\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 60, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/60\hfil Almost automorphy] {Almost automorphy of semilinear parabolic evolution equations} \author[M. Baroun, S. Boulite, G. M. N'Gu\'er\'ekata, L. Maniar \hfil EJDE-2008/60\hfilneg] {Mahmoud Baroun, Said Boulite, Gaston M. N'Gu\'er\'ekata, Lahcen Maniar} % in alphabetical order \address{Mahmoud Baroun, Said Boulite and Lahcen Maniar\newline D\'epartement de Math\'ematiques, Facult\'e des Sciences Semlalia, Universit\'e Cadi Ayyad\\ B.P. 2390, 40000 Marrakesh, Morocco} \email{m.baroun@ucam.ac.ma} \email{sboulite@ucam.ac.ma} \email{maniar@ucam.ac.ma} \address{Gaston M. N'Gu\'er\'ekata \newline Department of mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA} \email{gaston.n'guerekata@morgan.edu} \thanks{Submitted December 19, 2007. Published April 22, 2008.} \subjclass[2000]{34G10, 47D06} \keywords{Parabolic evolution equations; almost automorphy; \hfill\break\indent exponential dichotomy; Green's function} \begin{abstract} This paper studies the existence and uniqueness of almost automorphic mild solutions to the semilinear parabolic evolution equation $$ u'(t)=A(t)u(t)+f(t, u(t)), $$ assuming that the linear operators $A(\cdot)$ satisfy the 'Acquistapace--Terreni' conditions, the evolution family generated by $A(\cdot)$ has an exponential dichotomy, and the resolvent $R(\omega,A(\cdot))$, and $f$ are almost automorphic. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \section{Introduction} In this work we investigate the almost automorphy of the solutions to the parabolic evolution equations \begin{gather} u'(t) = A(t)u(t)+g(t),\quad t\in\mathbb{R},\label{ipe}\\ u'(t) =A(t)u(t)+f(t,u(t)),\quad t\in\mathbb{R},\label{ipcp} \end{gather} in a Banach space $X$, where the linear operators $A(t)$ satisfy the `Acquistapace--Terreni' conditions and that the evolution family $U$ generated by $A(\cdot)$ has an exponential dichotomy. The asymptotic behavior of these equations was studied by several authors. The most extensively studied cases are the autonomous case $A(t)=A$ and the periodic case $A(t+T)=A(t)$, see \cite{Am-Ma, ABHN, BHR, GR, GRS, Pr, Vu} for almost periodicity and \cite{Bo-Ma-Ng,diag1,Gold,Hino,Nguerekata2,Ngu2} for almost automorphy. Maniar and Schnaubelt \cite{Man-Sch} studied the general case, where some resolvent $R(\omega,A(\cdot))$ of $A(\cdot)$ is only almost periodic. In this paper, we follow the idea of \cite{Man-Sch} and assume that the function $t\mapsto R(\omega,A(t))\in \mathcal{L}(X)$, for $\omega\ge0$, is almost automorphic. We show first the almost automorphy of the Green's function corresponding to $U$, following the strategy of \cite{Man-Sch} which consists in using Yosida-approximations of $A(\cdot)$. This result will yield the existence of a unique almost automorphic mild solution $u:\mathbb{R}\to X$ of \eqref{ipe} given by \begin{equation}\label{mild} u(t)= \int_\mathbb{R}\Gamma(t,\tau)g(\tau)\,d\tau, \quad t\in\mathbb{R}, \end{equation} for every almost automorphic function $g$. Using an interpolation argument, as in \cite{Ba-Bo-Di-Ma}, we show that the solution $u$ of \eqref{ipe} given by \eqref{mild} is also almost automorphic in every time invariant interpolation space $X_\alpha, \,\, 0\leq \alpha<1$. Finally, by a fixed point technique, if the semilinear term $f:\mathbb{R}\times X_\alpha\to X$ is almost automorphic and globally small Lipschitzian; i.e., the Lipshitz constant is small, we show that there is a unique almost automorphic mild solution on $X_\alpha$ to the semilinear parabolic evolution problem \eqref{ipcp}. This is an extension of \cite[Theorem 3.1]{Nguerekata2}. To illustrate our results, we also study an example of a reaction diffusion equation with time-varying coefficients. If the coefficients and the semilinear term $f$ are almost automorphic, we show that the solutions are almost automorphic. \section{Prerequisites}\label{Pre} A set $U=\{U(t,s): t\ge s, \;t,s\in \mathbb{R}\}$ of bounded linear operators on a Banach space $X$ is called an \emph{evolution family} if \begin{itemize} \item[(E1)] $U(t,s)=U(t,r)U(r,s)$ and $U(s,s)=I$ for $t\ge r\ge s$ and \item[(E2)] $(t,s)\mapsto U(t,s)$ is strongly continuous for $t>s$. \end{itemize} We say that an evolution family $U$ has an \emph{exponential dichotomy} if there are projections $P(t)$, $t\in\mathbb{R}$, being uniformly bounded and strongly continuous in $t$ and constants $\delta>0$ and $N\ge1$ such that \begin{enumerate} \item $U(t,s)P(s) = P(t)U(t,s)$, \item the restriction $U_Q(t,s):Q(s)X\to Q(t)X$ of $U(t,s)$ is invertible (and we set $U_Q(s,t):=U_Q(t,s)^{-1}$), \item $\|U(t,s)P(s)\| \le Ne^{-\delta (t-s)}\;$ and $\;\|U_Q(s,t)Q(t)\|\le Ne^{-\delta (t-s)}$ \end{enumerate} for $t\ge s$ and $t,s\in \mathbb{R}$. Here and below we let $Q(\cdot)=I-P(\cdot)$. Exponential dichotomy is a classical concept in the study of the long--term behaviour of evolution equations; see e.g., \cite{CL,Co,EN,He,LZ,S1,S3}. If $U$ has an exponential dichotomy, then the operator family $$ \Gamma(t,s):=\begin{cases} U(t,s)P(s),& t\ge s,\; t,s\in \mathbb{R},\\ -U_Q(t,s)Q(s),& t0$ and $\gamma\in\mathbb{R}$ such that $\|U(t,s)\|\le Me^{\gamma(t-s)}$ for $t\ge s$. In the present work, we study operators $A(t)$, $t\in \mathbb{R}$, on $X$ subject to the following hypothesis introduced by P.~Acquistapace and B.~Terreni in \cite{AT}. \begin{itemize} \item[(H1)] There is an $\omega\ge0$ such that the operators $A(t)$, $t\in \mathbb{R}$, satisfy $\Sigma_\phi\cup\{0\}\subseteq \rho(A(t)-\omega)$, $ \|R(\lambda,A(t)-\omega)\|\le \frac{K}{1+|\lambda|}$, and $$ \|(A(t)-\omega)R(\lambda,A(t)-\omega)\,[R(\omega,A(t))-R(\omega,A(s))]\| \le L\, |t-s|^\mu|\lambda|^{-\nu} $$ for $t,s\in\mathbb{R}$, $\lambda \in\Sigma_\phi:= \{\lambda\in\mathbb{C}\setminus\{0\}:|\arg \lambda|\le\phi\}$, and constants $\phi\in(\frac{\pi}{2},\pi)$, $L,K\ge0$, and $\mu,\nu\in(0,1]$ with $\mu+\nu>1$. \end{itemize} This assumption implies that there exists a unique evolution family $U$ on $X$ such that $(t,s)\mapsto U(t,s)\in\mathcal{L}(X)$ is continuous for $t>s$, $U(\cdot,s)\in C^1((s,\infty),\mathcal{L}(X))$, $\partial_t U(t,s)=A(t)U(t,s)$, and \begin{align}\label{au} \|A(t)^k U(t,s)\|&\le C\,(t-s)^{-k} \end{align} for $0< t-s\le 1$, $k=0,1$, $0\le \alpha<\mu$, $x\in D((\omega-A(s))^\alpha)$, and a constant $C$ depending only on the constants in (H1). Moreover, $\partial_s^+ U(t,s)x=-U(t,s)A(s)x$ for $t>s$ and $x\in D(A(s))$ with $A(s)x \in \overline{D(A(s))}$. We say that $A(\cdot)$ \emph{generates} $U$. Note that $U$ is exponentially bounded by \eqref{au} with $k=0$. We further suppose that \begin{enumerate} \item[(H2)] the evolution family $U$ generated by $A(\cdot)$ has an exponential dichotomy with constants $N,\delta>0$, dichotomy projections $P(t)$, $t\in\mathbb{R}$, and Green's function $\Gamma$. \end{enumerate} For the sequel, we need the following estimates, see \cite{Ba-Bo-Di-Ma} for the proof. \begin{proposition}\label{pes} For every $ 0\leq \alpha \leq 1$, we have the following assertions: \begin{itemize} \item[(i)] There is a constant $c(\alpha)$, such that \begin{equation}\label{eq1.1} \|U(t,s)P(s)x\|_{\alpha}^t\leq c(\alpha)e^{- \frac{\delta}{2}(t-s)}(t-s)^{-\alpha} \|x\|; \end{equation} \item[(ii)] there is a constant $m(\alpha)$, such that \begin{equation}\label{eq2.1} \|\widetilde{U}_{Q}(s,t)Q(t)x\|_{\alpha}^s\leq m(\alpha)e^{-\delta (t-s)}\|x\| \end{equation} for every $x \in X$ and $t > s$. \end{itemize} \end{proposition} We need to introduce the following definition, and we refer to \cite{Ngu2} for more information. \begin{definition}[S. Bochner] \label{def2.2} \rm (i) A continuous function $f: \mathbb{R}\to X$ is called almost automorphic if for every sequence $(\sigma_n)_{n\in N}$ there exists a subsequence $(s_n)_{n\in N}\subset(\sigma_n)_{n\in {\mathbb N}}$ such that $$ \lim_{n,m\to +\infty}f(t+s_n-s_m)=f(t)\quad\mbox{ for each }t\in \mathbb{R}. $$ This is equivalent to $$ g(t):=\lim_{n\to +\infty}f(t+s_n) \quad \mbox{and}\quad f(t)=\lim_{n\to +\infty}g(t-s_n) $$ are well defined for each $t\in \mathbb{R}$. We note that $f\in AA(\mathbb{R},X)$. (ii) A function $f:\mathbb{R}\times Y\to X$ is said to be almost automorphic if it satisfies the following conditions: $f(\cdot,y)$ is almost automorphic for every $y\in Y$ and $f$ is continuous jointly in $(t,x)$. We note $f\in AA(\mathbb{R}\times Y,X)$. \end{definition} The function $g$ in the definition above is measurable, but not necessarily continuous. It is well-known that $ AA(\mathbb{R},X)$ is a Banach space under the sup-norm $\|f\|_{ AA(\mathbb{R},X)}=\sup_{t\in \mathbb{R}} \|f(t)\|$. \section{Main results} \label{Yos} In this section, we study the existence of almost automorphic solutions to the semilinear evolution equations \begin{equation}\label{see0} u'(t)=A(t)u(t)+f(t,u(t)),\quad t\in \mathbb{R}, \end{equation} where $A(t), t\in \mathbb{R}$, satisfy (H1) and (H2), and the following assumptions hold: \begin{itemize} \item[(H3)] $R(\omega,A(\cdot))\in AA(\mathbb{R},\mathcal{L}(X))$; \item[(H4)] there are $0\leq \alpha<\beta<1$ such $X_\alpha^t=X_\alpha$, $t\in \mathbb{R}$, $X_\beta^t=X_\beta$, $t\in \mathbb{R}$, with uniform equivalent norms; \item[(H5)] the function $f:\mathbb{R}\times X_{\alpha} \to X $ belongs to $AA( \mathbb{R}\times X_{\alpha},X)$ and is globally small Lipschitzian; i.e., there is a small $K_f>0$ such that \begin{equation*} \| f(t,u)-f(t,v)\| \leq K_f \| u-v\|_{\alpha} \quad \text{for all }t\in\mathbb{R}\text{ and }u,v\in X_{\alpha}. \end{equation*} \end{itemize} By a mild solution of \eqref{see0} we understand a continuous function $u:\mathbb{R}\to X_{\alpha} $, which satisfies the following variation of constants formula \begin{equation} u(t)=U(t,s)u(s)+\int_s^tU(t,\sigma )f(\sigma,u(\sigma) )d\sigma\quad \text{for all } t\geq s, \, t,s\in \mathbb{R}. \label{FVC} \end{equation} To achieve the goal of this section, we show some intermediate results. Let us define the Yosida approximations $A_n(t)=nA(t)R(n,A(t))$ of $A(t)$ for $n>\omega$ and $t\in\mathbb{R}$. These operators generate an evolution family $U_n$ on $X$. It has been shown in \cite[Lemma 3.1, Proposition 3.3, Corollary 3.4]{Man-Sch} that assumptions (H1) and (H2) are satisfied by $A_n(\cdot)$ with the same constants for every $n\geq n_0$. In the following lemma, we show that the Yosida approximations $A_n(\cdot)$ satisfy also assumption (H3) for large $n$. The formulas on the resolvent used in the proof are taken from \cite{Man-Sch}. \begin{lemma}\label{lemma1} If {\rm (H1)} and {\rm (H3)} hold, then there is a number $n_1\ge n_0$ such that $R(\omega, A_n(\cdot))\in AA(\mathbb{R},\mathcal{L}(X))$ for $n\ge n_1$. \end{lemma} \begin{proof} Let $(s'_l)_{l\in\mathbb{N}}$ be a sequence of real numbers, as $R(\omega,A(\cdot))$ is almost automorphic, there is a subsequence $(s_l)_{l\in \mathbb{N}}$ such that \begin{equation}\label{limit} \lim_{l,\;k\to +\infty}\|R(\omega,A(t+s_l-s_k))-R(\omega,A(t))\|=0, \end{equation} for each $t\in \mathbb{R}$ If $n\ge n_0$ and $|\arg(\lambda-\omega)|\le \phi$, we have \begin{equation} \label{res2} \begin{aligned} &R(\omega, A_n(t+s_l-s_k))-R(\omega, A_n(t)) \\ &=\frac{n^2}{(\omega+n)^2}\Big(R\Big(\frac{\omega n}{\omega+n},A(t+s_l-s_k)\Big) -R\Big(\frac{\omega n}{\omega+n},A(t)\Big)\Big) \\ &=\frac{n^2}{(\omega+n)^2}R(\omega, A(t+s_l-s_k)) \big[1-\frac{\omega^2}{\omega+n}R(\omega,A(t+s_l-s_k))\big]^{-1} \\ &\quad -\frac{n^2}{(\omega+n)^2} R(\omega, A(t))\big[1-\frac{\omega^2}{\omega+n}R(\omega, A(t))\big]^{-1}. \end{aligned} \end{equation} We can also see that $$ \big\|\frac{\omega^2}{\omega+n}R(\omega,A(s))\big\| \leq \frac{\omega^2}{\omega+n}\frac{K}{1+\omega}\\ \leq \frac{\omega K}{n}\leq \frac{1}{2} $$ for $n\geq n_1:=\max\{n_0, 2\omega K\}$ and $s\in\mathbb{R}$. In particular, \begin{equation}\label{2} \big\|\big[1-\frac{\omega^2}{\omega+n}R(\omega,A(s))\big]^{-1}\big\|\le 2. \end{equation} Hence, \eqref{res2} implies \begin{align*} &\|R(\omega, A_n(t+s_l-s_k))-R(\omega, A_n(t))\|\\ &\leq 2 \|R(\omega,A(t+s_l-s_k))-R(\omega, A(t))\|\\ &\quad +\frac{K}{1+\omega} \big\|\big[1-\frac{\omega^2}{\omega+n} R(\omega,A(t+s_l-s_k))\big]^{-1} -\big[1-\frac{\omega^2}{(\omega+n)^2}R(\omega,A(t))\big]^{-1}\big\|. \end{align*} Employing \eqref{2} again, we obtain \begin{align*} &\big\|\big[1-\frac{\omega^2}{\omega+n}R(\omega,A(t+s_l-s_k))\big]^{-1} - \big[1-\frac{\omega^2}{\omega+n}R(\omega,A(t))\big]^{-1}\big\| \\ &\le 4 \big\|\big[1-\frac{\omega^2}{\omega+n}R(\omega,A(t+s_l-s_k))\big]- \big[1-\frac{\omega^2}{\omega+n}R(\omega,A(t))\big]\big\|\\ &\leq 4\omega\, \|R(\omega,A(t+s_l-s_k))-R(\omega,A(t))\|. \end{align*} Therefore, \begin{equation} \begin{aligned} &\|R(\omega, A_n(t+s_l-s_k))-R(\omega, A_n(t))\| \\ &\leq (2+4K) \|R(\omega, A(t+s_l-s_k))-R(\omega, A(t))\| \end{aligned}\label{an-diff} \end{equation} for $n\ge n_1$ and $t\in \mathbb{R}$. The assertion thus follows from \eqref{limit}. \end{proof} The following technical lemma is also needed. \begin{lemma}\label{ggn} Assume that {\rm (H1)-- (H3)} hold. For every sequence $(s'_l)_{l\in \mathbb{N}} \in\mathbb{R}$, there is a subsequence $ (s_l)_{l\in \mathbb{N}}$ such that for every $\eta>0$, and $t,\; s \in \mathbb{R}$ there is $l(\eta,t,s)>0$ such that \begin{equation}\label{gamman} \|\Gamma_n(t+s_l-s_k,s+s_l-s_k)-\Gamma_n(t,s) \| \le c n^2\eta \end{equation} for a large $n$ and $l,\; k\geq l(\eta,t,s)$. \end{lemma} \begin{proof} Let a sequence $ (s'_l)_{l\in \mathbb{N}} \in\mathbb{R}$. Since $R(\omega, A(\cdot))\in AA(\mathbb{R},X)$, then we can extract a subsequence $(s_l)$ such that \begin{equation}\label{0} \|R(\omega,A(\sigma+s_l-s_k)) -R(\omega, A(\sigma))\|\to 0, \quad k,l\to \infty, \end{equation} for all $\sigma\in \mathbb{R}$. As in \cite{Man-Sch}, we have \begin{align*} %\label{formula-n} &\Gamma_n(t+s_l-s_k,s+ s_l-s_k)-\Gamma_n(t,s)\\ &=\int_{\mathbb{R}}\Gamma_n(t,\sigma)(A_n(\sigma)-\omega) [R(\omega,A_n(\sigma+s_l-s_k))- R(\omega, A_n(\sigma))] \\ &\quad\times (A_n(\sigma+s_l-s_k)-\omega) \Gamma_n(\sigma+s_l-s_k,s+s_l-s_k)\,d\sigma \end{align*} for $s, t\in \mathbb{R}$ and $l,k, \in \mathbb{N}$ and large $n$. This formula, the estimate \eqref{an-diff} and \cite[Corollary 3.4]{Man-Sch} imply \begin{align*} %\label{3} &\|\Gamma_n(t+s_l-s_k,s+s_l-s_k)-\Gamma_n(t,s) \| \\ &\le c n^2 \int_\mathbb{R} e^{-\frac{3\delta}{4}|t-\sigma|} e^{-\frac{3\delta}{4}|\sigma-s|}\|R(\omega,A_n(\sigma+s_l-s_k)) -R(\omega, A_n(\sigma))\|\,d\sigma \\ &\le c n^2(2+4K) \int_\mathbb{R} e^{-\frac{3\delta}{4}|t-\sigma|} e^{-\frac{3\delta}{4}|\sigma-s|}\|R(\omega,A(\sigma+s_l-s_k)) -R(\omega, A(\sigma))\|\,d\sigma \to 0, \end{align*} as $k,l\to \infty$, by \eqref{0} and the Lebesgue's Dominated Convergence Theorem. Hence, for $\eta >0$ there is $l(\eta,t,s)>0$ such that $$ \|\Gamma_n(t+s_l-s_k,s+s_l-s_k)-\Gamma_n(t,s) \|< c n^2\eta $$ for large $n$ and $l,\; k\geq l(\eta,t,s)$. \end{proof} The almost automorphy of the Green function $\Gamma$ is proved in the next proposition. An analogous result for the almost periodicity is shown in \cite{Man-Sch}. \begin{proposition}\label{gamma-ap} Assume that {\rm (H1)-- (H2)} hold. Let a sequence $(s'_l)_{l\in \mathbb{N}}\in\mathbb{R}$ there is a subsequence $(s_l)_{l\in \mathbb{N}}$ such that for every $h>0$ $$ \|\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma(t,s)\|\to 0, \quad k,l\to \infty $$ for $|t-s|\geq h$. \end{proposition} \begin{proof} Let $(s'_l)_{l\in \mathbb{N}}$ be a sequence in $\mathbb{R}$, and consider the subsequence $(s_l)$ given by Lemma \ref{ggn}. Let $\varepsilon>0$ and $h>0$. There is $t_\varepsilon > h$ such that $$ \|\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma(t,s)\| \leq \varepsilon $$ for $|t-s|\geq t_\varepsilon$ and $l,\;k \in \mathbb{N}$. For $ h \leq |t-s|\leq t_\varepsilon$, by \cite[Lemma 4.2]{Man-Sch} we have \begin{gather} \|\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma_n(t+s_l-s_k,s+s_l-s_k)\| \leq c(t_\varepsilon) n^{-\theta},\label{x}\\ \|\Gamma(t,s)-\Gamma_n(t,s)\| \leq c(t_\varepsilon) n^{-\theta}\label{y} \end{gather} for all $k,l$ and large $n$. Let $n_\varepsilon>0$ large enough such that $n^{-\theta}<\frac{\varepsilon}{4c(t_\varepsilon)}$ for $n\geq n_\varepsilon$. Take $0<\eta<\frac{\varepsilon}{2cn_\varepsilon^2}$. Hence, by \eqref{x}, \eqref{y} and Lemma \ref{ggn}, one has $$ \|\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma(t,s)\| \leq 2c(t_\varepsilon) n_\varepsilon^{-\theta}+ cn_\varepsilon^2\eta\leq \varepsilon $$ for all $k,l\geq l(\varepsilon,t,s)$. Consequently, $\|\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma(t,s)\|\to 0$ as $l,\,k \to +\infty$ for $|t-s|> h>0$. \end{proof} Using Proposition \ref{gamma-ap}, we show the existence of a unique almost automorphic solution to the inhomogeneous evolution equation \begin{equation}\label{ipe1} u'(t)=A(t)u(t)+g(t),\quad t\in \mathbb{R}. \end{equation} More precisely, we state the following main result. \begin{theorem} \label{main} Assume {\rm (H1)--(H4)}. Then, for every $g\in AA(\mathbb{R},X)$, the unique bounded mild solution $u(\cdot)=\int_{\mathbb{R}}\Gamma(\cdot,s)g(s)\,ds $ of \eqref{ipe1} belongs to $AA(\mathbb{R},X_{\alpha})$. \end{theorem} \begin{proof} First we prove that the mild solution $ u$ is almost automorphic in $X$. Let a sequence $ (s'_ l)_{l\in \mathbb{N}} $ and $h>0$. As $ g\in AA(\mathbb{R}, X)$ there exists a subsequence $(s_ l)_{l\in \mathbb{N}} $ such that $ \lim_{l,\;k\to +\infty}\|g(t+s_l-s_k)-g(t)\|\to 0$. Now, we write \begin{align*} &u(t+s_l-s_k)-u(t)\\ &= \int_{\mathbb{R}}\Gamma(t+s_l-s_k,s+s_l-s_k)g(s+s_l-s_k)\,ds -\int_{\mathbb{R}}\Gamma(t,s)g(s)\,ds\\ &=\int_{\mathbb{R}}\Gamma(t+s_l-s_k,s+s_l-s_k)(g(s+s_l-s_k)-g(s))\,ds\\ &\quad + \int_{|t-s|\geq h}(\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma(t,s))g(s)\,ds\\ &\quad +\int_{|t-s|\le h}(\Gamma(t+s_l-s_k,s+s_l-s_k)-\Gamma(t,s))g(s)\,ds. \end{align*} For $\varepsilon' >0$, we deduce from Proposition \ref{gamma-ap} and (H2) that $$ \|u(t+s_l-s_k)-u(t)\|\leq 2N\int_{\mathbb{R}}e^{-\delta|t-s|} \|g(s+s_l-s_k)-g(s)\|\,ds\, +(\tfrac{4}{\delta}\,\varepsilon' +4Nh)\|g\|_{\infty} $$ for $t\in\mathbb{R}$ and $l,\;k > l(\varepsilon,\; h)> 0$. Now, for $\varepsilon>0$, take $h$ small and then $\varepsilon' >0$ small such that $$ \|u(t+s_l-s_k)-u(t)\|\leq 2N\int_{\mathbb{R}}e^{-\delta|t-s|} \|g(s+s_l-s_k)-g(s)\|\,ds +\tfrac{\varepsilon}{2} $$ for $t\in\mathbb{R}$ and $l,\;k > l(\varepsilon)> 0$. Finally, by the Lebesgue's Dominated Convergence Theorem, $u$ is almost automorphic in $X$. Using the reiteration theorem, we obtain $X_{\alpha}=(X,X_{\beta})_{\theta}$, with $ \theta=\alpha/\beta$. By the property of interpolation, we have \begin{align*} &\|u(t+s_l-s_k)-u(t)\|_{\alpha}\\ &\leq c(\alpha,\beta)\|u(t+s_l-s_k)-u(t)\|^{\frac{\beta-\alpha}{\beta}} \|u(t+s_l-s_k)-u(t)\|_{\beta}^{\frac{\alpha}{\beta}}. \end{align*} Using estimates in Proposition \ref{pes} we can show that $u$ is bounded in $X_\beta$. Hence, \begin{equation} \label{bounded} \begin{aligned} \|u(t+s_l-s_k)-u(t)\|_{\alpha} &\leq c(\alpha,\beta) c^{\frac{\beta}{\alpha}}\|u(t+s_l-s_k)-u(t) \|^{\frac{\beta-\alpha}{\beta}} \\ &\leq c'\|u(t+s_l-s_k)-u(t)\|^{\frac{\beta-\alpha}{\beta}}. \end{aligned} \end{equation} Since $u$ is almost automorphic in $X$, $u(t+s_l-s_k)\to u(t)$, as $l,k\to \infty$, for $t\in \mathbb{R}$, and thus $x \in AA(\mathbb{R},X_{\alpha})$. \end{proof} As a consequence of Theorem \ref{main} and a fixed point technique, we achieve the aim of the paper. \begin{theorem} \label{thm3.5} Assume that {\rm (H1)--(H5)} hold. Then \eqref{see0} admits a unique mild solution $u$ in $AA(\mathbb{R},X_{\alpha})$. \end{theorem} \begin{proof} Consider $v\in AA(\mathbb{R},X_{\alpha})$ and $f\in AA(\mathbb{R}\times X_{\alpha}, X)$. Then, by \cite[Theorem 2.2.4, p. 21]{Ngu2}, the function $g(\cdot):=f(\cdot ,v(\cdot ))\in AA(\mathbb{R},X)$, and from Theorem \ref{main}, the inhomogeneous evolution equation \[ u'(t)=A(t)u(t)+g(t),\quad t\in \mathbb{R}, \] admits a unique mild solution $u\in AA(\mathbb{R},X)$ given by \[ u(t)=\int_{\mathbb{R} }\Gamma(t,s)f(s,v(s))ds ,\quad t\in \mathbb{R}. \] Let the operator $F:AA(\mathbb{R},X_{\alpha}) \to AA(\mathbb{R},X_{\alpha}) $ be defined by \[ (Fv)(t) :=\int_{\mathbb{R} }\Gamma(t,s)f(s,v(s))ds \quad \text{for all }t\in \mathbb{R}. \] Now we prove that $F$ has a unique fixed point. The estimates \eqref{eq1.1} and \eqref{eq2.1} yield \begin{align*} \| Fx(t)-Fy(t)\|_{\alpha} &\leq c(\alpha)\int_{-\infty }^t e^{-\delta(t-s)}(t-s)^{-\alpha}\|f(s,y(s))-f(s,x(s))\|ds\\ &\quad + c(\alpha)\int_t^{+\infty}e^{-\delta(t-s)}\|f(s,y(s)) -f(s,x(s))\|ds.\\ &\leq K_f c'(\alpha) \|x-y\| _\infty \end{align*} for all $t\in \mathbb{R}$ and $x$, $y\in AA(\mathbb{R},X_{\alpha})$. If we assume that $K_f c'(\alpha)<1$, then $F$ has a unique fixed poind $u\in AA(\mathbb{R},X_{\alpha})$. Thus $u$ is the unique almost automorphic solution to the equation \eqref{see0}. \end{proof} \begin{example} \label{exa3.6} \rm Consider the parabolic problem \begin{equation}\label{pde} \begin{gathered} \partial_t\,u(t,x) =A(t,x,D)u(t,x)+h(t,\nabla u(t,x)),\quad t\in \mathbb{R},\;x\in\Omega, \\ B(x,D) u(t,x) = 0,\quad t\in\mathbb{R},\;x\in\partial\Omega, \end{gathered} \end{equation} on a bounded domain $\Omega\subseteq\mathbb{R}^n$ with boundary $\partial\Omega$ of class $C^2$ and outer unit normal vector $\nu(x)$, employing the differential expressions \begin{gather*} A(t,x,D)=\sum_{k,l} a_{kl}(t,x)\partial_k\partial_l +\sum_k a_{k}(t,x)\,\partial_k+ a_0(t,x), \\ B(x,D)= \sum_k b_k(x)\,\partial_k +b_0(x). \end{gather*} We require that $a_{kl}=a_{lk}$ and $b_k$ are real--valued, $a_{kl},a_k,a_0\in C^\mu_b(\mathbb{R}, C(\overline{\Omega}))$, $b_k,b_0\in C^1(\partial\Omega)$, $$ \sum_{k,l=1}^n a_{kl}(t,x)\,\xi_k\,\xi_l\ge\eta |\xi|^2\,, \quad \text{and}\quad \sum_{k=1}^n b_k(x)\nu_k(x)\ge \beta $$ for constants $\mu\in(1/2,1)$, $\beta,\eta>0$ and all $\xi\in\mathbb{R}^n$, $k,l=1,\cdots,n$, $t\in \mathbb{R}$, $x\in\overline{\Omega}$ resp.\ $x\in \partial\Omega$. ($C_b^\mu$ is the space of bounded, globally H\"older continuous functions.) We set $X=C(\overline{\Omega})$, $$ D(A(t))=\{u\in \bigcap_{p>1}W^2_p(\Omega): A(t,\cdot,D)u\in C(\overline{\Omega}),\; B(\cdot,D)u = 0 \text{ on } \partial\Omega\} $$ for $t\in\mathbb{R}$. It is known that the operators $A(t)$, $t\in \mathbb{R}$, satisfy (H1), see \cite{Ac, Lun}, or \cite[Exa.2.9]{S2}. Thus $A(\cdot)$ generates an evolution family $U(\cdot,\cdot)$ on $X$. Let $\alpha\in(1/2,1)$ and $p>\frac{n}{2(1-\alpha)}$. Then $X_\alpha^t=X_\alpha=\{f\in C^{2\alpha}(\overline{\Omega}):B(\cdot,D)u = 0\}$ with uniformly equivalent constants due to \cite[Theorem~3.1.30]{Lun}, and $X_\alpha\hookrightarrow W^{2}_p(\Omega)$. It is clear that the function $f(t,u)(x):=h(t,\nabla u(x)), \,x\in \Omega$, is continuous from $\mathbb{R}\times X_\alpha$ to $X$, and if $h$ is small Lipschitzian and almost automorphic then $f$ is. 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