Pullback Attractors for a Nonlocal Nonautonomous Evolution Model in $\mathbb {R}^N$

Abstract

In this work we consider the nonlocal evolution equation with time-dependent terms which arises in models of phase separation in $\mathbb {R}^N$

$$\begin{aligned} \partial _t u=- u + g \left( \beta (J*u) +\beta h(t)\right) \end{aligned}$$

under some restrictions on h, growth restrictions on the nonlinear term g and $\beta >1$. We prove the existence, regularity and upper-semicontinuity of pullback attractors with respect to functional parameter h(t) in some weighted spaces.

Appendix

In this section we give a brief description of as arises the model (1.6). For this we faithfully follow Section 2.1 of [10].

A spin configuration can be described as follows: at all lattice sites there is a spin variable with two different values in $\{-1, 1\}$.The value of the spin at x is flipped at rates which depend on the value of the others spin. More precisely, we have the following definition.

Definition 8.1

A spin configuration is a map

$$\begin{aligned} \sigma : \mathbb {Z}^{d} \rightarrow \{-1,1\}, \end{aligned}$$

that is an element of $\{-1,1\}^{\mathbb {Z}^{d}}$. The value $\sigma (x)$ of the spin at x is a function of the   configuration $\sigma$, thus a random variable on the space of all the spin configurations $\{-1,1\}^{\mathbb {Z}^{d}}$. The restriction to $\Delta \subset \mathbb {Z}^{d}$ of a configuration $\sigma$, is denoted by $\sigma _{\Delta }$, which is therefore a function on $\Delta$ with values $\{-1,1\}$.

Definition 8.2

A Kac potential is a function $J_{\gamma }:{\mathbb {Z}^{d}} \times {\mathbb {Z}^{d}} \rightarrow \mathbb {R}$, which depends on a (scaling) parameter $\gamma$ and has the form

$$\begin{aligned} J_{\gamma }(x,y)=\gamma ^{d}J(\gamma [x-y]), \end{aligned}$$

(8.1)

where $\gamma$ varies in the set $\{2^{-n}, \, n\in \mathbb {Z_{+}}\}$.

As in [10], we assume that J is a smooth symmetric probability density (thus $J \geqslant 0$) with compact support.

Given (a magnetic field) $h\in \mathbb {R}$, we define the energy of the spin configuration $\sigma _{\Delta }$ as

$$\begin{aligned} H_{\gamma }(\sigma _{\Delta })=-h \sum _{x\in \Delta } - \frac{1}{2}\sum _{x\not =y\in \Delta } J_{\gamma }(x,y)\sigma (x)\sigma (y), \end{aligned}$$

(8.2)

while its energy inclusive of the interaction with the spins in the complement, $\Delta {c}$, of $\Delta$, is

$$\begin{aligned} H_{\gamma }(\sigma _{\Delta }-\sigma _{\Delta ^{c}})=H_{\gamma }(\sigma _{\Delta }) - \sum _{x\in \Delta , y\not \in \Delta } J_{\gamma }(x,y)\sigma (x)\sigma (y), \end{aligned}$$

(8.3)

Definition 8.3

Given (the “the inverse temperature”) $\beta >0$ and $\gamma >0$, we denote by Glauber dynamics the unique Markov process on $\{-1,1\}^{\mathbb {Z}^{d}}$, (given in [18]) whose pregenerator is the operator $L_{\gamma }$ with domain the set of all cylinder functions f on which it acts as

$$\begin{aligned} L_{\gamma }f(\sigma )=\sum _{x\in \mathbb {Z}^{d}}c_{\gamma }(x,\sigma )[f(\sigma {x})-f(\sigma )]. \end{aligned}$$

(8.4)

In (8.4), $\sigma ^{x}$ is the configuration obtained from $\sigma$ by flipping the spin at x, namely,

$$\begin{aligned} \sigma ^{x}(y)= \sigma (y) \, if \, y\not =x, \,\, \sigma ^{x}(y)= -\sigma (x) \, if \, y=x. \end{aligned}$$

(8.5)

The “flip rate” $c_{\gamma }(x, \sigma )$ of the spin at x in the configuration $\sigma$ is

$$\begin{aligned} C_{\gamma }(x,\sigma )= \frac{e^{-\beta h_{\gamma }(x)\sigma (x)}}{e^{-\beta h_{\gamma }(x)}- e^{\beta h_{\gamma }(x)}}. \end{aligned}$$

(8.6)

where

$$\begin{aligned} h_{\gamma }(x)=h+(J_{\gamma }\circ \sigma )(x) \,\, \text{ and } \,\, (J_{\gamma }\circ \sigma )(x)=\sum _{y\not =x}J_{\gamma }(x,y)\sigma (y). \end{aligned}$$

(8.7)

The space of realizations of Glauber dynamics is $D(\mathbb {R}_{+}, \{-1,1\}^{\mathbb {Z}^{d}})$, the Skorohod space of cadlag trajectories, (continuous from the right and with limits from the left). The value of the spin in x at time t is $\sigma (x,t)$ which is thus a random variable on $D(\mathbb {R}_{+}, \{-1,1\}^{\mathbb {Z}^{d}})$.

Note that

$$\begin{aligned} c_{\gamma }(x, \sigma )= Z_{\gamma }(\sigma _{x^{c}})^{-1}e^{\frac{-\beta }{2\Delta _{x}H_{\gamma }(\sigma )}} \end{aligned}$$

where $\Delta _{x}H_{\gamma }(\sigma )$ is the change of energy due to the spin flip at x, that is

$$\begin{aligned} \Delta _{x}H_{\gamma }(\sigma )=H_{\gamma }((\sigma ^{x})_{\Lambda })- H_{\gamma }((\sigma _{\Lambda }) \end{aligned}$$

where $\Lambda$ is any set which contains x and such that the spin at x does not interact with those in $\Lambda ^{c}$. Here $Z_{\gamma }(\sigma _{x^{c}})^{-1}$ is the denominator in (8.6), but, for what we say below, it may be any other function, provided it is independent of $\sigma (x)$, as implied by the notation. In fact, the important point about the rates is that they verify the “detailed balance” condition

$$\begin{aligned} e^{-\beta \Delta _{x}H_{\gamma }(\sigma )}=\frac{c_{\gamma }(x,\sigma ^{x})}{c_{\gamma }(x,\sigma )}. \end{aligned}$$

(8.8)

Thus the Glauber dynamics is intimately related to the notion of Gibbs measures.

Definition 8.4

The Gibbs measure $\mu _{\beta ,h,\gamma }$ is any probability on $\{-1,1\}^{\mathbb {Z}^{d}}$ such that, for $x\in {\mathbb {Z}}^{d}$, satisfies the equation

$$\begin{aligned} \mu _{\beta ,h,\gamma }\left( \sigma (x)=\pm 1 \, | \, \{\sigma (y), \, y\not =x\} \right) =\frac{e^{\pm \beta h_{\gamma }(x)}}{e^{- \beta h_{\gamma }(x)}+ e^{\beta h_{\gamma }(x)}}, \end{aligned}$$

(8.9)

where the left hand side is the probability that $\sigma (x)=\pm 1$ conditioned on the $\sigma$-algebra generated by all the spin $\sigma (y), \, y\not =x$.

In definition above, $\beta$ has the physical meaning of an inverse temperature and h of an external magnetic field, $J_{\gamma }$ of the spin-spin interaction strength. Notice that the left hand side of (8.9) is a function of $\sigma (x)$ and all $\sigma (y), \, y\not =x$, it is thus a function of the whole spin configuration $\sigma$. Then, from (8.8) and (8.9), it follows that

$$\begin{aligned} \mu _{\beta ,h,\gamma }\left( \sigma (x) \, | \, \{\sigma (y), \, y\not =x\} \right) c_{\gamma }(x,\sigma )= \mu _{\beta ,h,\gamma }\left( \sigma ^{x}(x) \, | \, \{\sigma (y), \, y\not =x\} \right) c_{\gamma }(x,\sigma ^{x}) \end{aligned}$$

so that the operator $L_{\gamma }^{(x)}$ defined by (8.4) after setting $c_{\gamma }(y,\sigma )=0$ for all $y\not =x$, is self adjoint in $L^{2}(\{-1,1\}^{\mathbb {Z}^{d}}, \mu _{\beta ,h,\gamma })$. It then follows that also the full generator of the Glauber dynamics is self adjoint and $\mu _{\beta ,h,\gamma }$ is stationary, the Glauber dynamics then being a reversible process. Finally we mention that (8.8) does not depend on the choice of $Z_{\gamma }$, which appears in the definition of $c_{\gamma }$, thus different choice of $Z_{\gamma }$ define other, equally acceptable, reversible evolutions. The choice (8.6) gives rise to a simpler limiting mesoscopic equation.

The “scale separation” between the two levels is specified by $\gamma$ in the transition micro-mesoscopic

$$\begin{aligned} (x,t) \rightarrow (r,t)=(\gamma x, t), \end{aligned}$$

thus, time is unchanged while space is shrunk by $\gamma$. The microscopic points $x\in \mathbb {Z}^{d}$ are represented in the mesoscopic space $\mathbb {R}^{d}$ by the lattice $\gamma \mathbb {Z}^{d}$. It is thus convenient to partition $\mathbb {R}^{d}$ into the “elementary squares”

$$\begin{aligned} \{r \, : \, [r]_{\gamma }=\gamma x\}, \end{aligned}$$

with $x\in \mathbb {R}^{d}$ and, denoting by $r=(r_1, \ldots , r_d)$, $x=(x_1, \ldots , x_n)$,

$$\begin{aligned}{}[r]_{\gamma }=\gamma x \quad \text{ if } \quad x\in \mathbb {Z}^{d} \, \text{ and } \quad \gamma x_i \le r_i < \gamma (x_i +1)\quad \text{ for } \text{ all } \quad i=1, \ldots , d. \end{aligned}$$

(8.10)

Definition 8.5

Let X be a measurable space. Denoting by $\mathbb {M}(X)$ the space of all the real valued, measurable functions on X. We define $\Gamma _{\gamma }: \mathbb {M}(\mathbb {Z}^{d}) \rightarrow \mathbb {M}(\mathbb {R}^{d})$ by

$$\begin{aligned} (\Gamma _{\gamma }(f))(r)=f(x), \,\, x=\gamma ^{-1}[r]_{\gamma } \, \text{ and } \, f\in \mathbb {M}(\mathbb {Z}^{d}), \end{aligned}$$

(8.11)

where $[r]_{\gamma }$ is defined in (8.10).

In particular, we denote by

$$\begin{aligned} \sigma _{\gamma }= \Gamma _{\gamma }(\sigma ),\quad \sigma _{\gamma , t}= \Gamma _{\gamma }(\sigma _{t}), \end{aligned}$$

(8.12)

$\sigma _{\gamma }(r)$ is thus the image of the spin configuration $\sigma$ in the mesoscopic representation.

Definition 8.6

We define, for any $0< \alpha < 1$ and $\gamma$ as in (8.1), the block spin transformation $f \rightarrow f^{(\alpha ,\gamma )}$, f and $f^{(\alpha , \gamma )}$ both in $\mathbb {M}(\mathbb {R}^{d})$, as

$$\begin{aligned} f^{(\alpha ,\gamma )}(r)= & {} N_{\gamma }^{-1} \int \mathbf{1} (\{ | [r]_{\gamma } - [r']_{\gamma } | \leqslant \gamma ^{1-\alpha } \}) f(r') dr' \end{aligned}$$

(8.13)

$$\begin{aligned} N_{\gamma }= & {} \int \mathbf{1} (\{ | [r]_{\gamma } - [r']_{\gamma } | \leqslant \gamma ^{1-\alpha } \}) dr' \end{aligned}$$

(8.14)

As well as in [10], we use the shorthand notation

$$\begin{aligned} \sigma _{\gamma }^{(\alpha )}{:=}(\sigma _{\gamma })^{(\alpha ,\gamma )};\quad \sigma _{\gamma , t}^{(\alpha )}{:=}(\sigma _{\gamma , t})^{(\alpha ,\gamma )} \end{aligned}$$

(8.15)

to avoid redundancy in the formulas, and in general we may omit the superscript $\gamma$ in $(\alpha , \gamma )$, when $\gamma$ already appears as a subscript.

The more familiar form of the block spin transformation is recovered when we apply the transformation to a function $g=\Gamma _{\gamma }(f)$, $f\in \mathbb {M}(\mathbb {Z}^{d})$. In that case $g^{\alpha , \gamma }(r)$, $[r]_{\gamma }{:=}\gamma x$, is given by

$$\begin{aligned} g^{\alpha , \gamma }(r)= \mathbf{A}_{\gamma ^{-\alpha },x}(f){:=}\frac{1}{|\mathbf{B}_{\gamma ^{-\alpha }}|} \sum _{y\in \mathbf{B}_{\gamma ^{-\alpha },x}} f(y) \end{aligned}$$

(8.16)

where

$$\begin{aligned} \mathbf{B}_{\gamma ^{-\alpha },x} = \{ |x-y|\leqslant \gamma ^{-\alpha } \}, \,\, | \mathbf{B}_{\gamma ^{-\alpha }} |= \text{ cardinality } \, \text{ of } \, \mathbf{B}_{\gamma ^{-\alpha },x}. \end{aligned}$$

(8.17)

We end this section with the following theorem, that it has been proved in [10] (see Sect. 2.1, Theorem 2.1.6).

Theorem 8.7

For any $\alpha \in (0,1)$ there are $\zeta$, a and b all positive and for any n and any $k^{*} \geqslant 2$, there is c so that the following holds: Given $\gamma$ small enough, for all $\sigma \in \{-1,1\}^{\mathbb {Z}^{d}}$ and $m\in [-1,1]^{\mathbb {Z}^{d}}$, $\Vert m\Vert _{\infty }\leqslant 1$, for which (see (8.12)–(8.15) for notation)

$$\begin{aligned} \sup _{|r|\le k^{*} \gamma ^{-1}} | \sigma _{\gamma }^{(\alpha )}(r) - m^{(\alpha , \gamma )}(r) | \leqslant \gamma ^{\zeta } \end{aligned}$$

we have that

$$\begin{aligned} \mathbb {P}_{\sigma }^{\gamma } \left( \sup _{t\le a \log \gamma ^{-1} |r| \le (k^{*}-1)\gamma ^{-1}} \left| \sigma _{\gamma , t}^{(\alpha )}(r) - m_{\gamma }^{(\alpha , \gamma )}(r,t) \right| \geqslant \gamma ^{b} \right) \leqslant c\gamma ^{n} \end{aligned}$$

where $\mathbb {P}_{\sigma }^{\gamma }$ is the law of the Glauber dynamics when the process starts at time 0 from $\sigma$;

$$\begin{aligned} m_{\gamma }^{(\alpha ,\gamma )}(\cdot ,t)=(m_{\gamma }(\cdot ,t))^{(\alpha ,\gamma )} \end{aligned}$$

and $m_{\gamma }(r,t)$ is the unique solution of the Cauchy problem (1.6) with initial datum $m_{\gamma }(r,0)=(\Gamma _{\gamma })(r)$.