_{\mathcal {F}^{\mathrm {loc}}(\mathbb {R}_+)}^+$ is said to be a uniform trajectory attractor of the translation semigroup {T(t)}_t≥0 on $\mathcal {K}_{\mathcal {F}^{\mathrm {loc}}(\mathbb {R}_+)}^+$ in the induced topology from $C^{\mathrm {loc}}(\mathbb {R}_+;H)$, if

• 1. $\mathcal {U}$ is a compact set in $C^{\mathrm {loc}}(\mathbb {R}_+;H)$ and bounded in $L_{\infty }(\mathbb {R}_+;H)$;

• 2. $\mathcal {U}$ is strictly invariant with respect to {T(h)}_h≥0, i.e. $T(h)\mathcal {U}=\mathcal {U}$ ∀h ≥ 0;

• 3. $\mathcal {U}$ is a minimal uniformly attracting set for $\mathcal {K}_{C^{\mathrm {loc}}(\mathbb {R}_+;H)}^+$ in the topology of $C^{\mathrm {loc}}(\mathbb {R}_+;H)$, i.e. $\mathcal {U}$ belongs to any compact uniformly attracting set $\mathcal {P}$ of $\mathcal {K}_{C^{\mathrm {loc}}(\mathbb {R}_+;H)}^+$: $\mathcal {U}\subseteq \mathcal {P}$.

Note that uniform trajectory attractor of the translation semigroup {T(t)}_t≥0 on $\mathcal {K}_{C^{\mathrm {loc}}(\mathbb {R}_+;H)}^+$ in the induced topology from $C^{\mathrm {loc}}(\mathbb {R}_+;H)$ coincides with the classical global attractor for the continuous semi-group {T(t)}_t≥0 defined on $\mathcal {K}_{C^{\mathrm {loc}}(\mathbb {R}_+;H)}^+.$

Assumptions I–IV are sufficient conditions for the existence of uniform trajectory attractor for weak solutions of Problem (21.12) in the topology of $C^{\mathrm {loc}}({\mathbb R}_+;H)$.

Theorem 21.3

Let Assumptions I–IV hold. Then there exists an uniform trajectory attractor $\mathcal {U}\subset \mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ of the translation semigroup {T(t)}_t≥0 on $\mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ in the induced topology from ${C^{\mathrm {loc}}({\mathbb R}_+;H)}$ . Moreover, there exists a compact in ${C^{\mathrm {loc}}({\mathbb R}_+;H)}$ uniformly attracting set $\mathcal {P}\subset C^{\mathrm {loc}}({\mathbb R}_+;H)\cap L_{\infty }(\mathbb {R}_+;H)$ for the extended united trajectory space $\mathcal {K}_{C^{\mathrm {loc}}({\mathbb R}_+;H)}^+$ of Problem (21.12) in the topology of $C^{\mathrm {loc}}({\mathbb R}_+;H)$ such that $\mathcal {U}$ coincides with ω-limit set of $\mathcal {P}$ :

$$\displaystyle \begin{aligned} \mathcal{U}=\bigcap\limits_{t\ge 0}{\mathrm{cl}}_{C^{\mathrm{loc}}({\mathbb R}_+;H)}\left[\bigcup\limits_{h\ge t}T(h)\mathcal{P}\right]. \end{aligned} $$

(21.20)

Proof

This statement directly follows from Gorban et al. [8, Theorem 3.1] and Theorem 21.1.

4 Conclusions

Asymptotic translation uniform integrability condition for a function acting from positive semi-axe of time line to a Banach space is equivalent to uniform integrability condition. As a result, we claim only asymptotic (as time t → +∞) assumptions of translation compactness for parameters of non-autonomous reaction-diffusion equations.