\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 79, pp. 1--31.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2018/79\hfil Well-posedness of integro-differential equations] {Well-posedness of degenerate integro-differential equations in function spaces} \author[R. Aparicio, V. Keyantuo \hfil EJDE-2018/79\hfilneg] {Rafael Aparicio, Valentin Keyantuo} \address{Rafael Aparicio \newline University of Puerto Rico, R\'io Piedras Campus, Statistical Institute and Computerized Information Systems, Faculty of Business Administration, 15 AVE Unviversidad STE 1501, San Juan, PR 00925-2535, USA} \email{rafael.aparicio@upr.edu} \address{Valentin Keyantuo \newline University of Puerto Rico, R\'io Piedras Campus, Department of Mathematics, Faculty of Natural Sciences, 17 AVE Universidad STE 1701, San Juan, PR 00925-2537, USA} \email{valentin.keyantuo1@upr.edu} \dedicatory{Communicated by Jerome A. Goldstein} \thanks{Submitted September 1, 2017. Published March 20, 2018.} \subjclass[2010]{45N05, 45D05, 43A15, 47D99} \keywords{Well-posedness; maximal regularity; $R$-boundedness; \hfill\break\indent operator-valued Fourier multiplier; Lebesgue-Bochner spaces; Besov spaces; \hfill\break\indent Triebel-Lizorkin spaces; H\"older spaces} \begin{abstract} We use operator-valued Fourier multipliers to obtain characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We treat periodic vector-valued Lebesgue, Besov and Trieblel-Lizorkin spaces. We observe that in the Besov space context, the results are applicable to the more familiar scale of periodic vector-valued H\"older spaces. The equation under consideration are important in several applied problems in physics and material science, in particular for phenomena where memory effects are important. Several examples are presented to illustrate the results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article, we consider the following problem which consists in a second order degenerate integro-differential equation with infinite delay in a Banach space: \begin{equation}{\label{eq1}} \begin{aligned} & (Mu')'(t)-\Lambda u'(t) -\frac{d}{dt}\int_{-\infty}^t c(t-s)u(s) ds\\ &=\gamma u(t)+ Au(t) +\int_{-\infty}^t b(t-s) Bu(s) ds +f(t),\quad 0\leq t\leq 2\pi, \end{aligned} \end{equation} and periodic boundary conditions $u(0)=u(2\pi)$, $(Mu')(0)=(Mu')(2\pi)$. Here, $A, B, \Lambda$ and $M$ are closed linear operators in a Banach space $X$ satisfying the assumption $D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$, $b, c\in L^1(\mathbb{R}_+)$, $f$ is an $X$-valued function defined on $[0, 2\pi]$, and $\gamma$ is a constant. In case $M=0$, the second boundary condition above becomes irrelevant and we are in the presence of a first order degenerate equation. Equations of the form \eqref{eq1} appear in a variety of applied problems. The case where the memory effect is absent has been studied by many authors. The monograph \cite{FY} by Favini and Yagi is devoted to these problems and contains meaningful applications to concrete problems. Recently applications to inverse problems and in the context of multivalued operators have been investigated (see e.g. \cite{FLT}). The book \cite{MF01} by Melnikova and Filinkov also treats abstract degenerate equations. Evolutionary integro-differential equations arise typically in mathematical physics by constitutive laws pertaining to materials for which memory effects are important, when combined with the usual conservation laws such as balance of energy or balance of momentum. For details concerning the underlying physical principles, we refer to Coleman-Gurtin \cite{CG}, Lunardi \cite{Lu}, Nunziato \cite{Nu}, and the monograph Pr\"uss \cite{Pr} (particularly Chapter II, Section 9) for work on the subject. The latter reference contains a wealth of results on general aspects of evolutionary integral equations and their relevance in concrete models from the physical sciences. Equations of first and second order in time are of interest. Typical examples for $\ b(\cdot)$ and $c(\cdot)$ are the completely monotonic functions $Ke^{-\omega t}t^\mu$ where $K\ge 0$, $\omega>0$ and $\mu>-1$, and linear combinations thereof. Several authors have considered particular cases of the above equation. Earlier papers: Lunardi \cite{Lu}, Da Prato-Lunardi \cite{PL, PL1}, Clement-Da Prato \cite{CPr}, Pr\"uss \cite{Pr1}, Nunziato \cite{Nu}, Alabau-Boussouira-Cannarsa-Sforza \cite{ACS08} and \cite{Sf} for example, use various techniques for the solvability of problems of this type. In the case of Hilbert spaces, the results obtained by these authors are complete. This is due to the fact that Plancherel's theorem is available in Hilbert space. When $X$ is not a Hilbert space, this is no longer the case because of Kwapien's theorem which states that the validity of Plancherel's theorem for $X$-valued functions requires $X$ to be isomorphic to Hilbert space (see for example Arendt-Bu \cite{AB}). Beginning with the papers by Weis \cite{W1,W2}, Arendt-Bu \cite{AB}, Arendt-Batty-Bu \cite{ABHN}, it became possible to completely characterize well-posedness of the problem in periodic vector-valued function spaces. Initially, Arendt and Bu \cite{AB} dealt with the problem $u'(t)=Au(t)+f(t)$, $u(0)=u(2\pi)$. Maximal regularity for the evolution problem in $L^p$ was treated earlier by Weis \cite{W1,W2} (see also \cite{CPr} for a different proof of the operator-valued Mikhlin multiplier theorem using a transference principle). The study in the $L^p$ framework (when $1
0$, it is proved in \cite{AB2} that $B_{pq}^{s}(0,2\pi,X)\subset L^p(0,2\pi,X)$ and the embedding is continuous; moreover, $f\in B_{pq}^{s+1}(0,2\pi,X)$ if and only if $f$ is differentiable a.e on $[0,2\pi]$ and $f'\in B_{pq}^{s}(0,2\pi,X)$. In the case where $p = q = \infty$ and $0 < s < 1$ we have that $B^s_{\infty\infty}(0,2\pi,X)$ corresponds to the space $C^s(0,2\pi, X)$ of H\"older continuous functions with equivalent norm $$ \|f\|_{C^s(0,2\pi;X)}=\sup_{t_1 \neq t_2}\dfrac{\|f(t_2)-f(t_1)\|_X}{|t_2-t_1|^s}+\|f\|_\infty. $$ \subsection*{Triebel-Lizorkin spaces} Let $\phi=(\phi_k)_{k\in\mathbb{N}_0}\in\Phi(\mathbb{R})$ be fixed with $\phi$ and $\Phi(\mathbb{R})$ as above. For $1\leq p<\infty$, $1\leq q\leq\infty$, $s\in\mathbb{R}$, the $X$-valued $2\pi$-periodic Triebel-Lizorkin space with parameters $s$, $p$ and $q$ is denoted by $F_{pq}^s(0,2\pi;X)$ and defined by the set \[ \Big\{f\in\mathcal{D}'(0,2\pi,X): \|f\|_{pq}^s:= \big\|\big(\sum_{j\geq0}2^{sjq}\|\sum_{k\in\mathbb{Z}} e_k\otimes\phi_j(k)\hat{f}(k)\|_X^q\big)^{1/q}\big\|_p<\infty\Big\} \] with the usual modification if $q=\infty$. It is known that set $F_{pq}^s(0,2\pi,X)$ is independent of the choice of $\phi$, and again, different choices of $\phi$ lead to equivalent norms $\|\cdot\|_{pq}^s$. Equipped with the norm $\|\cdot\|_{pq}^s$, $F_{pq}^s(0,2\pi,X)$ is a Banach space. It is also known that if $s_1\leq s_2$, then $F_{pq}^{s_2}(0,2\pi,X)\subset F_{pq}^{s_1}(0,2\pi, X)$ and the embedding is continuous \cite{BK}. When $s>0$, it is show in \cite{BK} that $F_{pq}^{s}(0,2\pi,X)\subset L^p(0,2\pi,X)$ and the embedding is continuous; moreover, $f\in F_{pq}^{s+1}(0,2\pi,X)$ if and only if $f$ is differentiable a.e on $[0,2\pi]$ and $f'\in F_{pq}^{s}(0,2\pi,X)$. The exceptional case $p=\infty$ will not be considered in this paper. We refer to Schmeisser-Triebel \cite[ Section 3.4.2]{ST} for a discussion. Note that $F^s_{pp}((0, 2\pi); X) = B^s_{pp}((0, 2\pi); X)$ by an inspection of the definitions. We give the definition of operator-valued Fourier multipliers in each of the cases that will be of interest to us. First, in the case of Lebesgue spaces, we have: (See \cite{AB,AB2, BK}). \begin{definition} \label{def2.1}\rm Let $X$ and $Y$ be Banach spaces. For $ 1 \leq p \leq \infty, $ we say that a sequence $ ( M_k)_{k \in \mathbb{Z}} \subset \mathcal{L}(X,Y)$ is an $L^p$-multiplier, if for each $ f \in L^p (0,2\pi; X)$ there exists $ u \in L^p (0,2 \pi ; Y) $ such that $$ \hat u (k) = M_k \hat f (k) \mbox{ for all } k \in \mathbb{Z}. $$ \end{definition} In the case of Besov spaces, we have the following concept. \begin{definition} \label{def2.2} \rm Let $X$ and $Y$ be Banach spaces. For $1 \leq p , q\leq \infty$, $ s>0$, we say that a sequence $( M_k )_{k \in \mathbb{Z}}\subset \mathcal{L}(X,Y)$ is an $B^{s}_{pq}$-multiplier, if for each $ f \in B^{s}_{pq}(0,2\pi; X)$ there exists $ u \in B^{s}_{pq}(0,2\pi ; Y) $ such that $$ \hat u (k) = M_k \hat f (k) \mbox{ for all } k \in \mathbb{Z}. $$ \end{definition} Finally, in the case of Triebel-Lizorkin spaces, we have the following concept. \begin{definition} \label{def2.3} \rm Let $X$ and $Y$ be Banach spaces. For $1\leq p<\infty$, $1\leq q\leq \infty$, $s>0$, and let $ ( M_k )_{k \in \mathbb{Z}} \subset \mathcal{L}(X,Y)$, we say that a sequence $( M_k )_{k \in \mathbb{Z}}\subset \mathcal{L}(X,Y)$ is an $F^{s}_{pq}$-multiplier, if for each $ f \in F^{s}_{pq}(0,2\pi; X)$ there exists $ u \in F^{s}_{pq}(0,2\pi ; Y) $ such that $$ \hat u (k) = M_k \hat f (k) \mbox{ for all } k \in \mathbb{Z}. $$ \end{definition} From the uniqueness theorem of Fourier series, it follows that $u$ is uniquely determined by $f$ in each of the above mentioned cases. We denote by $\mathcal{Y}=\mathcal{Y}(X)$ any of the following spaces of $X$-valued functions: $L^p (0,2\pi; X)$, $1 \leq p \leq \infty $; $B^{s}_{pq}(0,2\pi; X)$, $1 \leq p , q\leq \infty$, $ s>0$; $F^{s}_{pq}(0,2\pi; X)$, $1\leq p<\infty$, $1\leq q\leq \infty$, $s>0$. We define the sets \begin{gather*} \mathcal{Y}^{[1]}=\{u \in \mathcal{Y}: u \text{ is almost everywhere differentiable and }u'\in \mathcal{Y}\}, \\ \mathcal{Y}^{[1]}_{\rm per}=\{u \in \mathcal{Y}: \exists v\in\mathcal{Y}, \text{ such that }\hat v(k) = ik \hat u(k)\text{ for all }k \in\mathbb{Z} \} \end{gather*} In the case that $\mathcal{Y}=L^{p}(0,2\pi;X)$, $\mathcal{Y}^{[1]}$ is denoted by $W^{1,p}(0,2\pi;X)$ and $\mathcal{Y}^{[1]}_{\rm per}$ by $W^{1,p}_{\rm per}(0,2\pi;X)$. In the case that $\mathcal{Y}=B_{pq}^{s}(0,2\pi;X)$, $\mathcal{Y}^{[1]}=B_{pq}^{s+1}(0,2\pi;X)$. In the case that $\mathcal{Y}=F_{pq}^s(0,2\pi;X)$, $\mathcal{Y}^{[1]}=F_{pq}^{s+1}(0,2\pi;X)$. \begin{remark} \label{RB} \rm Using integration by parts, the fact that $\mathcal{Y}\subset L^1(0,2\pi, X)$ and the uniqueness theorem of Fourier coefficients, we have \begin{equation}\label{per} \begin{gathered} \mathcal{Y}^{[1]}_{\rm per}=\{u\in\mathcal{Y}^{[1]}:u(0)=u(2\pi)\},\\ \mathcal{Y}^{[1]}_{\rm per}=\{u\in\mathcal{Y}^{[1]}:\widehat{u}'(k) = ik \hat u(k)\text{ for all }k \in\mathbb{Z}\}. \end{gathered} \end{equation} Therefore, if $u\in\mathcal{Y}^{[1]}_{\rm per}$, then $u$ has a unique continuous representative such that $u(0)=u(2\pi)$. We always identify $u$ with this continuous function. \end{remark} \begin{remark} \label{R4} \rm It is clear from the definitions that: \begin{itemize} \item[(a)] if $(M_k)_{k\in\mathbb{Z}}, (N_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ are $\mathcal{Y}$-Fourier multipliers and $\alpha,\beta$ are constants, then $(\alpha M_k+\beta N_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ is a $\mathcal{Y}$-Fourier multiplier as well. \item[(b)] if $(M_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ and $(N_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(Y,Z)$ are $\mathcal{Y}$-Fourier multipliers, then $(N_kM_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Z)$ is a $\mathcal{Y}$-Fourier multiplier as well. In particular, when $X=Y=Z$, if $(M_k)_{k\in\mathbb{Z}}, \, (N_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, then $(N_kM_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier as well. \end{itemize} \end{remark} \begin{proposition}[{\cite[Fejer's Theorem]{AB}}]\label{Fejer's} Let $f\in L^p(0,2\pi;X))$, then one has $$ f=\lim_{n\to\infty}\frac{1}{n+1}\sum_{m=0}^n\sum_{k=-m} ^me_k\hat{f}(k) $$ with convergence in $L^p(0,2\pi;Y))$. \end{proposition} \begin{remark} \label{R5} \rm (a) If $(kM_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier, then $(M_k)_{k\in\mathbb{Z}}$ is also a $\mathcal{Y}$-Fourier multiplier. (b) If $(M_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ is a $\mathcal{Y}$-Fourier multiplier, then there exists a bounded linear operator $T\in\mathcal{L}(\mathcal{Y}(X),\mathcal{Y}(Y))$ satisfying $\widehat{(Tf)}(k)=M_k\hat{f}(k)$ for all $k\in\mathbb{Z}$. This implies in particular that the sequence $(M_k)_{k\in\mathbb{Z}}$ must be bounded. \end{remark} For $ j \in \mathbb{N}$, denote by $r_j$ the $j$-th Rademacher function on $[0,1]$, i.e. $ r_j (t) = sgn(\sin(2^j \pi t))$. For $x \in X$ we denote by $ r_j \otimes x $ the vector valued function $ t \to r_j(t)x$. The important concept of $R$-bounded for a given family of bounded linear operators is defined as follows. \begin{definition} \rm A family $ \mathbf{T} \subset \mathcal{L}(X,Y)$ is called $R$-bounded if there exists $ c_q \geq 0$ such that \begin{equation}{\label{eq4}} \| \sum_{j=1}^n r_j \otimes T_j x_j \|_{L^q(0,1;X)} \leq c_q \| \sum_{j=1}^n r_j \otimes x_j \|_{L^q(0,1;X)} \end{equation} for all $ T_1,\dots, T_n \in \mathbf{T}, x_1,\dots,x_n \in X$ and $ n \in \mathbb{N}, $ where $ 1 \leq q < \infty$. We denote by $R_q(\mathbf{T})$ the smallest constant $c_q$ such that \eqref{eq4} holds. \end{definition} \begin{remark} {\label{R1}} \rm Several useful properties of $R$-bounded families can be found in the monograph of Denk-Hieber-Pr\"uss \cite[Section 3]{DHP}, see also \cite{A, AB, CPSW, PW, KW}. We collect some of them here for later use. \begin{itemize} \item[(a)] Any finite subset of $\mathcal{L}(X)$ is is $R$-bounded. \item[(b)] If $\mathbf{S}\subset \mathbf{T}\subset\mathcal{L}(X)$ and $\mathbf{T}$ is $R$-bounded, then $\mathbf{S}$ is $R$-bounded and $R_p(\mathbf{S})\leq R_p(\mathbf{T})$. \item[(c)] Let $ \mathbf{S}, \mathbf{T} \subset \mathcal{L}(X)$ be $R$-bounded sets. Then $ \mathbf{S} \cdot \mathbf{T} := \{ S \cdot T : S \in \mathbf{S}, T \in \mathbf{T} \}$ is $R$-bounded and $$ R_p (\mathbf{S} \cdot \mathbf{T}) \leq R_p ( \mathbf{S}) \cdot R_p( \mathbf{T}).$$ \item[(d)] Let $ \mathbf{S}, \mathbf{T} \subset \mathcal{L}(X)$ be $R$-bounded sets. Then $ \mathbf{S} + \mathbf{T} := \{ S + T : S \in \mathbf{S}, T \in \mathbf{T} \}$ is $R$- bounded and $$ R_p (\mathbf{S} + \mathbf{T}) \leq R_p ( \mathbf{S}) + R_p( \mathbf{T}).$$ \item[(e)] If $\mathbf{T}\subset\mathcal{L}(X)$ is $R$- bounded, then $\mathbf{T}\cup \{0\}$ is $R$-bounded and $R_p(\mathbf{T}\cup \{0\})= R_p(\mathbf{T})$. \item[(f)] If $\mathbf{S},\mathbf{T}\subset\mathcal{L}(X)$ are $R$- bounded, then $\mathbf{T}\cup \mathbf{S}$ is $R$-bounded and $$ R_p(\mathbf{T}\cup \mathbf{S})\leq R_p(\mathbf{S})+R_p(\mathbf{T}). $$ \item[(g)] Also, each subset $M \subset \mathcal{L}(X) $ of the form $ M = \{\lambda I : \lambda \in \Omega \}$ is $R$-bounded whenever $ \Omega \subset \mathbb{C}$ is bounded ($I$ denotes the identity operator on $X$). \end{itemize} The proofs of (a), (e), (f), and (g) rely on Kahane's contraction principle. We sketch a proof of (f). Since we assume that $\mathbf{S},\mathbf{T}\subset\mathcal{L}(X)$ are $R$-bounded, it follows from (e) (which is a consequence of Kahane's contraction principle) that $\mathbf{S}\cup \{0\}$ and $\mathbf{T}\cup \{0\}$ are $R$-bounded. We now observe that $\mathbf{S}\cup \mathbf{T} \subset \mathbf{S}\cup \{0\} +\mathbf{T}\cup \{0\}$. Then using (d) and (b) we conclude that $\mathbf{S}\cup \mathbf{T}$ is $R$-bounded. We make the following general observation which will be valid throughout the paper, notably in Section 4. Whenever we wish to establish $R$-boundedness of a family of operators $(M_k)_{k\in\mathbb{Z}}$, if at some point we make an exception such as $(k\ne 0)$, $(k\notin \{-1,0\})$ and so on, then later we recover the property for the entire family using items (a), (c) and (f) of the foregoing remark. The corresponding observation for boundedness is clear. \end{remark} \begin{remark} \label{R2} \rm If $X=Y$ is a $UMD$ space and $ M_k = m_k I $ with $ m_k \in \mathbb{C}$, then the Marcinkiewicz condition $\sup_{k}{|m_k|} + \sup_k |k (m_{k+1} - m_k)|< \infty $ implies that the set $ \{ M_k \}_{k \in \mathbb{Z}} $ is an $L^p$-multiplier. (see \cite{AB} or \cite[Theorem 4.4.3]{Am}). \end{remark} Another important notion in Banach space theory is that of Fourier type for a Banach space. Conditions for Fourier multipliers are simplified when the Banach spaces involved satisfy this condition. The Hausdorff-Young inequality states that for $1\le p\le 2$, the Fourier transform maps $L^p(\mathbb{R}):= L^p(\mathbb{R};\mathbb{C})$ continuously into $L^{p'} (\mathbb{R})$ where $ \frac{1}{p}+\frac{1}{p'}=1$, with the convention that $p'=\infty$ when $p=1$. In particular, when $p=2$, Plancherel's theorem holds. When $X$ is a Banach space and one considers $L^p(\mathbb{R};X)$, the situation is no longer the same. It is known that Plancherel's theorem (here we mean $L^2-$continuity of the $X-$valued Fourier transform) holds if and only if $X$ is isomorphic to a Hilbert space (see e.g. \cite{Am,ABHN,AB,GW2}). For every Banach space, the Hausdorff-Young theorem holds with $p=1$. A Banach space is said to have non-trivial Fourier type if the Hausdorff-Young theorem holds true for some $p\in (1, 2]$. By a result of Bourgain \cite{Bo, Bo2}, $UMD$ spaces are examples of spaces with nontrivial Fourier type (see \cite{GW2,AB1}). More generally, $B$-convex spaces, in particular superreflexive Banach spaces have nontrivial Fourier type (\cite[Proposition 3]{Bo2}). However, there exist non reflexive Banach spaces with nontrivial Fourier type. The implications of the property of having non trivial Fourier type are studied in Giradi-Weis \cite{GW2}. For Banach spaces with non trivial Fourier type, in particular for $UMD$ spaces, the conditions for the validity of operator-valued Fourier multiplier theorems are greatly simplified. \section{Characterization in terms of Fourier multipliers} In this section, we characterize the well-posedness of the problem \begin{equation} \label{eP} \begin{gathered} \begin{aligned} &(Mu')'(t)-\Lambda u'(t) -\frac{d}{dt}\int_{-\infty}^t c(t-s)u(s) ds\\ &=\gamma u(t)+ Au(t)+\int_{-\infty}^t b(t-s) Bu(s) ds +f(t),\,\,\, 0\leq t\leq 2\pi, \end{aligned} \\ u(0)=u(2\pi)\quad\text{and}\quad (Mu)'(0)=(Mu)'(2\pi) \end{gathered} \end{equation} in the vector-valued Lebesgue, Besov, and Triebel-Lizorkin spaces. Here $A, B, \Lambda$ and $M$ are closed linear operators in a Banach space $X$ satisfying $D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$, $b, c\in L^1(\mathbb{R}_+)$, $f$ is an $X$-valued function defined on $[0, 2\pi]$, and $\gamma$ is a constant. The results are in terms of operator-valued Fourier multipliers. Let $b,c$ be complex valued functions and $\gamma$ a constant. We define the $M,\Lambda$-resolvent set of $A$ and $B$, $\rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$, associated to \eqref{eP} by \[ \{\lambda\in\mathbb{C} \vert\mathcal{M}(\lambda):D(A)\cap D(B)\to X\text{ is bijective and } [\mathcal{M}(\lambda)]^{-1}\in\mathcal{L}(X)\} \] where $\mathcal{M}(\lambda)=\lambda^2M-A- \tilde{b}(\lambda)B-\lambda \Lambda-\lambda \tilde{c}(\lambda)I-\gamma I$. Thus, $\lambda\in\rho_{\Lambda,M,\tilde{a},\tilde{b},\tilde{c}}(A,B)$ if and only if $[\mathcal{M}(\lambda)]^{-1}$ is a linear continuous isomorphism from $X$ onto $D(A)\cap D(B)$. Here we consider $D(A)$, $D(B)$, $D(\Lambda)$ and $D(M)$ as normed spaces equipped with their respective graph norms. These are Banach space since all the operators are closed. For $a\in L^1(\mathbb{R}_+)$, $u\in \mathcal{Y}$, we denote by $a*u$ the function \begin{equation} (a*u)(t):=\int_{-\infty}^ta(t-s)u(s)ds \end{equation} Since $\mathcal{Y}\subset L^1(0,2\pi;X)$, it follows that $a*u\in L^1(0,2\pi;X)$ and $(a*u)(0)=(a*u)(2\pi)$ by \eqref{eq2.2}. With this notation we may rewrite \eqref{eq1} in the following way: \begin{equation*} (Mu')'(t)-\Lambda u'(t)-\frac{d}{dt}(c*u)(t)\\ =\gamma u(t)+ Au(t)+(b*Bu)(t)+f(t), \quad 0\leq t\leq 2\pi. \end{equation*} If $b$, $c\in L^1(\mathbb{R}_+)$ and $u \in L^1(0,2\pi;D(A))\cap L^1(0,2\pi;D(B))$, then $c*u$, $b*Bu\in L^1(0,2\pi;X)$ by \eqref{eq2.2} and $\widehat{(c*u)}(k)=\tilde{c}(ik)\hat{u}(k)$, $\widehat{(a*Au)}(k)=\tilde{a}(ik)A\hat{u}(k)$ and $\widehat{(b*Bu)}(k)=\tilde{b}(ik)B\hat{u}(k)$ by \eqref{eq3.3}. If additionally we have that $ \frac{d}{dt}(c*u)\in L^1(0,2\pi;X)$, then $c*u\in W^{1,1}(0,2\pi;X)$ and $(c*u)(0)=(c*u)(2\pi)$. Then $\widehat{\frac{d}{dt}(c*u)}(k)=ik\tilde{c}(ik)\hat{u}(k)$ by \eqref{per}. In what follows, we adopt the following notation: \begin{equation} \label{eq2} b_k:=\tilde{b}(ik), c_k:=\tilde{c}(ik) \end{equation} \begin{remark} \label{R3} \rm By the Riemann-Lebesgue lemma, the sequences $(b_k)_{k\in\mathbb{Z}}$ and $(c_k)_{k\in\mathbb{Z}}$ so defined are bounded. In fact $\lim_{|k|\to\infty}b_k=0$, and similarly for $(c_k)_{k\in\mathbb{Z}}$. Moreover, $(b_kI)_{k\in\mathbb{Z}}$ and $(c_k I)_{k\in\mathbb{Z}}$ define a $\mathcal{Y}$-Fourier multiplier. \end{remark} We now give the definition of solutions of \eqref{eP} in our relevant cases. \begin{definition}\label{def3.2} \rm A function $u\in \mathcal{Y}$ is called a {\it strong $\mathcal{Y}$-solution} of \eqref{eP} if $u\in \mathcal{Y}(D(A))\cap \mathcal{Y}(D(B))\cap \mathcal{Y}^{[1]}_{\rm per}$, $u'\in \mathcal{Y}(D(\Lambda))\cap \mathcal{Y}(D(M))$, $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$, and equation \eqref{eq1} holds for almost all $t\in [0,2\pi]$. \end{definition} \begin{lemma}\label{Fourier} Let $X$ be a Banach space, and $A$, $B$, $\Lambda$, $M$ be closed linear operators in $X$ such that $D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a constant, $b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2}. Assume that $u$ is a strong $\mathcal{Y}$-solution of \eqref{eP}. Then $$ [-k^2M-A -b_kB-ik\Lambda-ikc_kI-\gamma I]\hat{u}(k)=\hat{f}(k). $$ for all $k\in\mathbb{Z}$. \end{lemma} \begin{proof} Let $k\in\mathbb{Z}$. Since $u$ is a strong $\mathcal{Y}$-solution of \eqref{eP}, $u\in \mathcal{Y}(D(A))\cap \mathcal{Y}(D(B))\cap \mathcal{Y}^{[1]}_{\rm per}$, $ u'\in \mathcal{Y}(D(\Lambda))\cap \mathcal{Y}(D(M))$, $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$ and \begin{align*} &(Mu')'(t)-\Lambda u'(t)-\frac{d}{dt}(c*u)(t)\\ &=\gamma u(t)+Au(t)+ (b*Bu)(t)+f(t), \quad \text{for a.e } t\in[0, 2\pi]. \end{align*} Since $u\in \mathcal{Y}(D(A))\cap \mathcal{Y}(D(B))$, we have $$ \hat{u}(k)\in D(A)\cap D(B)\quad\text{and}\quad \widehat{Au}(k)=A\hat{u}(k),\hat{Bu}(k)=B\hat{u}(k). $$ by \cite[Lemma 3.1]{AB}. Since $u\in\mathcal{Y}^{[1]}_{\rm per}$, we have $\widehat{u}'(k)=ik\hat{u}(k)$ by \eqref{per}. Since $ u'\in \mathcal{Y}(D(\Lambda))\cap \mathcal{Y}(D(M))$, it follows that $\widehat{(\Lambda u')}=\Lambda \widehat{u}'(k)=ik\Lambda\hat{u}(k)$, $\widehat{M u'}=M\widehat{u}'(k)=ikM\hat{u}(k)$ by \cite[Lemma 3.1]{AB}. Since $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$, it follows that $\widehat{(Mu')'}=ik\widehat{M u'}(k)=-k^2M\hat{u}(k)$ by \eqref{per}. Since $u\in\mathcal{Y}(D(A))\subset L^1(0,2\pi;D(A))$, $u\in\mathcal{Y}(D(B))\subset L^1(0,2\pi;D(B))$ and $b$, $c\in L^1(\mathbb{R}_+)$, it follows that $c*u$, $b*Bu\in L^1(0,2\pi;X)$, $(c*u)(0)=(c*u)(2\pi)$ by \eqref{eq2.2} and $\widehat{(c*u)}(k)=\tilde{c}(ik)\hat{u}(k)$, $\widehat{(b*Bu)}(k)=\tilde{b}(ik)B\hat{u}(k)$ by \eqref{eq3.3}. Since $\mathcal{Y}\subset L^1(0,2\pi;X)$, we have $u$, $\Lambda u'$, $(Mu')'$ and $f\in L^1(0,2\pi;X)$. So $u$, $Au$, $Bu$, $b*Bu$, $\Lambda u'$, $(Mu')'$ and $f$ all belong to $L^1(0,2\pi;X)$. Then $\frac{d}{dt}(c*u)$ must be in $L^1(0,2\pi;X)$. Therefore $c*u\in W^{1,1}_{\rm per}(0,2\pi;X)$ and $\widehat{\frac{d}{dt}(c*u)}(k)=ik\tilde{c}(ik)\hat{u}(k)$ by \eqref{per}. Taking Fourier series on both sides of \eqref{eq1} we obtain $$ [-k^2M-A -b_kB-ik\Lambda-ikc_kI-\gamma I]\hat{u}(k)=\hat{f}(k), \quad k\in\mathbb{Z}. $$ \end{proof} When \eqref{eP} is $\mathcal{Y}$ well-posed, the map $\mathcal{S}:\mathcal{Y}\to\mathcal{Y}$, $f\mapsto u$ where $u$ is the unique strong solution, is linear. We adopt the following definition of well-posedness. \begin{definition}\label{DefWell} \rm We say that \eqref{eP} is $\mathcal{Y}$-well-posed, if for each $f\in \mathcal{Y}$, there exists a unique strong $\mathcal{Y}$-solution $u$ of \eqref{eP} which depends continuously on $f$ in the sense that the operator $\mathcal{S}:\mathcal{Y}\to \mathcal{Y}$ defined by $\mathcal{S}(f)=u$ where $u$ is the unique strong $\mathcal{Y}$-solution of \eqref{eP} is continuous. \end{definition} \begin{remark} \label{WD} \rm We note that, according to Section 2, \cite{AB,AB2,BK}, all the spaces of vector-valued functions $\mathcal{Y}$ concerned in this paper are continuously embedded in $L^1(0,2\pi,X)$. It follows that: If $f_n \to f$ in $\mathcal{Y}$, then $f_n\to f$ in $L^1(0,2\pi,X)$ and consequently for each $k\in\mathbb{Z}$, $\lim_{n\to\infty}\hat{f_n}(k)=f(k)$ in $X$. \end{remark} Our definition imposes an additional condition to that given in the previous works such as \cite{Bu2}, \cite{LP} that allows us to establish the following characterization of well-posed of \eqref{eP} in terms of Fourier multipliers. Actually, the above definition stems from the Hadamard concept of well-posedness in partial differential equations. We refer for example to \cite{FAT} and \cite{ABHN} for the presentation of this fundamental concept. \begin{theorem}\label{t2} Let $X$ be a Banach space and $A$, $B$, $\Lambda$, $M$ be closed linear operators in $X$ such that $ D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a constant, $b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2}. Then the following assertions are equivalent. \begin{itemize} \item[(i)] \eqref{eP} is $\mathcal{Y}$-well-posed. \item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$ and $(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$, $(kN_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, where $$ N_k=[k^2M+A +b_kB+ik\Lambda+ikc_kI+\gamma I]^{-1} $$ \end{itemize} In this case the following maximal regularity property holds: The unique strong $\mathcal{Y}$-solution $u$ is such that $Au$, $b*Bu$, $\Lambda u$, $\Lambda u'$, $c*u$, $\frac{d}{dt}(c*u)$, $Mu$, $Mu'$ and $(Mu')'$ all belong to $\mathcal{Y}$ and there exists a constant $C>0$ independent of $f\in \mathcal{Y}$ such that \begin{align*} &\|u\|_\mathcal{Y} +\|Au\|_\mathcal{Y}+\|b*Bu\|_\mathcal{Y} +\|\Lambda u\|_\mathcal{Y}+\|\Lambda u'\|_\mathcal{Y}+\|c*u\|_\mathcal{Y}\\ &+\|\frac{d}{dt} (c*u)\|_\mathcal{Y}+\|Mu\|_\mathcal{Y} +\|Mu'\|_\mathcal{Y}+\|(Mu')'\|_\mathcal{Y}\leq C\|f\|_\mathcal{Y} \end{align*} \end{theorem} \begin{proof} (i) $\Rightarrow$ (ii). Let $k\in\mathbb{Z}$ and $y\in X$. Define $f(t)=e^{ikt}y$. Then $\hat{f}(k)=y$. By assumption, there exists a unique strong $\mathcal{Y}$-solution $u$ of \eqref{eP}. By Lemma \ref{Fourier}, we have that for all $k\in\mathbb{Z}$, $$ [-k^2M-A -b_kB-i k\Lambda-ikc_kI-\gamma I]\hat{u}(k)=y $$ It follows that $$[-k^2M-A-b_kB-i k\Lambda-ikc_kI-\gamma I]$$ is surjective for each $k\in\mathbb{Z}$. Next we prove that for each $k\in\mathbb{Z}$, $$ [-k^2M-A -b_kB-ik\Lambda-ikc_kI-\gamma I] $$ is injective. Let $x\in D(A)\cap D(B)$ such that \begin{equation}\label{injec} [-k^2M-A -b_kB-ik\Lambda-ikc_kI-\gamma I]x=0 \end{equation} Define $u(t)=e^{ikt}x$ when $t\in[0,2\pi]$. Then $\hat{u}(k)=x$ and $\hat{u}(n)=0$ for all $n\in\mathbb{Z}$, $n\neq k$. By \eqref{injec} we have \begin{align*} \widehat{(Mu')'}(n)-\widehat{\Lambda u'}(n)-\widehat{\frac{d}{dt}(c*u)}(n)&=\gamma \hat{u}(n)+ \widehat{Au}(n)+\widehat{(b*Bu)}(n), \end{align*} for all $n\in\mathbb{Z}$. From uniqueness theorem of Fourier coefficients, we conclude that $u$ satisfies \[ (Mu')'(t)-\Lambda u'(t)-\frac{d}{dt}(c*u)(t) =\gamma u(t)+ Aw(t)+ (b*Bu)(t) \] for almost all $t\in[0,2\pi]$. Thus $u$ is a strong $\mathcal{Y}$-solution of \eqref{eP} with $f=0$. We obtain $x=0$ by the uniqueness assumption. We have shown that $$ [-k^2M-A -b_kB-ik\Lambda-ikc_kI-\gamma I] $$ is injective for each $k\in\mathbb{Z}$. Now we show that $$ N_k=[k^2M+A +b_kB+ik\Lambda+ikc_kI+\gamma I]^{-1}\in\mathcal{L}(X) $$ Let $k\in\mathbb{Z}$ and $(x_n)_{n\in\mathbb{N}}$ be a sequence in $X$ such that $x_n\to x$. For each $n\in\mathbb{N}$ we define $f_n(t)=e^{ikt}x_n$ and $f(t)=e^{ikt}x$. Then $f_n,f\in \mathcal{Y}$, for every $n\in\mathbb{N}$ and $f_n\to f$ in $\mathcal{Y}$. Since \eqref{eP} is $\mathcal{Y}$-well-posed, for each $f_n, f\in\mathcal{Y}$ there exists a unique strong $\mathcal{Y}$-solution $\mathcal{S}(f_n)=u_n$, $\mathcal{S}(f)=u$. Since $f_n\to f$ in $\mathcal{Y}$, we have $u_n\to u$ in $\mathcal{Y}$ by continuity of $\mathcal{S}$. Therefore $\hat{u}_n(k)\to \hat{u}(k)$ by Remark \ref{WD}. Since $$ -k^2M-A -b_kB-ik\Lambda-ikc_kI-\gamma I $$ is bijective, we obtain $\hat{u}_n(k)=-N_kx_n, \hat{u}(k)=-N_kx$ by Lemma \ref{Fourier}; then $N_kx_n\to N_kx$. Thus by the Closed Graph Theorem, $N_k\in\mathcal{L}(X)$. Thus $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$. We now set for each $k\in\mathbb{Z}$: \begin{gather*} M_k=k^2MN_k\quad B_k=AN_k\\ S_k=BN_k\quad H_k=kN_k. \end{gather*} Next we show that $(M_k)_{k\in\mathbb{Z}}$, $(B_k)_{k\in\mathbb{Z}}$, $(S_k)_{k\in\mathbb{Z}}$, and $(H_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers. Since $N_k\in\mathcal{L}(X)$, $B$, $\Lambda$, $M$ are closed, $M_k$, $B_k$, $H_k$ and $S_k$ are bounded for all $k\in\mathbb{Z}$. Now let $f\in\mathcal{Y}$, then there exists a strong $\mathcal{Y}$-solution $u$ of \eqref{eP}. Then $\hat{u}(k)=-N_k\hat{f}(k)$ for all $k\in\mathbb{Z}$ by Lemma \ref{Fourier}. Therefore $$ \hat{u}(k)\in D(A)\cap D(B)\subset D(\Lambda)\cap D(M), $$ for all $k\in\mathbb{Z}$. Since $B$ is closed, \begin{equation*} \widehat{Bu}(k)=B\hat{u}(k)=-BN_k\hat{f}(k)=-B_k\hat{f}(k) \end{equation*} for all $k\in\mathbb{Z}$ by \cite[Lemma 3.1]{AB}. Since $\Lambda$, $M$ are closed, $u\in\mathcal{Y}^{[1]}_{\rm per}$, $u'\in \mathcal{Y}(D(\Lambda))\cap\mathcal{Y}(D(M))$, and $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$, we have \begin{gather*} \widehat{u}'(k)=ik\hat{u}(k)=-ikN_k\hat{f}(k)=-iH_k\hat{f}(k), \\ \widehat{\Lambda u'}(k)=\Lambda \widehat{u}'(k) =ik\Lambda\hat{u}(k)=-ik\Lambda N_k\hat{f}(k)=-iS_k\hat{f}(k), \\ \widehat{(Mu')'}(k)=ik\widehat{Mu'}(k)=ikM\widehat{u}'(k)=-k^2M\hat{u}(k) =k^2MN_k\hat{f}(k)=M_k\hat{f}(k) \end{gather*} for all $k\in\mathbb{Z}$ by \eqref{per} and \cite[Lemma 3.1]{AB}. It follows that $(M_k)_{k\in\mathbb{Z}}$, $(B_k)_{k\in\mathbb{Z}}$, $(S_k)_{k\in\mathbb{Z}}$, and $(H_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers. Therefore the implication (i) $\Rightarrow$ (ii) is true. \smallskip (ii) $\Rightarrow$ (i). Since $$ k^2MN_k+AN_k+b_kBN_k+ik\Lambda N_k+ikc_kN_k+\gamma N_k =I, $$ we have $$ AN_k=I-\left(k^2MN_k+AN_k +b_kBN_k+ikc_kN_k+\gamma N_k\right) $$ for each $k\in\mathbb{Z}$. Therefore, $(AN_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier by Remarks \ref{R4}, \ref{R5}, and \ref{R3}. Since $(k^2MN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$, $(kN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$, and $(AN_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, it follows that $(N_k)_{k\in\mathbb{Z}}$, $(ikc_kN_k)_{k\in\mathbb{Z}}$, $(c_kN_k)_{k\in\mathbb{Z}}$, $(ikN_k)_{k\in\mathbb{Z}}$, $(ik\Lambda N_k)_{k\in\mathbb{Z}}$, $(\Lambda N_k)_{k\in\mathbb{Z}}$, $(-k^2MN_k)_{k\in\mathbb{Z}}$ $(ikMN_k)_{k\in\mathbb{Z}}$, and $(MN_k)_{k\in\mathbb{Z}}$ are also $\mathcal{Y}$-Fourier multipliers again by Remarks \ref{R4}, \ref{R5}, and \ref{R3}. From the fact that $(AN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$, $(\Lambda N_k)_{k\in\mathbb{Z}}$, $(MN_k)_{k\in\mathbb{Z}}$, and $(c_kN_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, then for all $f\in \mathcal{Y}$, we conclude that exist $u$, $v_1$, $v_2$, $v_3$, $v_4$, and $v_5\in \mathcal{Y}$ such that \begin{equation}\label{D} \hat{u}(k)=N_k\hat{f}(k), \end{equation} and \begin{equation}\label{Y} \begin{gathered} \hat{v}_1(k)= AN_k\hat{f}(k)=A\hat{u}(k)=\widehat{Au}(k),\\ \hat{v}_2(k)= BN_k\hat{f}(k)=B\hat{u}(k)=\widehat{Bu}(k),\\ \hat{v}_3(k)= \Lambda N_k\hat{f}(k)=\Lambda\hat{u}(k) =\widehat{\Lambda u}(k),\\ \hat{v}_4(k)= MN_k\hat{f}(k)=M\hat{u}(k)=\widehat{Mu}(k),\\ \hat{v}_5(k)= c_kN_k\hat{f}(k)=c_k\hat{u}(k)=\widehat{c*u}(k), \end{gathered} \end{equation} for all $k\in\mathbb{Z}$ by the closedness of $A$, $B$, $\Lambda$, $M$, and \eqref{eq3.3}. Since $i\mathbb{Z}\subset\rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$, it follows that $$ \hat{u}(k)\in D(A)\cap D(B)\subset D(\Lambda)\cap D(M), $$ for all $k\in\mathbb{Z}$ by \eqref{D}. Since $A$, $B$, $\Lambda$, and $M$ are closed, $$ u(t)\in D(A)\cap D(B) $$ and $Au(t)=v_1(t)$, $Bu(t)=v_2(t)$, $\Lambda u(t)=v_3(t)$, $Mu(t)=v_4(t)$ and $(c*u)(t)=v_5(t)$ a.e.\ $t\in[0,2\pi]$ by \eqref{Y} and \cite[Lemma 3.1]{AB} (here we also use the fact that $\mathcal{Y}\subset L^p(0,2\pi,X)$). Therefore $$ u\in \mathcal{Y}(D(A))\cap\mathcal{Y}(D(B)), $$ and $c*u$, $\Lambda u$, $Mu\in\mathcal{Y}$. Since $(ikN_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier, there exists $v_6\in\mathcal{Y}$ such that \begin{equation}\label{Y3} \hat{v}_6(k)=ikN_k\hat{f}(k)=ik\hat{u}(k)\in D(\Lambda)\cap D(M). \end{equation} for all $k\in\mathbb{Z}$. Therefore by \eqref{per} and \eqref{Y3}, $u\in\mathcal{Y}^{[1]}_{\rm per}$, $\widehat{u}'(k)=ik\hat{u}(k)$ and $$ \widehat{u}'(k)\in D(\Lambda)\cap D(M), $$ for all $k\in\mathbb{Z}$. Since $(ik\Lambda N_k)_{k\in\mathbb{Z}}$ and $(ikMN_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, there exist $v_7$, $v_9\in\mathcal{Y}$ such that \begin{equation}\label{Y1} \begin{gathered} \hat{v_7}(k)=ik\Lambda N_k\hat{f}(k)=\Lambda(ik\hat{u}(k)) =\Lambda\widehat{u}'(k)=\widehat{\Lambda u'}(k),\\ \hat{v}_8(k)=ikMN_k\hat{f}(k)=M(ik\hat{u}(k)) =M\widehat{u}'(k)=\widehat{Mu'}(k), \end{gathered} \end{equation} for al $k\in\mathbb{Z}$. Since $\Lambda$ and $M$ are closed, $$ u'(t)\in D(\Lambda)\cap D(M) $$ and $\Lambda u'(t)=v_7(t)$, $Mu'(t)=v_8(t)$ a.e.\ $t\in[0,2\pi]$ by \eqref{Y1} and \cite[Lemma 3.1]{AB} (here again, we also use the fact that $\mathcal{Y}\subset L^p(0,2\pi,X)$). Therefore $$ u'\in\mathcal{Y}( D(\Lambda))\cap \mathcal{Y}(D(M)). $$ Since $(-k^2MN_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier, there exists $v_9\in\mathcal{Y}$ such that \begin{equation}\label{Y2} \hat{v}_9(k)=-k^2kMN_k\hat{f}(k)=ik(ikM\hat{u}(k)) =ikM\widehat{u}'(k)=ik\widehat{Mu}'(k), \end{equation} for al $k\in\mathbb{Z}$ by \eqref{Y1}. Then $Mu'\in\mathcal{Y}^{[1]}_{\rm per}$. Since $(ikc_kN_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier, there exists $v_{10}\in\mathcal{Y}$ such that \begin{equation} \hat{v}_{10}(k)=ikc_kN_k\hat{f}(k)=ikc_k\hat{u}(k)=ik\widehat{(c*u)}(k), \end{equation} for al $k\in\mathbb{Z}$ by \eqref{Y}. Then $c*u\in\mathcal{Y}^{[1]}_{\rm per}$ by \eqref{per}. Since $\hat{u}(k)=N_k\hat{f}(k)$, we have $$ [-k^2M-A -b_kB-i k\Lambda-ikc_kI-\gamma I](-\hat{u}(k))=\hat{f}(k), $$ this means that \[ (\widehat{M}w')'(k)-\widehat{\Lambda w'}(k)-\widehat{\frac{d}{dt}(c*w)}(k) =\gamma \hat{w}(k)+\widehat{Aw}(k)+\widehat{(b*Bw)}(k)+\hat{f}(k), \] for all $k\in\mathbb{Z}$ where $w=-u$. From the uniqueness theorem of Fourier coefficients, we conclude that $w$ satisfies \[ (Mw')'(t)-\Lambda w'(t)-\frac{d}{dt}(c*w)(t) =\gamma w(t)+ Aw(t)+(b*Bw)(t) +f(t) \] for almost all $t\in[0,2\pi]$. Thus $w$ is a strong $\mathcal{Y}$-solution of \eqref{eP}. To prove uniqueness, let $u$ be a strong $\mathcal{Y}$-solution of \eqref{eP} with $f=0$. Then $$ [-k^2M-A -b_kB-i k\Lambda-ikc_kI-\gamma I]\hat{u}(k)=0 $$ for all $k\in\mathbb{Z}$ by Lemma \ref{Fourier}. Since $ik\in \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$ for all $k\in\mathbb{Z}$, it follows that $\hat{u}(k)=0$ for all $k\in\mathbb{Z}$. From the uniqueness theorem of Fourier coefficients we have that $u=0$. Now we show the continuous dependence of $u$ on $f$. Let $f\in \mathcal{Y}$, then the unique strong $\mathcal{Y}$-solution of \eqref{eP}, $u$, is such that $\hat{u}(k)=-N_k\hat{f}(k)$ for all $k\in\mathbb{Z}$ by Lemma \ref{Fourier} and $i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$. Since $N_k$ is a $\mathcal{Y}$-Fourier multiplier, there exists a bounded linear operator $T\in\mathcal{L}(\mathcal{Y},\mathcal{Y})$ such that $\widehat{Tf}(k)=\hat{u}(k)$ for all $k\in\mathbb{Z}$ by Remark \ref{R5}. Then $Tf=u$, so $u$ depends continuously on $f$. The last assertion of the theorem is a direct consequence of the fact that $Au$, $b*Bu$, $\Lambda u$, $\Lambda u'$, $c*u$, $\frac{d}{dt}(c*u)$, $Mu$, $Mu'$ and $(Mu')'\in \mathcal{Y}$ are defined through the following operator valued Fourier multipliers $(-AN_k)_{k\in\mathbb{Z}}$, $(-b_kBN_k)_{k\in\mathbb{Z}}$, $(-\Lambda N_k)_{k\in\mathbb{Z}}$, $(-k\Lambda N_k)_{k\in\mathbb{Z}}$, $(-c_kN_k)_{k\in\mathbb{Z}}$, $(-kc_kN_k)_{k\in\mathbb{Z}}$, $(-MN_k)_{k\in\mathbb{Z}}$, $(kMN_k)_{k\in\mathbb{Z}}$, $(k^2MN_k)_{k\in\mathbb{Z}}$ (here we use the Remarks \ref{R4}, \ref{R5}, and \ref{R3}). \end{proof} The last assertion of the previous theorem is known as the {\it maximal regularity} property for \eqref{eP}. \begin{remark}\label{rmk3.7} \rm We can construct the solution $ u(\cdot)$ given by the above theorems using Proposition \ref{Fejer's} and the fact that $\mathcal{Y}$ is continuously embedded in $L^p(0,2\pi; X) $. More precisely, \begin{equation} u(\cdot) = -\lim_{n\to\infty} \frac{1}{n+1} \sum_{m=0}^n \sum_{k= -m}^m e_k (\cdot) N_k \hat f(k), \end{equation} with convergence in $L^p(0,2\pi;X)$. \end{remark} \begin{remark}\label{Auto} \rm If at most one operator of those that appear in \eqref{eq1} is unbounded, then the additional condition in our definition of well-posedness is obtained automatically. In that case the operators $$ -k^2M-A-b_kB-ik\Lambda-ikc_kI-\gamma I $$ are closed for all $k\in\mathbb{Z}$ and once we show that they are bijective, continuity follows from the Closed Graph Theorem. \end{remark} \section{Concrete characterization on periodic Lebesgue, Besov and Triebel-Lizorkin spaces} In this section, we give concrete conditions that allow us to apply Theorem \ref{t2}. Specifically we obtain conditions under which the sequences $(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$, and $(kN_k)_{k\in\mathbb{Z}}$ are Fourier multipliers in the scale of spaces under consideration by use of the operator valued multiplier theorems established in \cite{AB1,AB,AB2,BK}. Versions of the multiplier theorems on the real line can be found in \cite{Am1,GW1,GW2} (the reference \cite{GW2} contains concrete criteria for $R$-boundedness of operator families), \cite{W1,W2}. The $L^p$-case is much different from the other scales of spaces in that it involves the notion of $R$-boundedness and one has to restrict consideration to $UMD$ Banach spaces. Fortunately, many Banach spaces, for example $L^p(\Omega,\mu)$, $1
0$, $\nu>-1$
and $C$ is a constant. We give a class of functions which discriminate between
the above conditions in the following example.
\begin{example} \label{examp4.1} \rm
Let $\beta>0$, $\omega>0$, $c\in\mathbb{R}$ and consider the family of
functions
$$
b(t)=\begin{cases}
0 &\text{if }0 0$ and $1\le p, \, q\le \infty$, the following statements are equivalent.
\begin{itemize}
\item[(i)] \eqref{eP} is $B^s_{pq}$-well-posed.
\item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}\}$,
and $\{kN_k:k\in\mathbb{Z}\}$ are bounded, where
$$N_k=[k^2M+A
+b_kB+i k\Lambda+ikc_kI+\gamma I]^{-1}$$
\end{itemize}
\end{theorem}
\begin{proof} (i) $\Rightarrow$ (ii). Assume that
\eqref{eP} is $B^s_{pq}$-well-posed. Then by Theorem \ref{t2},
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$ and
$(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$
and $(kN_k)_{k\in\mathbb{Z}}$ are $B^s_{pq}$-Fourier multipliers. The
boundedness of $\{k^2MN_k:k\in\mathbb{Z}\}$,
$\{BN_k:k\in\mathbb{Z}\}$, $\{kN_k:k\in\mathbb{Z}\}$,
and $\{kN_k:k\in\mathbb{Z}\}$ now
follows from Remark \ref{R5}.
\smallskip
(ii) $\Rightarrow$ (i). By Theorem \ref{t2}, it suffices
to show that the families $(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
and $(kN_k)_{k\in\mathbb{Z}}$ are $B_{pq}^s$-Fourier multipliers. Let
$M_k=k^2MN_k$, $B_k=BN_k$, $H_k=kN_k$,
and $S_k=k\Lambda N_k$. Since (H2) implies (H1), the verification of the
Marcinkiewicz condition of order one is similar
to what was done in the proof of Theorem \ref{tfmp}.
It remains to prove that
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 M_k\|<\infty$,
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 B_k\|<\infty$,
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 S_k\|<\infty$, and
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 H_k\|<\infty$.
We recall from the proof of Theorem \ref{tfmp} that the family
$(T_k)_{k\in\mathbb{Z}}$ defined through
$$
T_k=\frac{2k+1}{k^2}M_k+\Delta b_kB_k+i\Delta c_kH_k+i\frac
{c_{k+1 }}{k}H_k+i\frac{1}{k}S_k, \, k\neq0
$$
is such that
$N_{k+1}^{-1}N_k=I+T_k$, $Q_k=-kT_k$, $k\Delta
N_k=N_{k+1}Q_k$ for all
$k\in\mathbb{Z}$, $k\neq0$, and $\{kT_k:k\in\mathbb{Z}\}$ is bounded.
We observe that
\begin{align*}
\Delta T_k&=\Delta(\frac{2k+1}{k^2}M_k)+\Delta(\Delta
b_k)B_k+i\Delta(\Delta(c_k)H_k)\\
&\quad +i\Delta(\frac {c_{k+1}}{k}H_k)+i\Delta(\frac{1}{k}S_k))
\end{align*}
However,
\begin{align*}
\Delta(\frac{2k+1}{k^2}M_k)
&=\frac{2k+3}{(k+1)^2}M_{k+1}-\frac{2k+1}{k^2}M_k\\
&=\frac{2k+3}{(k+1)^2}M_{k+1}-\frac{2k+3}{(k+1)^2}M_{k}+\frac{2k+3}{(k+1)^2}M_{
k}-\frac{2k+1}{k^2}M_k\\
&=\frac{2k+3}{(k+1)^2}\Delta M_k-\frac{2k^2+4k+1}{k^2(k+1)^2}M_k\\
&=\frac{2k+3}{k(k+1)^2}(k\Delta M_k)-\frac{2k^2+4k+1}{k^2(k+1)^2}M_k,
\end{align*}
\begin{align*}
\Delta(\frac{1}{k}S_k)&=\frac{1}{k+1}S_{k+1}-\frac{1}{k}S_k\\
&=\frac{1}{k+1}S_{k+1}-\frac{1}{k+1}S_{k}+\frac{1}{k+1}S_{k}-\frac{1}{k}S_k\\
&=\frac{1}{k+1}\Delta S_k-\frac{1}{k(k+1)}S_k\\
&=\frac{1}{k(k+1)}(k\Delta S_k)-\frac{1}{k(k+1)}S_k,
\end{align*}
\begin{align*}
\Delta(\frac {c_{k+1}}{k} H_k)
&=\frac {c_{k+2}}{k+1}H_{k+1}-\frac {c_{k+1}}{k}H_k\\
&=\frac {c_{k+2}}{k+1}H_{k+1}-\frac {c_{k+2}}{k+1}H_{k}+\frac
{c_{k+2}}{k+1}H_{k}-\frac {c_{k+2}}{k}H_{k}+\frac {c_{k+2}}{k}H_{k}-\frac
{c_{k+1 }}{k}H_k\\
&=\frac {c_{k+2}}{k+1}\Delta H_k+\frac{\Delta
c_{k+1}}{k}H_k-\frac{c_{k+2}}{k(k+1)}H_k\\
&=\frac {c_{k+2}}{k(k+1)}(k\Delta H_k)+\frac{(k+1)\Delta
c_{k+1}}{k(k+1)}H_k-\frac{c_{k+2}}{k(k+1)}H_k,
\end{align*}
\begin{align*}
\Delta[k(\Delta b_k)B_k]
&=(\Delta b_{k+1})B_{k+1}-(\Delta b_k)B_k\\
&=(\Delta b_{k+1})B_{k+1}-(\Delta b_{k+1})B_{k}+(\Delta b_{k+1})B_{k}-(\Delta
b_k)B_k\\
&=(\Delta b_{k+1})\Delta B_{k}+(\Delta^2 b_{k})B_k\\
&=\frac{1}{k(k+1)}((k+1)\Delta b_{k+1})(k\Delta B_{k})+\frac{1}{k^2}(k^2\Delta^2
b_{k})B_k,
\end{align*}
and
\begin{align*}
\Delta((\Delta c_k)H_k))
&=(\Delta c_{k+1})H_{k+1}-(\Delta c_k)H_k\\
&=(\Delta c_{k+1})H_{k+1}-(\Delta c_{k+1})H_{k}+(\Delta c_{k+1})H_{k}-(\Delta
c_k)H_k\\
&=(\Delta c_{k+1})\Delta H_{k}+(\Delta^2 c_k)H_k\\
&=\frac{1}{k(k+1)}((k+1)\Delta c_{k+1})(k\Delta H_{k})+\frac{1}{k^2}(k^2\Delta^2
c_k)H_k
\end{align*}
for all $k\in\mathbb{Z}$, $k\neq 0,-1$. Since
$\{b_k:k\in\mathbb{Z}\}$ and
$\{c_k:k\in\mathbb{Z}\}$ satisfy (H2), we have $(M_k)_{k\in\mathbb{Z}}$,
$(B_k)_{k\in\mathbb{Z}}$, $(S_k)_{k\in\mathbb{Z}}$, $(H_k)_{k\in\mathbb{Z}}$
satisfy the Marcinkiewicz condition of order one, and $\{c_k:k\in\mathbb{Z}\}$
is bounded by Remark \ref{R3}.
It follows that $\sup_{k\in\mathbb{Z}}\{k^2\|\Delta T_k\|\}<\infty$.
We observe that from \eqref{Tk} we have
\begin{align*}
k^2\Delta^2 N_k
&=k^2[\Delta N_{k+1}-\Delta N_k]\\
&=k^2[-N_{k+2}T_{k+1}+N_{k+1}T_k]\\
&=-k^2N_{k+2}[T_{k+1}-N_{k+2}^{-1}N_{k+1}T_k]\\
&=-k^2N_{k+2}[T_{k+1}-(I+T_{k+1})T_k]\\
&=-k^2N_{k+2}[T_{k+1}-T_k-T_{k+1}T_k]\\
&=-k^2N_{k+2}[\Delta T_k-T_{k+1}T_k]\\
&=-N_{k+2}[k^2\Delta T_k-\frac{k}{k+1}Q_{k+1}Q_k]
=N_{k+2}R_k
\end{align*}
where we have set $R_k=-[k^2\Delta T_k-\frac{k}{k+1}Q_{k+1}Q_k]$ for all
$k\in\mathbb{Z}$, $k\neq0,-1$. Since
$\{Q_k:k\in\mathbb{Z}\}$ and $\{k^2\Delta T_k:k\in\mathbb{Z}\}$ are
bounded, $\{R_k:k\in\mathbb{Z}\}$ is bounded.
Now, we have
\[
k^2\Delta^2 B_k=k^2\Delta^2( BN_k)=B(k^2\Delta^2 N_k)=BN_{k+2}R_k=B_{k+2}R_k,
\]
\begin{align*}
k^2\Delta^2 H_k
&=k^2\Delta^2( kN_k)\\
&=k^2[(k+2)N_{k+2}-2(k+1)N_{k+1}+kN_k]\\
&=k^2[kN_{k+2}-2kN_{k+1}+kN_k]+2k^2N_{k+2}-2k^2N_{k+1}\\
&=k^3\Delta^2N_k+2k^2\Delta N_{k+1}\\
&=k(k^2\Delta^2N_k)+\frac{2k^2}{k+1}[(k+1)\Delta N_{k+1}]\\
&=kN_{k+2}R_k+\frac{2k^2}{k+1}N_{k+2}Q_{k+1}\\
&=\frac{k}{k+2}H_{k+2}R_k+\frac{2k^2}{(k+1)(k+2)}H_{k+2}Q_{k+1},
\end{align*}
\begin{align*}
k^2\Delta^2 S_k
&=k^2\Delta^2(k\Lambda N_k)=k^2\Lambda\Delta^2
(kN_k)=\Lambda(k^2\Delta^2 H_k)\\
&=\Lambda\Big(\frac{k}{k+2}H_{k+2}R_k+\frac{2k^2}{(k+1)(k+2)}H_{k+2}Q_{
k+1}\Big)\\
&=\frac{k}{k+2}S_{k+2}R_k+\frac{2k^2}{(k+1)(k+2)}S_{k+2}Q_{k+1}.
\end{align*}
Finally,
\begin{align*}
k^2\Delta^2 M_k
&=k^2\Delta^2( k^2MN_k)\\
&=k^2[(k+2)^2MN_{k+2}-2(k+1)^2MN_{k+1}+k^2MN_k]\\
&=k^2[k^2MN_{k+2}-2k^2MN_{k+1}+k^2MN_k]+k^2(4k+4)MN_{k+2}\\&\hspace{.5cm}-2k^2(2k+1)MN_{k+1}
\\
&=k^2M(k^2\Delta^2N_k)+\frac{2k^2(2k+1)}{k+1}M[(k+1)\Delta
N_{k+1}]+2k^2MN_{k+2}\\
&=k^2MN_{k+2}R_k+\frac{2k^2(2k+1)}{k+1}MN_{k+2}Q_{k+1}+2k^2MN_{k+2}\\
&=\frac{k^2}{(k+2)^2}M_{k+2}R_k+\frac{2k^2(2k+1)}{(k+1)(k+2)^2}M_{k+2}Q_{
k+1}+\frac { 2k^2 }{(k+2)^2}M_{k+2}
\end{align*}
for all $k\in\mathbb{Z}$, $k\neq0,-1,-2$. Since
$\{B_k:k\in\mathbb{Z}\}$,
$\{S_k:k\in\mathbb{Z}\}$, $\{H_k:k\in\mathbb{Z}\}$, $\{M_k:k\in\mathbb{Z}\}$,
$\{Q_k:k\in\mathbb{Z}\}$, and
$\{R_k:k\in\mathbb{Z}\}$ are bounded, $\{k^2\Delta^2
B_k:k\in\mathbb{Z}\}$, $\{k^2\Delta^2
H_k:k\in\mathbb{Z}\}$, $\{k^2\Delta^2
S_k:k\in\mathbb{Z}\}$ and $\{k^2\Delta^2
M_k:k\in\mathbb{Z}\}$ are bounded. This completes the proof.
\end{proof}
From the proof of Theorem \ref{tfmg} and using \cite[Theorem 3.2]{BK}, we
deduce the following result for
$F^s_{pq}$-solutions in the case that $1 0$.
\begin{theorem}\label{tfm1}
Let $X$ be a Banach space and $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that
$D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a constant,
$b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2} such that
$(b_k)_{k\in\mathbb{Z}}$
and $(c_k)_{k\in\mathbb{Z}}$ satisfy {\rm (H2)}. Then for
$s>0$ and $1 0$. For this, assumption (H2) is no longer sufficient.
A condition which implies that $(M_k)_{k\in\mathbb{Z}}$ is a Fourier multiplier
for the scale $F_{pq}^s$, $s\in\mathbb{R}$, $1 0$ and $1\leq p<\infty$, $1\leq q\le \infty$, the following assertion are
equivalent.
\begin{itemize}
\item[(i)] \eqref{eP} is $F^s_{p,q}$-well-posed.
\item[(i)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$,
$\{k\Lambda N_k:k\in\mathbb{Z}\}$,
and $\{kN_k:k\in\mathbb{Z}\}$ are bounded, where
$$
N_k=[k^2M+A +b_kB+i k\Lambda+ikc_kI+\gamma I]^{-1}
$$
\end{itemize}
\end{theorem}
\begin{proof} (i) $\Rightarrow$ (ii). Assume that
\eqref{eP} is $F^s_{pq}$-well-posed. Then by Theorem \ref{t2},
$i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$ and
$(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
and $(kN_k)_{k\in\mathbb{Z}}$ are $F^s_{pq}$-Fourier multipliers. The
boundedness of $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
and $\{kN_k:k\in\mathbb{Z}\}$ follows of Remark \ref{R5}.
\smallskip
(ii) $\Rightarrow$ (i). In view of Theorem \ref{t2}, it suffices
to show that the families $(k^2MN_k)_{k\in\mathbb{Z}}$,
$(BN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$,
and $(kN_k)_{k\in\mathbb{Z}}$ are $F_{pq}^s$-Fourier multipliers. Let
$M_k=k^2MN_k$, $B_k=BN_k$, $H_k=kN_k$
and $S_k=k\Lambda N_k$. Since (H3) implies (H2) and (H2) implies (H1),
the verification of the Marcinkiewicz condition of order two and one is equal
to what was done in the proof of Theorem \ref{tfmg}.
It remains to prove the following inequalities:
\begin{gather*}
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 M_k\|<\infty,\quad
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 B_k\|<\infty,\\
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 S_k\|<\infty, \quad
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 H_k\|<\infty.
\end{gather*}
But we obtain this using the same technique as used in the proof
of the previous theorems.
\end{proof}
The following remark concerns the independence on the parameters
regarding the results of Section 4.
\begin{remark} \rm
$\bullet$ In Theorem \ref{tfmp}, if the problem is well-posed for
some $p\in (1,\infty)$, then it well-posed for all $p\in (1,\,\infty)$.
$\bullet$ Likewise, in Theorems \ref{tfm}, \ref{tfmg}, \ref{tfm1},
and \ref{tfmf}, if the problem under consideration is well-posed for
one set of parameters in the range afforded by the corresponding
theorem then it is well-posed for any set of parameters in that range.
This is a direct consequence of statement (ii) in each of the mentioned
theorems.
\end{remark}
\section{Examples and applications}
A large number of partial differential equations arising in physics and in
applied sciences can be written in the form of equation \eqref{eq1}; among them
there are some famous examples, such as the pseudo-parabolic equations and the
Sobolev type equations. Sobolev type equations have the form
\begin{equation}\label{Sob}
\Lambda u'=Au+f,
\end{equation}
generally denoting equations or systems in which spatial derivatives are
mixed with the time derivative of highest
order. Showalter \cite{Sh791, Sh792} studied Sobolev type equations of
the first and second order in time.
Specifically, Equation \ref{Sob} is called {\it strongly regular} if
$\Lambda^{-1}A$ is continuous, {\it weakly regular } if $\Lambda$ is invertible
but does not dominate $A$ and { \it degenerate } if $\Lambda$ is not invertible.
Strongly regular Sobolev type equations
are also widely known as pseudoparabolic. The Sobolev type equations are of
interest not only for the sake of generalizations but also because they arise
naturally in a variety
of applications (e.g. in acoustics, electromagnetics, viscoelasticity, heat
conduction etc., see e.g. \cite{LSY}).
A general theory in the context of generalized semigroups is developed
in the monograph \cite{MF01}.
For the periodic case initially, Arendt and Bu \cite{AB} deal with the
problem $u'(t)=Au(t)+f(t)$, $u(0)=u(2\pi)$.
This problem corresponds to \eqref{eP} with $M=B=0$, $\Lambda=-I$,
$c=0$, and $\gamma=0$. The additional condition of our definition of
well-posedness is obtained automatically by Remark \ref{Auto}. In this case
their result are equivalent to our result by Remarks \ref{R4} and \ref{R5}.
Arendt and Bu \cite{AB} (see also the review paper \cite{A}) consider the
problem $u''(t)=Au(t)+f(t)$, $u(0)=u(2\pi)$, $u'(0)=u'(2\pi)$. This problem
corresponds to \eqref{eP} with $M=I$, $\Lambda=B=0$
$c=0$, and $\gamma=0$. Here again the additional condition of our
definition of well-posedness is obtained automatically by Remark \ref{Auto}. In
this case their result are equivalent to our result by Remarks \ref{R5}.
Keyantuo and Lizama \cite{KL1, KL2} considered well-posedness of \eqref{eP}
when $B=M=0$ and $\Lambda$ is a scalar operator. As noted earlier, this problem is
relevant for viscoelasticity and was previously studied in the framework of
periodic solutions by Da Prato-Lunardi \cite{PL} among other references,
and on the real line by \cite{ CP, PL1}.
Second order equations are considered in this context in \cite{KL3, Pb}
The additional condition of our definition of well-posedness is obtained
automatically by Remark \ref{Auto}. Their results can be deduced from ours.
Some additional papers on the subject are Bu \cite{Bu1, Bu2, BY1}.
Delay equations are considered in \cite{BY2, PP}
with the method of operator-valued Fourier multipliers.
Bu \cite{Bu2} considered the well-posedness of \eqref{eP} when $B=\Lambda=0$,
$c=0$, and $\gamma$. His results follow from ours. With our
definition of well-posedness we do not need the a priori the estimate
\cite[(2.2)]{Bu2}. Thus, in the reference \cite{Bu2}, the author considers the problem
\begin{gather*}
(Mu')'(t)= Au(t)+f(t),\quad 0\leq t\leq 2\pi,\\
u(0)=u(2\pi), \quad (Mu')(0)=(Mu')(2\pi).
\end{gather*}
It follows from Theorem \ref{tfmp} that this problem is $L^p$-well-posed if and
only if $i\mathbb{Z}\subset
\rho_{0,M,\tilde{0},\tilde{0}}(A,0)=\rho_{M}(A)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$
and $\{kN_k:k\in\mathbb{Z}\}$ are $R$-bounded, where
$N_k=(k^2M+A)^{-1}$.
In a similar way, we deduce the results in $B^s_{p,q}$ and $F^s_{p,q}$ using
Theorem \ref{tfmg} and Theorem \ref{tfmf} respectively.
We introduce some facts on uniformly elliptic operators on domains of
$\mathbb{R}^n$ to discuss the examples that follow. Let
$\Omega\subset\mathbb{R}^n$ be open, $n\geq1$.
We consider measurable functions
$\alpha_{ik}$, $\beta_{k}$, $ \gamma_k$, and $\alpha_0$
$(1\leq j,k\leq n)$ on $\Omega$. We assume that the following
uniform ellipticity condition holds: The functions $\alpha_{kj}$, $\beta_{k}$,
$ \gamma_k$, $\alpha_0$ are bounded on $\Omega$, i.e., $\alpha_{kj}$, $\beta_{k}$,
$ \gamma_k$, $\alpha_0\in L^{\infty}(\Omega,\mathbb{C})$ for
$1\leq j,k\leq n$ and the principal part is
elliptic; i.e., there exists a constant $\eta>0$ such that
\begin{equation}\label{EL}
\operatorname{Re}(\sum_{j,k=1}^n\alpha_{kj}
(x)\xi_j\overline{\xi_k})\geq\eta|\xi|^2\quad \text{for all }\xi\in\mathbb{C}^n,
\text{ a.e. } x\in \Omega.
\end{equation}
The largest possible $\eta$ in \eqref{EL} is called the ellipticity constant
of the matrix $(\alpha_{jk})_{1\leq j,k\leq n}$.
Then we consider the elliptic operator
$L:W_{\rm loc}^{1,2}(\Omega) \to \mathcal{D}(\Omega)'$ given by
$$
Lu=-\sum_{k,j=1}^nD_j(\alpha_{kj}D_ku)+\sum_{k=1}
^n(\beta_kD_ku-D_k(\gamma_ku))+\alpha_0u.
$$
With the help of bilinear forms we will define various realizations of
$L \in L^2(\Omega)$ corresponding to
diverse boundary conditions. Let $V$ be a closed subspace of
$W^{1,2}(\Omega)$ containing $W_0^{1,2}(\Omega)$. We define the form
$\alpha_V:V\times V\to\mathbb{C}$ by
$$
\alpha_V(u,v)=\int_{\Omega}\Big[\sum_{k,j=1}^n\alpha_{kj}D_ku\overline
{(D_jv)}+ \sum_{k=1}^n(\beta_k\overline{v}D_ku+\gamma_ku\overline{D_kv}
)+\alpha_0u\overline{v}\Big]dx.
$$
Then $\alpha_V$ is densely defined,
accretive, and closed sesquilinear form on $L^2(\Omega)$
(see \cite[Chapter 4 p. 100-101]{Ouh}). Denote by $A_V$ the operator
on $L^2(\Omega)$ associated with $\alpha_V$. Then $-A_V$ generates a
$C_0$-semigroup $T_V$ on $L^2(\Omega)$ (see \cite[Proposition 1.51]{Ouh}).
It follows from the definition of the associated
operator that $A_V u = Lu$ for all $u \in D(A_V )$.
We will say that we have:
\begin{itemize}
\item {Dirichlet boundary conditions} if $V=W_0^{1,2}(\Omega)$;
\item {Neumann boundary conditions} if $V=W^{1,2}(\Omega)$;
\end{itemize}
We consider Dirichlet boundary conditions with $\Omega$ bounded
and we assume the following additional conditions: $\alpha_{kj}$ is real-valued
with $\alpha_{kj}=\alpha_{jk}$, $\beta_k=\gamma_k=0$, $\alpha_0\geq0$. Then,
in this case the semigroup $T_V$ is positive,
$\|T_V(t)\|_{\mathcal{L}(L^2(\Omega))}\leq 1$ for all $t\geq0$, and $T_V$ is
given by an integral kernel $p_V(t,x,y)$ such that there exist constants $C>0$,
$b>0$, and $\delta>0$ such that
\begin{equation}\label{Ker}
|p_V(t,x,y)|\leq Ct^{-n/2}e^{-\delta t}e^{-\frac{|x-y|^2}{4bt}}
\end{equation}
for every $t>0$ and a.e. $x,y\in\Omega$, see
\cite[Theorem 4.2, Corollary 6.14 and Theorem 4.28]{Ouh}
and \cite{Dav}. For every $r\in(1,\infty)$, the
$C_0$-semigroup $T_V$ extends to a bounded $C_0$-semigroup $T_r$ on
$L^r(\Omega)$ with $\|T_r(t)\|_{\mathcal{L}(L^r(\Omega))}\leq 1$ for all
$t\geq0$, by \cite[Theorem 4.28]{Ouh}. By \eqref{Ker} there exist $M_r>0$, and
$\delta_r>0 $ depending only on $r$ such that
$\|T_r(t)\|_{\mathcal{L}(L^r(\Omega))}\leq M_re^{-\delta_r t}$ for all $t>0$ and
$r\in(1,\infty)$. Denote now by $-A_r$ the corresponding infinitesimal generator
on $L^r(\Omega)$. If $\lambda\in\mathbb{C}$, $\operatorname{Re}\lambda>-\delta$,
then $\lambda\in \rho(-A_r)$ and
\begin{equation}\label{RB2}
R(\lambda,-A_r)u=\int_0^\infty e^{-\lambda t}T_r(t)udt\text{ for all }u\in
L^r(\Omega),
\end{equation}
by \cite[Theorem 3.1.7]{ABHN}.
Let $r\in(1,\infty)$. The $C_0$-semigroup $T_r$ extends to a bounded
holomorphic semigroup on the sector $\Sigma_{\pi/2}$, where $\Sigma_\theta$ is
the sector in the complex right half plane of angle $\theta\in(0,\pi]$. By
\cite[Theorem 3.7.11]{ABHN} we have that $\Sigma_\pi\subset\rho(-A_r)$ and
$\sup_{\lambda\in\Sigma_{\pi-\varepsilon}}\|\lambda
R(\lambda,-A_r)\|<\infty$ for all $\varepsilon>0$.
Denote by $\sigma(A_r)$ the spectrum of the operator $A_r$ on
$L^r(\Omega)$. By \cite[Theorem 7.10]{Ouh}, we have that
$\sigma(A_r)=\sigma(A_2)\subset(0,\infty)$ for all
$r\in(1,\infty)$.
By \cite[Section 7.2.6]{A} we have that $\lambda R(\lambda,-A_r)$ is
$R$-bounded for all $\lambda\in \Sigma_{\pi/2+\theta_r}$ with
$0<\theta_r\leq\pi/2$.
Since $\lambda\to R(\lambda,-A_r)$ is analytic on
$\Sigma_\pi\cup\{\lambda\in\mathbb{C}:\operatorname{Re}\lambda>-\delta_r\}$,
it follows that $R(\lambda,-A_r)$ is $R$-bounded on every compact subset of
$\Sigma_\pi\cup\{\lambda\in\mathbb{C}:\operatorname{Re}\lambda>-\delta_r\}$ by
\cite[Proposition 3.10]{DHP}. By Remark \ref{R1}, we have that $R(\lambda,-A_r)$
and $\lambda R(\lambda,-A_r)$ are $R$-bounded on
$\Sigma_{\pi/2+\theta_r}\cup\{\lambda\in\mathbb{C}:\operatorname{Re}
\lambda\geq-\delta_r/2\}$. Using Kahane's principle, we obtain that
$R(\lambda,A_r)$ and $\lambda R(\lambda,A_r)$ are $R$-bounded on
$\mathbb{C}\setminus\Sigma_{\theta_r}\cup \{\lambda\in\mathbb{C}:
\operatorname{Re}\lambda\leq\delta_r/2\}$.
We conclude, with some examples using uniformly elliptic operators in
$L^r(\Omega)$ just discussed. General references on uniformly elliptic operators
in $L^p-$spaces and the associated heat kernel estimates are \cite{Dav} and \cite{Ouh}.
\begin{example}\label{E} \rm
Let us consider the boundary value problem (in which $L$ is a uniformly elliptic operator as defined above)
\begin{equation}\label{E1}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})+L\frac{\partial u(t,x)}{\partial t}\\
&=-Lu(t,x)-\int_{-\infty}^t{b(t-s)L
u(s,x)}ds+f(t,x), \quad (t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial
u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
where $f\in L^p(0,2\pi;L^r(\Omega))$ for $1 0$, $1\leq p,q\leq\infty$
by Theorem \ref{tfmg}. Observe that here we include the scale of vector-valued
H\"older spaces $C^s$, $0 0$
and $\nu>0$ satisfies the
required conditions for $b_k$ for all the cases.
\end{example}
\begin{example}\label{EII} \rm
Let us consider the boundary value problem
\begin{equation}\label{EII1}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})+L\frac{\partial u(t,x)}{\partial t}\\
&=Lu(t,x)+\int_{-\infty}^t{b(t-s)L
u(s,x)}ds+f(t,x), \, (t,x)\in[0,2\pi]\times\Omega,
\end{aligned} \\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad
m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial u(2\pi,x)}{\partial t},
\quad x\in\Omega,
\end{gathered}
\end{equation}
where $f\in L^p(0,2\pi;L^r(\Omega))$ for $1 -1$ for all $k\in\mathbb{Z}$, then
$\frac{k^2m(x)}{1+b_k-ik}\in\Sigma_{\pi/2+\theta_r}\cup\{0\}$ for all
$x\in\Omega$ and all $k\in\mathbb{Z}$. Therefore
$i\mathbb{Z}\subset \rho_{-A_r,M,\tilde{b},\tilde{0}}(A_r,A_r)$ and
$\{(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$,
$\{\frac{k^2}{1+b_k-ik}M(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$ are
$R$-bounded, here $M$ is the multiplication operator by $m$. By Remarks
\ref{R1} and \ref{R3}, we have that
$\{\frac{k}{1+b_k-ik}(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$ is also
$R$-bounded. Since
\begin{align*}
&\frac{k}{1+b_k-ik}A_r(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}\\
&=\frac{k}{1+b_k-ik}I
-\frac{k}{1+b_k-ik}\frac{k^2}{1+b_k-ik}M(\frac{k^2}{1+b_k-ik}M+A_r)^{-1},
\end{align*}
it follows that
$\{\frac{k}{1+b_k-ik}A_r(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$ is
$R$-bounded as well by Remark \eqref{R1}. Since
$N_k= \frac{1}{1+b_k-ik}(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}$, we have shown that
$\{k^2MN_k:k\in\mathbb{Z}\}$, $\{kA_rN_k:k\in\mathbb{Z}\}$ and
$\{kN_k:k\in\mathbb{Z}\}$ are $R$-bounded. Therefore, by Theorem \ref{tfmp}, we
have that \eqref{E12} is
$L^p(0,2\pi;L^r(\Omega))$-well-posed for all $1 -1$ for all $k\in\mathbb{Z}$, then we have that \eqref{E12} is
$B^s_{pq}(0,2\pi;L^r(\Omega))$-well-posed for all $s>0$, $1\leq p,q\leq\infty$
by Theorem \ref{tfmg}. Observe that here we include the scale of vector-valued
H\"older spaces $C^s$, $0 0$ and
$\nu>0$ that fulfills the required conditions for $b_k$ in all the cases.
\end{example}
\begin{example}\label{Ex21} \rm
Consider another initial-boundary value problem.
\begin{equation}\label{E2}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m_2(x)\frac{\partial u(t,x)}{\partial
t})-m_1(x)\frac{\partial u(t,x)}{\partial t}\\
&=Lu(t,x)+\int_{-\infty}^t{b(t-s)L u(s,x)}ds+f(t,x), \quad
(t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m_2(x)\frac{\partial u(0,x)}{\partial
t}=m_2(x)\frac{\partial u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
where $m_1$ and $m_2$ are real-valued measurable functions on $\Omega$ such
that $m\in L^\infty(\Omega)$, $m_2(x)\geq 0$, $\tau<|m_1(x)|\leq \mu$ for some
$\tau, \mu>0$, and $f\in L^p(0,2\pi;L^r(\Omega))$ for $1 -1$ for all $k\in\mathbb{Z}$, then
$\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}\notin(-\infty,0)$ for all $x\in\Omega$ and all
$k\in\mathbb{Z}$. Therefore
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{0}}(A_r,A_r)$, where
$\Lambda$ and $M$ are the
multiplication operators by $m_1$ and $m_2$ respectively. By Remark \ref{R3}, we
have that there exists $N\in\mathbb{N}$ such that
$\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}\in\Sigma_{\pi/2+\theta_r}$ for all
$x\in\Omega$ and all $k\in\mathbb{Z}$ whit $|k|\geq N$.
Then
$\{\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}
:k\in\mathbb{Z}, |k|\geq N, x\in\Omega\}$ are $R$-bounded. Since
$\{\frac{1}{km_2(x)+im_1(x)}:k\in\mathbb{Z}, x\in\Omega\}$ is bonded,
$\{\frac{k}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}:k\in\mathbb{Z},
|k|\geq N, x\in\Omega\}$ are $R$-bounded by Remark \ref{R1}. Since $m_1$ is
bounded, by Remark \ref{R1} we have that
$\{\frac{km_1(x)}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}:k\in\mathbb{
Z}, |k|\geq N, x\in\Omega\}$ are $R$-bounded. Therefore,
$\{\frac{k^2m_2(x)}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}
:k\in\mathbb{Z}, |k|\geq N, x\in\Omega\}$ are $R$-bounded. Since
$N_k=\frac{1}{b_k+1}(\frac{k^2}{b_k+1}M+i\frac{k}{b_k+1}\Lambda+A_r)^{-1}$,
we have show that $\{kN_k:k\in\mathbb{Z}, |k|\geq N\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}, |k|\geq N\}$ and $\{k^2MN_k:k\in\mathbb{Z}, |k|\geq N\}$ are
$R$-bounded. By Remark \ref{R1}, we have that $\{kN_k:k\in\mathbb{Z}\}$,
$\{k\Lambda N_k:k\in\mathbb{Z}\}$ and $\{k^2MN_k:k\in\mathbb{Z}\}$ are
$R$-bounded. Under the same conditions over $b_k$ in the Example \ref{EII} and
$f\in\mathcal{Y}$ we can apply Theorems \ref{tfmp}, \ref{tfmg} and \ref{tfmf}
to obtain that the \eqref{E21} is $\mathcal{Y}$-well-posed.
\end{example}
\newpage
\begin{example}\label{E31}\end{example}
Let us now consider the boundary-value problem
\begin{equation}\label{E3}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m_2(x)\frac{\partial u(t,x)}{\partial
t})-m_1(x)\frac{\partial u(t,x)}{\partial t}-\frac{\partial}{\partial
t}\int_{-\infty}^tc(t-s)u(s,x)ds\\
&=Lu(t,x)+\int_{-\infty}^t{b(t-s)L u(s,x)}ds \\
&\quad +\int_{-\infty}^tb(t-s)m_0(x)u(s,x)ds +f(t,x), \quad
(t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m_2(x)\frac{\partial u(0,x)}{\partial
t}=m_2(x)\frac{\partial u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
where $m_0$, $m_1$, and $m_2$ are real-valued measurable functions on $\Omega$
such that $0\leq m_0(x)\leq \mu$, $\tau<| m_1(x)|\leq\mu$, $0\leq m_2(x)$, for
some $\mu, \tau>0$, all $x\in\Omega$, and $f\in L^p(0,2\pi;L^r(\Omega))$ for
$1 0$. With $f\in\mathcal{Y}$ and the
appropriate $b$, and $c$ we can obtain that the \eqref{E32} is
$\mathcal{Y}$-well-posed.
In the case of Neumann boundary conditions, the
operator $A_r$ is not invertible. To apply the results to this case, we
can add in the right side of each of the above equations the term
$\eta u(t,x)$ for some $\eta>0$. Then the above conclusions hold in this case as
well.
\begin{example}\label{P} \rm
The following equation is a modification of
the one studied by Chill and Srivastava \cite{ChS}. Here we have include
memory term.
\begin{equation}
\begin{gathered}
u''(t)+\alpha A^{\frac{1}{2}} u'(t)=-Au(t)+\int_{-\infty}^t{b(t-s)A
u(s,x)}ds+f(t), \\
t\in[0,2\pi],\quad u(0)=u(2\pi),\quad u'(0)=u'(2\pi),
\end{gathered}
\end{equation}
where $A$ is a invertible sectorial operator in a Banach space $X$ which admits
a bounded $H^\infty$ functional calculus of angle $\beta$ (see for
example \cite{ChS}, \cite{DHP}) with
$\beta\in(0,\pi-2\tan^{-1}\frac{\sqrt{4-\alpha^2}}{\alpha})$ if
$0<\alpha<2$ or $\beta\in(0,\pi)$ if $\alpha\geq2$, $f\in B^s_{pq}(0,2\pi;X)$,
($1\leq p,q\leq\infty,\, s>0$), and $b\in {L}^1(\mathbb{R}_+)$ is such that
$b_k=\tilde{b}(ik)$ satisfies $|CQb_k|<\frac{1}{2}$ where $Q$ is a constant
provided by \cite[Lemma 4.1]{ChS}
and $C$ is a constant provided by the $H^\infty$ functional calculus.
In the same way as in the proof of theorem \cite[Theorem 4.1]{ChS} we have
that for $k\in\mathbb{Z}$,
$\|k^2(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq CP$,
$\|kA^{\frac{1}{2}}(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq CP$, and
$\|A(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq CP$. In this case for
$k\in\mathbb{Z}$, we have that
$N_k=(k^2-\alpha kiA^{\frac{1}{2}}-A+b_kA)^{-1}$. Since
$\|b_kA(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq \frac{1}{2}$, we have
$$
N_k=(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\sum_{n=0}^\infty (-1)^n
\Big(b_kA(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\Big)^n,
$$
which implies that $\|k^2N_k\|\leq CP$ and $\|\alpha
kA^{\frac{1}{2}}N_k\|\leq CP$ for $k\in\mathbb{Z}$. Now if
$b_k$ satisfy (H2), then we have that the problem \ref{P} is
$B^s_{pq}$-well-posed. This gives in particular well-posedness in
the H\"older spaces $C^s(0,2\pi ; X)$, $00$, $1\leq p<\infty$,
$1\leq q\leq\infty$, by Theorem \ref{tfmf}. Observe that if $s>$, $1-1$
for all $k\in\mathbb{Z}$, then we have that \eqref{E12} is
$F^s_{pq}(0,2\pi;L^r(\Omega))$-well-posed for all $s>0$, $1\leq p<\infty$,
$1\leq q\leq\infty$, by Theorem \ref{tfmf}. Observe that if $s>0$,
$10$.
\end{example}
\subsection*{Acknowledgments}
This work was partially supported by the Air Force Office
of Scientific Research (AFOSR) under the Award No: FA9550-15-1-0027.
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\end{document}