\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 130, pp. 1--20.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/130\hfil Variation of constants formula] {Variation of constants formula for functional parabolic partial differential equations} \author[A. Carrasco, H. Leiva\hfil EJDE-2007/130\hfilneg] {Alexander Carrasco, Hugo Leiva} % in alphabetical order \address{Alexander Carrasco \newline Universidad Centroccidental Lisandro Alvarado \\ Decanato de Ciencias, Departamento de Matem\'atica\\ Barquisimeto 3001, Venezuela} \email{acarrasco@ucla.edu.ve} \address{Hugo Leiva \newline Universidad de Los Andes Facultad de Ciencias, Departamento de Matem\'atica \\ M\'erida 5101, Venezuela} \email{hleiva@ula.ve} \thanks{Submitted July 2, 2007. Published October 5, 2007.} \subjclass[2000]{34G10, 35B40} \keywords{Functional partial parabolic equations; variation of constants formula; \hfill\break\indent strongly continuous semigroups} \begin{abstract} This paper presents a variation of constants formula for the system of functional parabolic partial differential equations \begin{gather*} \frac{\partial u(t,x)}{\partial t} = D\Delta u+Lu_t+f(t,x), \quad t>0,\; u\in \mathbb{R}^n \\ \frac{\partial u(t,x)}{\partial \eta} = 0, \quad t>0, \; x\in \partial\Omega \\ u(0,x) = \phi(x) \\ u(s,x) = \phi(s,x), \quad s\in[-\tau,0),\; x\in\Omega\,. \end{gather*} Here $\Omega$ is a bounded domain in $\mathbb{R}^n$, the $n\times n$ matrix $D$ is block diagonal with semi-simple eigenvalues having non negative real part, the operator $L$ is bounded and linear, the delay in time is bounded, and the standard notation $u_{t}(x)(s) = u(t+s,x)$ is used. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this paper we find a variation of constants formula for the system of functional parabolic partial differential equations $$\label{non-homogeneous} \begin{gathered} \frac{\partial u(t,x)}{\partial t} = D\Delta u+Lu_t+f(t,x), \quad t>0,\; u\in \mathbb{R}^n \\ \frac{\partial u(t,x)}{\partial \eta} = 0, \quad t>0, \; x\in \partial\Omega \\ u(0,x) = \phi(x) \\ u(s,x) = \phi(s,x), \quad s\in [-\tau,0),\; x\in\Omega \end{gathered}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, the $n\times n$ matrix $D$ is non diagonal with semi-simple eigenvalues having non negative real part, and $f:\mathbb{R} \times\Omega\to \mathbb{R}^n$ is an smooth function. The standard notation $u_{t}(x)$ defines a function from $[-\tau,0]$ to $\mathbb{R}^n$ by $u_{t}(x)(s) = u(t+s,x)$, $-\tau \leq s \leq 0$ (with $x$ fixed). Here $\tau\geq 0$ is the maximum delay, which is suppose to be finite. We assume the operator $L : L^{2}([-\tau,0];Z) \to Z$ is linear and bounded with $Z = L^{2}(\Omega)$ and $\phi_{0}\in Z$, $\phi\in L^{2}([-\tau,0];Z)$. The variational constant formula plays an important role in the study of the stability, existence of bounded solutions and the asymptotic behavior of non linear ordinary differential equations. The variation of constants formula is well known for the finite dimensional semi-linear ordinary differential equation $$\label{non-homo-ode} \begin{gathered} x'(t) = A(t) + f(t,x), \quad x \in \mathbb{R}^n \\ x(0) = x_{0}, \end{gathered}$$ and it gives the solution $$x(t) = \Phi(t)x_{0} + \int^{t}_{0}\Phi(t)\Phi^{-1}(s)f(s,x(s))ds$$ where $\Phi(\cdot)$ is the fundamental matrix of the system $$\label{homogeneous-ode} x'(t) = A(t)x.$$ Due to the importance of this formula, for semi linear ordinary differential equations, in 1961 the Russian mathematician Alekseev \cite{Alek} found a formula for the nonlinear ordinary differential equation $$\label{sistema-semilineal} y'(t) = f(t,y) + g(t,y), \quad y(t_{0}) = y_{0},$$ which is given by $$y(t,t_{0},y_{0}) = x(t,t_{0},y_{0}) + \int^{t}_{t_{0}}\Phi(t,s,y(s))g(s,y(s))ds,$$ where $x(t,t_{0},y_{0})$ is the solution of the initial value problem $$\label{sistema-no-autonomo} x'(t) = f(t,x), \quad x(t_{0}) = y_{0},$$ and $$\Phi(t,s,\xi) = \frac{\partial x(t,t_{0},y_{0})}{\partial y_{0}}.$$ This formula is used to compare the solutions of (\ref{sistema-semilineal}) with the solutions of (\ref{sistema-no-autonomo}). In fact, it was used in \cite{LG}. In infinite dimensional Banach spaces $Z$, we have the following general situation. If $A$ is the infinitesimal generator of strongly continuous semigroup $\{T(t) \}_{t \geq 0}$ in $Z$ and $f: [0, \beta] \to Z$ is a suitable function, then the solution of the initial value problem $$\label{sistemainfinito} \begin{gathered} z'(t) = Az(t) + f(t), \quad t> 0, \; z \in Z \\ z(0) = z_{0}, \end{gathered}$$ is given by the variation constant formula $$\label{milds} z(t) = T(t)z_{0} + \int^{t}_{0}T(t - s)f(s)ds , \quad t\in[0,\infty ).$$ Therefore, any solution of the problem (\ref{sistemainfinito}) is also solution of the integral equation (\ref{milds}). However, the converse may not be true, since a solution of (\ref{milds}) is not necessarily differentiable. We shall refer to a continuous solution of (\ref{milds}) as a mild solution of problem (\ref{sistemainfinito}); a mild solution is thus a kind of generalized solution. However, if $\{T(t) \}_{t \geq 0}$ is an analytic semigroup and the function $f$ satisfies the following H\"{o}lder condition $$\|f(s) -f(t) \| \leq L|s-t|^\theta, \quad s,t \in [0, \beta],$$ with $L>0$, $\theta \geq 1$, then the mild solution (\ref{milds}) is also solution of the initial value problem (\ref{sistemainfinito}). Our work and many others are motivated by the legendary paper by Borisovic and Turbabin \cite{BT}; there they found a variational constants formula for the system of nonhomogeneous differential equation with delay $$\label{sistemadelay} \begin{gathered} z'(t) = Lz_t + f(t), \quad t> 0, \; z \in \mathbb{R}^n \\ z(0) = z_{0},\\ z(s) = \phi(s), \quad s \in [-\tau, 0), \end{gathered}$$ where $f:\mathbb{R}^+ \to \mathbb{R}^n$ is a suitable function. The standard notation $z_{t}$ defines a function from $[-\tau,0]$ to $\mathbb{R}^n$ by $z_{t}(s) = z(t+s), -\tau \leq s \leq 0$. Here $\tau\geq 0$ is the maximum delay, which is suppose to be finite. We assume that the operator $L : L^{p}([-\tau,0];\mathbb{R}^n) \to \mathbb{R}^n$ is linear and bounded, and $z_{0}\in \mathbb{R}^n,\;\phi\in L^{p}([-\tau,0];\mathbb{R}^n)$. Under some conditions they prove the existence and the uniqueness of solutions for this system and associate to it a strongly continuous semigroup $\{T(t) \}_{t \geq 0}$ in the Banach space $\mathbb{M}_{p}([-\tau,0];\mathbb{R}^n) = \mathbb{R}^n \oplus L_{p}([-\tau,0];\mathbb{R}^n)$. Therefore, system (\ref{sistemadelay}) is equivalent to the following system of ordinary differential equations, in $\mathbb{M}_{p}$, $$\label{ecuacion3 ordinaria no homogenea} \begin{gathered} \frac{dW(t)}{dt} = \Lambda W(t) + \Phi(t),\quad t>0, \\ W(0) = W_{0} = (z_{0},\phi(\cdot)) \end{gathered}$$ where $\Lambda$ is the infinitesimal generator of the semigroup $\{T(t)\}_{t\geq0}$ and $\Phi(t) = (f(t),0)$. Hence, the solution of system (\ref{sistemadelay}) is given by the variational constant formula or mild solution $$\label{variacion3 de parametro} W(t) = T(t)W_{0} + \int_{0}^{t}T(t-s)\Phi(s)ds.$$ Finally, the formula we found here is valid for those system of PDEs that can be rewritten in the form ${\partial \over \partial t}u =D\Delta u$, like damped nonlinear vibration of a string or a beam, thermoplastic plate equation, etc. For more information about this, see the paper by Oliveira \cite{LO2}. To the best of our knowledge, there are variational constant formulas for reaction diffusion equations, functional equations and neutral equations \cite{IV}, but for functional partial parabolic equations we are not aware of results similar to the one presented here. At the same time, if we change the Neumann boundary condition by Dirichlet boundary condition, the result follows trivially. \section{Abstract Formulation of the Problem} In this section we choose a Hilbert Space where system \eqref{non-homogeneous} can be written as an abstract functional differential equation. To this end, we consider the following hypothesis. \begin{itemize} \item[(H1)] The matrix $D$ is semi simple (block diagonal) and the eigenvalues $d_{i} \in \mathbb{C}$ of $D$ satisfy $\mathop{\rm Re}(d_{i}) \geq 0$. Consequently, if $0 = \lambda_{1}< \lambda_{2}<\dots < \lambda_{n} \to \infty$ are the eigenvalues of $-\Delta$ with homogeneous Neumann boundary conditions, then there exists a constant $M \geq 1$ such that : \\ $\|e^{-\lambda_{n}Dt}\|\leq M$, \;\;$t\geq 0, \;\;\;n=1,2,3,\dots$\\ H2). For all $I>0$ and $z \in L^{2}_{\rm loc}([-\tau, 0); Z)$ we have the following inequality $$\int^{t}_{0}| Lz_{s}| ds \leq M_{0}(t)| z |_{L^{2}([-\tau,t),Z)},\quad \forall t\in[0,I],$$ where $M_{0}(\cdot)$ is a positive continuous function on $[0, \infty)$. \end{itemize} Consider $H = L^{2}(\Omega,\mathbb{R})$ and $0 = \lambda_{1}<\lambda_{2}<\dots <\lambda_{n}\to \infty$ the eigenvalues of $-\Delta$, each one with finite multiplicity $\gamma_{n}$ equal to the dimension of the corresponding eigenspace. Then \begin{itemize} \item[(i)] There exists a complete orthonormal set $\{\phi_{n,k} \}$ of eigenvectors of $-\Delta$. \item[(ii)] For all $\xi \in D(-\Delta)$ we have $$\label{p3} -\Delta \xi = \sum_{n = 1}^{\infty} \lambda_{n} \sum_{k = 1}^{\gamma_n} \langle \xi, \phi_{n,k}\rangle \phi_{n,k} =\sum_{n = 1}^{\infty} \lambda_{n}E_{n}\xi,$$ where $\langle\cdot, \cdot\rangle$ is the inner product in $H$ and $$\label{p4} E_{n}x = \sum_{k = 1}^{\gamma_n} \langle \xi, \phi_{n,k}\rangle \phi_{n,k}.$$ So, $\{ E_n \}$ is a family of complete orthogonal projections in $H$ and $\xi = \sum_{n = 1}^{\infty}E_{n} \xi$, $\xi \in H$. \item[(iii)] $\Delta$ generates an analytic semigroup $\{ T_{\Delta}(t) \}$ given by $$\label{p5} T_{\Delta}(t)\xi = \sum_{n = 1}^{\infty} e^{-\lambda_{n}t}E_{n}\xi.$$ \end{itemize} Now, we denote by $Z$ the Hilbert space $L^{2}(\Omega,\mathbb{R}^n )$ and define the following operator $$A : D(A)\subset Z\to Z , \quad A\psi = -D\Delta\psi$$ with $D(A)=H^{2}(\Omega,\mathbb{R}^n )\cap H^{1}_{0}(\Omega,\mathbb{R}^n )$.\\ Therefore, for all $z\in D(A)$ we obtain \begin{gather*} Az = \sum^{\infty}_{n = 1}\lambda_{n}DP_{n}z, \quad z = \sum^{\infty}_{n = 1}P_{n}z , \quad \| z \|^{2} = \sum^{\infty}_{n = 1}\| P_{n}z \|^{2} , \quad z\in Z \end{gather*} where $P_{n} = \mathop{\rm diag}(E_{n},E_{n},\dots ,E_{n})$ is a family of complete orthogonal proyections in $Z$. Consequently, system \eqref{non-homogeneous} can be written as an abstract functional differential equation in $Z$: $$\label{ecuacion diferencial abstracta no homogenea} \begin{gathered} \frac{dz(t)}{dt} = -Az(t) + Lz_{t} + f^{e}(t),\quad t>0 \\ z(0) = \phi_{0} \\ z(s) = \phi(s),\quad s\in[-\tau,0) \end{gathered}$$ Here $f^{e}: (0,\infty)\to Z$ is a function defined as follows: $$f^{e}(t)(x) = f(t,x),\quad t>0, \; x\in\Omega.$$ \section{Preliminaries Results} For the rest of this article, we will use the following generalization of lemma 2.1 from \cite{HLe}. \begin{lemma} \label{LG} Let $Z$ be a separable Hilbert space, $\{S_{n}(t)\}_{n\geq 1}$ a family of strongly continuous semigroups and $\{P_{n}\}_{n\geq 1}$ a family of complete orthogonal projection in $Z$ such that $$\Lambda_{n}P_{n} = P_{n}\Lambda_{n}, \quad n\geq 1,2,\dots$$ where $\Lambda_{n}$ is the infinitesimal generator of $S_{n}$. Define the family of linear operators $$S(t)z = \sum_{n = 1}^{\infty}S_{n}(t)P_{n}z,\quad t\geq0.$$ Then: \begin{itemize} \item[(a)] $S(t)$ is a linear and bounded operator if $\|S_{n}(t) \| \leq g(t)$, $n = 1,2,\dots$, with $g(t)\geq 0$, continuous for $t\geq0$. \item[(b)] $\{S(t)\}_{t \geq 0}$ is an strongly continuous semigroup in the Hilbert space $Z$ whose infinitesimal generator $\Lambda$ is given by $$\Lambda z = \sum_{n = 1}^{\infty}\Lambda_{n}P_{n}z, \quad z\in D(\Lambda)$$ with $$D(\Lambda) = \big\{z\in Z \: / \: \sum_{n = 1}^{\infty}\| \Lambda_{n}P_{n}z \|^{2} < \infty\big\}$$ \item[(c)] the spectrum $\sigma(\Lambda)$ of $\Lambda$ is given by $$\label{L6} \sigma(\Lambda) = \overline{\cup_{n=1}^{\infty} \sigma(\bar{\Lambda}_n)},$$ where $\bar{\Lambda}_{n} = \Lambda_{n}P_n :\mathcal{R}(P_n) \to \mathcal{R}(P_n)$. \end{itemize} \end{lemma} \begin{proof} First, from Hille-Yosida Theorem, $S_{n}(t)P_{n}=P_{n}S_{n}(t)$ since $\Lambda_{n}P_{n}= P_{n}\Lambda_{n}$. So that $\{S_{n}(t)P_{n}z\}_{n\geq1}$ is a family of orthogonal vectors in $Z$. Then \begin{align*} \| S(t)z \|^{2} & = \langle S(t)z,S(t)z \rangle \\ & = \Big\langle \sum_{n = 1}^{\infty}S_{n}(t)P_{n} z,\sum_{m = 1}^{\infty}S_{m}(t)P_{m}z \Big\rangle \\ & = \sum_{n = 1}^{\infty}\| S_{n}(t)P_{n}z \|^{2} \\ & \leq (g(t))^{2}\sum_{n = 1}^{\infty}\| P_ {n}z \|^{2} \\ & = (g(t)\| z \|)^{2} \end{align*} Therefore, $S(t)$ is a bounded linear operator. Second, we have the following relations: (i) \begin{align*} S(t)S(s)z & = \sum_{n = 1}^{\infty}S_{n} (t)P_{n}S(s)z \\ & = \sum_{n = 1}^{\infty}S_{n}(t)P_{n} \Big(\sum_{m = 1}^{\infty}S_{m}(s)P_{m}z\Big) \\ & = \sum_{n = 1}^{\infty}S_{n}(t + s)P_ {n}z \\ & = S(t + s)z \end{align*} (ii) $$S(0)z = \sum_{n = 1}^{\infty}S_{n}(0)P_{n}z = \sum_{n = 1}^{\infty}P_{n}z = z$$ (iii) \begin{align*} \| S(t)z - z \|^{2} & = \| \sum_{n = 1}^{\infty}S_{n}(t)P_{n}z - \sum_{n = 1}^{\infty}P_{n}z \|^{2} \\ & = \sum_{n = 1}^{\infty}\|(S_{n}(t) - I)P_{n}z\|^{2} \\ & = \sum_{n = 1}^{N}\| (S_{n} (t) - I)P_{n}z)\|^{2} + \sum_{n = N + 1}^{\infty}\| (S_ {n}(t) -I)P_{n}z \|^{2}\\ & \leq \sup_{1\leq n \leq N} \| (S_{n}(t) - I)P_{n}z \|^{2}\sum_{n = 1}^{N} + K\sum_{n = N + 1}^{\infty}\| P_{n}z \|^{2}, \end{align*} where $K = \sup_{0\leq t \leq 1 ;\;n \geq 1}\| (S_{n}(t) - I)\|^{2}\leq (g(t) + 1)^{2}$. Since $\{S_{n}(t)\}_ {t\geq 0}$ $(n =1,2,\dots )$ is an strongly continuous semigroup and $\{P_n \}_{n \geq 1}$ is a complete orthogonal projections, given an arbitrary $\epsilon >0$ we have, for some natural number $N$ and $0From here we obtain that$\lambda\in \sigma(\overline{\Lambda_{n_{0}}})$, and therefore$\lambda\in \overline{\cup_{n=1}^{\infty}\sigma(\overline{\Lambda_{n}})}$. \noindent(2) If$\overline{R(\Lambda-\lambda I)}\neq Z$, then there exists$z_{0}\in Z$non zero such that $$\langle z_{0},(\Lambda-\lambda I)z\rangle = 0, \ \ \forall z\in D(A).$$ But,$z= \sum_{n=1}^{\infty}P_{n}z$, so $$\langle z_{0},\sum_{n=1}^{\infty}(\overline{\Lambda_{n}}-\lambda I)P_{n}z\rangle = 0.$$ Now, if$z_{0}\neq 0$, then there is$n_{0}\in\mathbf{N}$such that$P_{n_{0}}z_{0} \neq0$. Hence, $0 = \langle z_{0},\sum_{n=1}^{\infty}(\overline{\Lambda_{n}}-\lambda I)P_{n}z\rangle = \langle z_{0},(\overline{\Lambda_{n_{0}}}-\lambda I)P_{n_{0}}z\rangle = \langle P_{n_{0}}z_{0},(\overline{\Lambda_{n_{0}}}-\lambda I)P_{n_{0}}z\rangle$ So,$R(\overline{\Lambda}_{n_{0}}-\lambda I)\neq P_{n_{0}}Z$. Therefore,$\lambda\in \sigma(\overline{\Lambda}_{n_{0}}) \subset\overline{\cup_{n=1}^{\infty}\sigma(\overline{\Lambda_{n}})}$. \noindent (3) Assume that$(\Lambda-\lambda I)$is injective,$\overline{R(\Lambda-\lambda I)}= Z$and$R(\Lambda-\lambda I)\subseteq Z$. For the purpose of getting a contradiction, we suppose that$\lambda\in \left(\overline{\cup_{n=1}^{\infty}\sigma(\overline{\Lambda_ {n}})}\right)^{C}$.\\ However, $$\Big(\overline{\cup_{n=1}^{\infty}\sigma(\overline{\Lambda_{n}})}\Big)^{C} \subset \Big(\bigcup_{n=1}^{\infty}\sigma (\overline{\Lambda_{n}})\Big)^{C} \\ = \bigcap_{n\geq1}\big(\sigma(\overline{\Lambda_{n}})\big)^{C} \\ = \bigcap_{n\geq1}\rho(\overline{\Lambda_{n}}),$$ which implies that,$\lambda\in \rho(\overline{\Lambda_{n}})$, for all$n\geq1$. Then we get that $$(\overline{\Lambda_{n}}-\lambda I):R(P_{n})\to R(P_{n})$$ is invertible, with$(\overline{\Lambda_{n}}-\lambda I)^{-1}$bounded. Hence, for all$z \in D(\Lambda)$we obtain $$P_{j}(\Lambda-\lambda I)z =(\overline{\Lambda_{j}}-\lambda I)P_{j}z, \quad j=1,2, \dots;$$ i.e., $$(\overline{\Lambda_{j}}-\lambda I)^{-1}P_{j}(\Lambda-\lambda I)z =P_{j}z, \quad j=1,2, \dots$$ Now, since$D(A)$is dense in$Z$, we may extend the operator$(\overline{\Lambda_{j}}-\lambda I)^{-1}P_{j}(\Lambda-\lambda I)$to a bounded operator$T_j$defined on$Z$. Therefore, it follows that $$T_{j}z = P_{j}z, \quad \forall z \in Z, \; j=1,2, \dots,$$ and $$\|T_{j}\| = \|P_{j}\| \leq 1, \quad j=1,2, \dots.$$ Since$\overline{R(\Lambda-\lambda I)}= Z$, we get $$\label{milagro} \|(\overline{\Lambda_{j}}-\lambda I)^{-1}\| \leq 1, \quad j=1,2, \dots.$$ Now we shall see that${R(\Lambda-\lambda I)}= Z$. In fact, given$z \in Z$we define$y$as $$y= \sum_{j=1}^{\infty}(\overline{\Lambda_{j}}-\lambda I)^{-1}P_{j}z.$$ >From (\ref{milagro}) we get that$y$is well defined. We shall see now that$y \in D(\Lambda)$and$(\Lambda-\lambda I)y = z$. In fact, we know that $$y \in D(\Lambda) \iff \sum_{j=1}^{\infty} \|\Lambda_{j}P_{j}y\|^2 < \infty.$$ On the other hand, we have $$\sum_{j=1}^{\infty} \|\overline{\Lambda}_{j}P_{j}y\|^2 = \sum_{j=1}^{\infty} \|\Lambda_{j}(\overline{\Lambda_{j}}-\lambda I)^{-1}P_{j}z\|^2 =\sum_{j=1}^{\infty} \|\{I + \lambda(\overline{\Lambda_{j}}-\lambda I)^{-1}\} P_{j}z\|^2.$$ So, $$\sum_{j=1}^{\infty} \|\Lambda_{j}P_{j}y\|^2 \leq \sum_{j=1}^{\infty} \|(1 + |\lambda|)^2 \|P_{j}z\|^2 = (1 + |\lambda|)^2 \|z\|^2 < \infty.$$ Then,$y \in D(\Lambda)$and$(\Lambda-\lambda I)= z$. Therefore$R(\Lambda-\lambda I)= Z$, which is a contradiction that came from the assumption:$\lambda\in \big(\overline{\cup_{n=1}^{\infty}\sigma(\overline{\Lambda_{n}})}\big)^{C}$. \end{proof} \begin{lemma}\label{LG1} Let$Z$be a separable Hilbert space,$\{S_{n}(t)\}_{t\geq0}$a family of strongly continuous semigroups with generators$\Lambda_{n}$and$\{P_{n}\}_{n\geq1}$a family of complete orthogonal projections such that $$\label{conmutatividad} \Lambda_{n}P_{m}=P_{m}\Lambda_{n},\quad n,m=1,2,\dots$$ If the operator $$\Lambda z = \sum_{n=1}^{\infty}\Lambda_{n}P_{n}z,\quad z\in D(\Lambda)$$ with $$D(\Lambda) = \{z\in Z: \sum_{n=1}^{\infty}\| \Lambda_{n}P_{n}z\|^{2}<\infty\}$$ generates a strongly continuous semigroup$\{S(t)\}_{t\geq0}$, then $$S(t)z = \sum_{n=1}^{\infty}S_{n}(t)P_{n}z,\quad z\in Z.$$ \end{lemma} \begin{proof} If$z_{0}\in Z$, then$P_{n}z_{0}\in D(\Lambda)$and the mild solution of the problem $$\label{conver1} \begin{gathered} z'(t) = \Lambda z(t) \\ z(0) = P_{n}z_{0} \end{gathered}$$ is given by$z_{n}(t) = S(t)P_{n}z_{0}$and it is a classic solution. Using (\ref{conmutatividad}) and the Hille-Yosida Theorem, we get$P_{n}S(t) = S(t)P_{n}$, which implies $$\label{conver2} S(t)z_{0} = \sum_{n=1}^{\infty}P_{n}S(t)z_{0}= \sum_{n=1}^{\infty}S(t)P_{n}z_{0}.$$ On the other hand, since$z_{n}(t)is a classic solution of (\ref{conver1}), we obtain \begin{align*} z'_{n}(t) & = \Lambda z_{n}(t)\\ & = \Lambda S(t)P_{n}z_{0}\\ & = \sum_{m=1}^{\infty}\Lambda_{m}P_{m}S(t)P_{n}z_{0}\\ & = \Lambda_{n}P_{n}S(t)P_{n}z_{0} \\ & = \Lambda_{n}S(t)P_{n}z_{0} = \Lambda_{n}z_{n}(t) \end{align*} So that,z_{n}(t) = S_{n}(t)P_{n}z_{0} = S(t)P_{n}z_{0}$and from (\ref{conver2}) we get $$S_{n}(t)z_{0} = \sum_{n=1}^{\infty}S_{n}(t)P_{n}z_{0}.$$ \end{proof} Now, applying Lemma \ref{LG} we can prove the following result. \begin{theorem}\label{T1} The operator$-A$is the infinitesimal generator of a strongly continuous semigroup$\{ T_{A}(t)\}_{t \geq 0}$in the space$Z$, given by $$\label{damp} T_{A}(t)z =\sum_{n=1}^{\infty}e^{-\lambda_{n}D t}P_nz% , \quad z\in Z , \; t \geq 0.$$ \end{theorem} \subsection{Existence and Uniqueness of Solutions} In this part we study the existence and the uniqueness of the solutions for system (\ref{ecuacion diferencial abstracta no homogenea}) in case$f^{e} \equiv 0$. That is, we analyze the homogeneous system $$\label{ecuacion diferencial abstracta homogenea} \begin{gathered} \frac{dz(t)}{dt} = -Az(t) + Lz_{t} ,\quad t>0 \\ z(0) = \phi_{0}=z_0 \\ z(s) = \phi(s), \quad s\in[-\tau,0) \end{gathered}.$$ \begin{definition}\label{mildsoluction} \rm A function$z(\cdot)$define on$[-\tau, \alpha )$is called a Mild Solution of (\ref{ecuacion diferencial abstracta homogenea}) if $$z(t) = \begin{cases} \phi(t) &-\tau\leq t <0, \\ T_{A}(t)z_{0} + \int^{t}_{0}T_{A}(t - s)Lz_{s}ds , & t\in[0,\alpha ) \end{cases}$$ \end{definition} \begin{theorem}\label{existencia} Problem (\ref{ecuacion diferencial abstracta homogenea}) admits only one mild solution defined on$[-\tau, \infty)$. \end{theorem} \begin{proof} Consider the initial function $\varphi(s) = \begin{cases} \phi(s), & -\tau\leq s <0 \\ T_{A}(s)z_{0} & s\geq 0 \end{cases}$ which belongs to$L^{2}_{\rm loc}([-\tau,\infty),Z)$. For a moment we shall set the problem on$[-\tau, I]$,$I>0$and denote by$G$the set $$G = \{\psi : \psi\in L^{2}[[-\tau,\alpha],Z] \quad\text{and} \quad | \psi - \varphi |_{L^{2}}\leq \rho, \quad \rho>0\},$$ where$\alpha >0$is a number to be determine. It is clear that$G$endowed with the norm of$L^{2}([-\tau,\alpha];Z)$is a complete metric space. Now, we consider the application$S:G \to Z$, for$z\in G$, given by $(Sz)(t) = Sz(t) = \begin{cases} \phi(t), & -\tau\leq t <0 \\ T_{A}(t)z_{0} + \int^{t}_{0}T_{A}(t - s)Lz_{s}ds, & t\in[0,\alpha] \end{cases}$ \noindent {\bf Claim 1.} There exists$\alpha> 0$such that \begin{itemize} \item[(i)]$Sz\in G$, for all$z\in G$. \item[(ii)]$S$is a contraction mapping. \end{itemize} In fact, we prove (i) as follows: $| Sz(t) -\varphi(t)| \leq \int^{t}_{0}| T_{A}(t - s)Lz_{s}| ds \leq \int^{\alpha}_{0}M| Lz_{s}| ds \leq M M_{0}(\alpha)| z |_{L^{2}([-\tau,\alpha),Z)}.$ Integrating, we have $$| Sz - \varphi |_{L^{2}} \leq K \alpha^{1/2}|z |_{L^{2}}$$ where$K = max\{M M_{0}(\alpha)/ \alpha\in[0,I]\}$. >From here we get $$| Sz - \varphi |_{L^{2}} \leq K\alpha^{1/2} (|\varphi |_{L^{2}} + \rho) , \quad z \in G.$$ Taking $$\alpha < \Big(\frac{\rho}{K(|\varphi |_{L^{2}} + \rho)}\Big)^2$$ we obtain that$Sz \in G$, for all$z \in G$. To prove (ii), we use the linearity of$L$to obtain: $$| Sz - Sw|_{L^{2}} \leq K\alpha^{1/2}| z - w|_{L^{2}}, \quad \forall z,w \in G.$$ Next, to prove that$S$it is a contraction and$S(G)\subset G$it is sufficient to choose$\alpha$so that $$\alpha < \min\big\{\Big(\frac{1}{K}\Big)^{2},\Big(\frac{\rho}{K(| \varphi |_{L^{2}} + \rho)}\Big)^{2}\big\}$$ Therefore,$S$is a contraction mapping. So, if we apply the contraction mapping Theorem, there exists a unique point$z \in G$such that$Sz = z$. i.e., $z(t) = Sz(t) = \begin{cases} \phi(t) , & -\tau\leq t <0 \\ T_{A}(t)z_{0} + \int^{t}_{0}T_{A}(t - s)Lz_{s}ds , & t\in[0,\alpha], \end{cases}$ which proves the existence and the uniqueness of the mild solution of the initial value problem (\ref{ecuacion diferencial abstracta homogenea}) on$[-\tau,\alpha]$. \noindent {\bf Claim 2.}$\alpha$could be equal to$\infty$. In fact, let$z$be the unique mild solution define in a maximal interval$[-\tau,\delta)(\delta\geq\alpha)$. \noindent By contradiction, let us suppose that$\delta < \infty$. Since$z$is a mild solution of (\ref{ecuacion diferencial abstracta homogenea}), we have that $$z(t) =T_{A}(t)z_{0} + \int_{0}^{t}T_{A}(t - s)Lz_{s}ds,\quad t \in [0,\delta).$$ Consider the sequence$\{t_{n}\}$such that$t_{n}\to \delta^{-}$. Let us prove that$\{z(t_{n})\}is a Cauchy sequence. In fact, \begin{align*} &| z(t_{n}) - z(t_{m}) |\\ & = | T_{A}(t_{n})z_{0} - T_{A}(t_ {m})z_{0}+\int_{0}^{t_{n}}T_{A}(t_{n} - s)Lz_{s}ds - \int_{0}^{t_{m}}T_{A}(t_{m} - s)Lz_{s}ds | \\ & \leq |(T_{A}(t_{n}) - T_{A}(t_{m}))z_{0}|+| \int_{0}^ {t_{n}}T_{A}(t_{n}-s)Lz_{s}ds - \int_{0}^{t_{m}}T_{A}(t_{m}-s)Lz_{s}ds | \end{align*} But, \begin{align*} &| \int_{0}^{t_{n}}T_{A}(t_{n}-s)Lz_{s}ds - \int_{0}^{t_{m}}T_{A}(t_{m}-s)Lz_{s}ds | \\ & \leq |\int_{0}^{t_{m}}(T_{A}(t_{n}-s)-T_{A}(t_{m}-s))Lz_{s}ds | + | \int_{t_{n}}^{t_{m}}T_{A}(t_{n}-s)Lz_{s}ds | \end{align*} Now, forz\in L^{2}([-\tau,\delta])$we obtain $$\int_{0}^{t_{m}}| (T_{A}(t_{n}-s)-T_{A}(t_{m}-s)) Lz_{s}| ds\leq \int_{0}^{\delta}| (T_{A}(t_{n}-s)-T_{A}(t_{m}-s)) Lz_{s}| ds$$ We know that \begin{gather*} \lim_{n,m \to \infty}| (T_{A}(t_{n}-s)-T_{A}(t_{m}-s)) Lz_{s}|=0, \\ |(T_{A}(t_{n}-s)-T_{A}(t_{m}-s)) Lz_{s}|\leq 2M| Lz_{s}| \end{gather*} But, from the hypothesis (H1), we obtain $$\int_{0}^{\delta}2M| Lz_{s}| ds\leq 2MM_{0}(\delta)| z|_{L^{2}([-\tau,\delta);Z)}$$ Therefore, applying the Lebesgue Dominated Convergence Theorem, we obtain $$\lim_{n,m \to \infty}\int_{0}^{\delta}| (T_{A}(t_{n}-s)-T_{A}(t_{m}-s)) Lz_{s}| ds=0$$ Then, since the family$\{T_{A}(t)\}_{t\geq 0}$is strongly continuous and$t_{n},t_{m}\to \delta^{-}$when$n,m\to\infty$, the sequence$\{z(t_{n})\}$is a Cauchy sequence and therefore there exists$B\in Z$such that $$\lim_{n\to \infty}z(t_{n}) = B.$$ Now, for$t\in[0,\delta)we obtain that \begin{align*} | z(t) - B | & \leq | z(t) - z(t_{n}) | + | z(t_{n}) - B |\\ & \leq | (T_{A}(t) - T_{A}(t_{n}))z_{0} | \;+\; | z(t_{n}) - B | \\ & \quad + | \int_{0}^{t_{n}}T_{A}(t_{n}-s)Lz_{s}ds - \int_{0}^{t}T_{A}(t-s) Lz_{s}ds | \end{align*} However, \begin{align*} &\big| \int_{0}^{t_{n}}T_{A}(t_{n}-s)Lz_{s}ds - \int_{0}^{t}T_{A}(t-s)Lz_{s}ds \big| \\ & \leq \int_{0}^{t_{n}}|(T_{A}(t-s)- T_{A}(t_{n}-s))Lz_{s}|ds + \int_{t}^{t_{n}}| T_{A}(t-s)Lz_{s}| ds. \end{align*} On the other hand, forz\in L^{2}([-\tau,\delta])$we get the estimate $$\int_{0}^{t_{n}}| (T_{A}(t-s)- T_{A}(t_{n}-s))Lz_{s}| ds\leq \int_{0}^{\delta}| (T_{A}(t-s)- T_{A}(t_{n}-s))Lz_{s}| ds$$ Therefore, applying the Lebesgue Dominated Convergence Theorem, we obtain $$\lim_{n \to \infty}\int_{0}^{\delta}| (T_{A}(t-s)-T_{A}(t_{n}-s)) Lz_{s}|=0$$ Then, since the family$\{T_{A}(t)\}_{t\geq 0}$is strongly continuous and$t_{n}\to \delta^{-}$when$n\to\infty$, it follows that$z(t)\to B$as$t\to \delta^{-}$. The function $\varphi(s) = \begin{cases} z(s) , & \delta-\tau\leq s < \delta \\ T_{A}(s)B , & s\geq\delta \end{cases}$ belongs to$ L^{2}_{\rm loc}([\delta-\tau,\infty),Z)$. So, if we apply again the contraction mapping Theorem to the Cauchy problem $$\label{ecuacion diferencial abstracta homogenea2} \begin{gathered} \frac{dy(t)}{dt} = -Ay(t) + Ly_{t}, \quad t>\delta \\ y(\delta) = B \\ y(s) = z(s),\quad s\in[\delta-\tau,\delta) \end{gathered}$$ where$z(\cdot)$is the unique solution of the system (\ref{ecuacion diferencial abstracta homogenea}), then we get that (\ref{ecuacion diferencial abstracta homogenea2}) admits only one solution$y(\cdot)$on the interval$[\delta-\tau,\delta + \epsilon]$with$\epsilon >0$. Therefore, the function $\widetilde{z}(s) = \begin{cases} z(s) & -\tau \leq s < \delta \\ y(s), & \delta \leq s < \delta + \epsilon \end{cases}$ is also a mild solution of (\ref{ecuacion diferencial abstracta homogenea}) which is a contradiction. So,$\delta = \infty$. \end{proof} \section{The Variation Of Constants Formula} Now we are ready to find the formula announced in the title of this paper for the system (\ref{ecuacion diferencial abstracta no homogenea}), but first we need to write this system as an abstract ordinary differential equation in an appropriate Hilbert space. In fact, we consider the Hilbert space$\mathbb{M}_{2}([-\tau,0];Z) = Z \oplus L_{2}([-\tau,0];Z)$with the usual innerproduct given by $$\Big\langle \begin{pmatrix} \phi_{01} \\ \phi_{1} \end{pmatrix}, \begin{pmatrix} \phi_{02} \\ \phi_{2} \end{pmatrix} \Big\rangle = \langle\phi_{01},\phi_{02}\rangle_{Z} + \langle\phi_{1},\phi_{2}\rangle_{L_{2}}.$$ Define the operators$T(t)$in the space$\mathbb{M}_{2}$for$t\geq 0$by $$\label{Operador Semigrupo} T(t)\begin{pmatrix} \phi_{0} \\ \phi(.) \end{pmatrix} = \begin{pmatrix} z(t) \\ z_t \end{pmatrix}$$ where$z(\cdot)$is the only mild solution of the system (\ref{ecuacion diferencial abstracta homogenea}). \begin{theorem}\label{semigroup} The family of operators$\{T(t)\}_{t\geq 0}$defined by (\ref {Operador Semigrupo}) is an strongly continuous semigroup on$\mathbb{M}_{2}$such that $$\label{semigrupo} T(t)W = \sum_{n = 1}^{\infty}T_{n}(t)Q_{n}W,\quad W\in \mathbb{M}_{2} ,\;t\geq 0,$$ where $$Q_{n} = \begin{pmatrix} P_{n} & 0 \\ 0 & \widetilde{P}_{n} \end{pmatrix},$$ with$(\widetilde{P}_{n}\phi)(s)=P_{n}\phi(s)$,$\phi\in L^{2}([-\tau,0];Z)$,$s\in[-\tau,0]$, and$\{\{T_{n}(t)\}_{t\geq0},\ n = 1,2.3,\dots \}$is a family of strongly continuous semigroups on$\mathbb{M}_{2}^{n}=Q_{n}\mathbb{M}_{2}$given in the same way as in \cite[Theorem 2.4.4]{CP3} and defined by $$T_{n}(t) \begin{pmatrix} w^{0}_{n} \\ w_{n}\end{pmatrix} = \begin{pmatrix} W^{n}(t)\\ W^{n}(t + \cdot) \end{pmatrix}, \quad \begin{pmatrix} w^{0}_{n} \\ w_{n}\end{pmatrix} \in \mathbb{M}_{2}^{n},$$ where$W^{n}(\cdot)$is the unique solution of the initial value problem $$\label{ec. con retardo} \begin{gathered} \frac{dw(t)}{dt} = -\lambda_{n}Dw(t) + L_{n}w_{t} ,\quad t>0 \\ w(0) = w_{n}^{0} \\ w(s) = w_{n}(s),\quad s\in[-\tau,0) \end{gathered}$$ and$L_{n}=L\widetilde{P}_{n}= P_{n}L$, as it is in most the case practical problems. \end{theorem} \begin{proof}[Proof of Theorem \ref{semigroup}] First, we shall prove that $$T(t)W = \sum_{n = 1}^{\infty}T_{n}(t)Q_{n}W,\quad W\in \mathbb{M}_ {2},\; t\geq0.$$ In fact, let$W= \begin{pmatrix} w_{1} \\ w_{2} \end{pmatrix} \in \mathbb{M}_{2}. \begin{align*} & \sum_{n = 1}^{\infty}T_{n}(t)Q_{n}W\\ & = \sum_{n = 1}^{\infty}T_{n}(t) \begin{pmatrix} P_{n} & 0 \\ 0 & \widetilde{P}_{n} \end{pmatrix} \begin{pmatrix} w_{1} \\ w_{2} \\ \end{pmatrix}\\ & = \sum_{n = 1}^{\infty}T_{n}(t) \begin{pmatrix} P_{n}w_{1} \\ \widetilde{P}_{n}w_{2} \end{pmatrix}\\ & = \sum_{n = 1}^{\infty} \begin{pmatrix} z^{n}(t) \\ z^{n}(t + \cdot) \end{pmatrix} \quad z^{n}(\cdot) \mbox{ is the only mild solution of (\ref{ec. con retardo})}\\ & = \sum_{n = 1}^{\infty} \begin{pmatrix} e^{\mathcal{A}_{n}t}P_{n}w_{1} + \int_{0}^{t}e^{\mathcal{A}_{n}(t-s)}L_{n} (\widetilde{P}_{n}z^{n}(s + \cdot))ds \\ (\widetilde{P}_{n}z(t + \cdot)) \end{pmatrix} \\ & = \begin{pmatrix} \sum_{n = 1}^{\infty}e^{\mathcal{A}_{n}t}P_{n}w_{1} + \int_{0}^{t}\sum_{n = 1}^{\infty}e^{\mathcal{A}_{n}(t-s)}P_{n} \Big(L\sum_{m = 1}^{\infty}(\widetilde{P}_{m}z(s + \cdot)) \Big)ds \\ \sum_{n = 1}^{\infty}(\widetilde{P}_{n}z(t + \cdot)) \end{pmatrix} \\ & = \begin{pmatrix} T_{\mathcal{A}}(t)w_{1} + \int_{0}^{t}T_{\mathcal{A}}(t-s)L z(s + \cdot)ds \\ z(t + \cdot) \end{pmatrix} \\ & = \begin{pmatrix} z(t) \\ z_{t}(\cdot) \end{pmatrix}, \quad z(\cdot)\mbox{ is the only mild solution of (\ref{ecuacion diferencial abstracta homogenea})} \\ & = T(t)W. \end{align*} In the same way as in \cite[Theorem 2.4.4]{CP3} we can prove that the infinitesimal generator of\{T_{n}(t)\}_{t\geq 0}$is given by $$\Lambda_{n}\begin{pmatrix} w_{n}^{0} \\ w_{n}(\cdot) \end{pmatrix} =\begin{pmatrix} -\Lambda_{n}Dw_{n}^{0} + L_{n}w_{n}(\cdot) \\ \frac{\partial w_{n}(\cdot)}{\partial s} \end{pmatrix}$$ with $$D(\Lambda_{n}) =\{ \begin{pmatrix} w_{n}^{0} \\ w_{n}(\cdot)\end{pmatrix} \in \mathbb{M}_{2}^{n}: w_{n} \text{ is a.c., } \frac {\partial w_{n}(\cdot)}{\partial s}\in L_{2}([-\tau,0];Q_{n}Z),\; w_{n}(0)=w_{n}^{0}\}.$$ Furthermore, the spectrum of$\Lambda_n$is discrete and given by $$\label{spectrum} \sigma(\Lambda_n) = \sigma_{p}(\Lambda_n) = \{ \lambda \in \mathbb{C} : \det (A_{n}(\lambda)) = 0 \},$$ where$A_{n}(\lambda)$is given by $$\Lambda_{n}(\lambda)z = \lambda z + \lambda_n Dz - L_{n}e^{\lambda(\cdot)}z, \ \ z \in Z_n = P_n Z,$$ which can be considered a matrix since$\mbox{dim}(Z_n) <\infty$. On the other hand,$\{Q_n \}_{n \geq 1}$is a family of complete orthogonal projection on$\mathbb{M}_{2}and $$\Lambda_{n}Q_n = Q_n \Lambda_{n}, \quad n=1,2,3, \dots$$ In fact, \begin{align*} \Lambda_{n}Q_n \begin{pmatrix} w_{n}^{0} \\ w_{n}(\cdot)\end{pmatrix} & = \Lambda_{n}\begin{pmatrix} P{n}w_{n}^{0} \\ \widetilde{P_{n}}w_{n}(\cdot) \end{pmatrix} = \begin{pmatrix} -\Lambda_{n}DP_{n}w_{n}^{0} + L_{n}\widetilde{P_{n}}w_{n}(\cdot) \\ \frac{\partial \widetilde{P_{n}}w_{n}(\cdot)}{\partial s} \end{pmatrix} \\ & = \begin{pmatrix} -\Lambda_{n}DP_{n}w_{n}^{0} + L\widetilde{P_{n}}\widetilde{P_{n}}w_ {n}(\cdot) \\ \widetilde{P_{n}}\frac{\partial w_{n}(\cdot)}{\partial s} \end{pmatrix} \\ &= \begin{pmatrix} -\Lambda_{n}DP_{n}w_{n}^{0} + P_{n}L_{n}w_{n}(\cdot) \\ \widetilde{P_{n}}\frac{\partial w_{n}(\cdot)}{\partial s} \end{pmatrix}\\ & =\begin{pmatrix} P_{n} & 0 \\ 0 & \widetilde{P}_{n} \end{pmatrix} \begin{pmatrix} -\Lambda_{n}Dw_{n}^{0} + L_{n}w_{n}(\cdot) \\ \frac{\partial w_{n}(\cdot)}{\partial s} \end{pmatrix} \\ &=Q_{n}\Lambda_{n}\begin{pmatrix} w_{n}^{0} \\ w_{n}(\cdot)\end{pmatrix} \end{align*} Now, we shall check condition (a) of Lemma \ref{LG}. To this end we need to prove the following claim. \noindent {\bf Claim.} IfW^{n}(t)$is the solution of (\ref{ec. con retardo}), then the following inequalities hold \begin{gather}\label{ineq1} \| W^{n}(t) \|_{Z} \leq c_ {2}e^{c_{1}t} \|w_{n}^{0} \|, \quad t \geq 0, \\ \int_{0}^{t}\| W^{n}(u) \|_{Z}du\leq ke^{c_{2}t}\|w_{n}^{0} \|, \quad t \geq 0. \end{gather} In fact, if we put$M_{1}=\max\{M,\| L\|\}, then we get $$\| W^{n}(t + \theta)\|_{Z} \; \leq \;M_{1}\|w_{n}^{0} \|+M_{1}^{2} \int_{0}^{t}\| W_{s}^{n}\|_{L^{2}}ds;\quad \theta\in [-\tau,0],$$ this implies $$\| W^{n}(t + \theta)\|_{Z}^{2}\leq \Big( M_{1}\|w_{n}^{0} \| + M_{1}^{2}\int_{0}^{t}\| W_{s}^{n}\|_{L^{2}}ds\Big)^{2}.$$ Next, \begin{align*} \int_{-\tau}^{0}\| W^{n}(t + \theta)\|_{Z} ^{2}d\theta & \leq \int_{-\tau}^{0}\Big( M_{1}\|w_{n}^{0} \|+M_{1}^{2}\int_{0}^{t}\| W_{s}^{n}\|_{L^{2}}ds\Big)^{2}d\theta \\ & \leq \int_{-\tau}^{0}2^{2}\Big( M_{1}^{2}\|w_{n}^{0} \|^2 +M_{1}^{4}\Big( \int_{0}^{t}\|W_{s}^{n}\|_{L^{2}}ds\Big)^{2}\Big)d\theta \\ & = 2^{2}\tau M_{1}^{2}\|w_{n}^{0} \|^2+M_{1}^{4} \Big(\int_{0}^{t}\| W_{s}^{n}\|_{L^{2}}ds\Big)^{2} \int_{-\tau}^{0}d\theta \\ & = c_{2}^2 \|w_{n}^{0} \|^2 +c_{1}^2 \Big( \int_{0}^{t}\|W_{s}^{n}\|_{L^{2}}ds\Big)^{2} \\ & \leq \Big( c_{2}\|w_{n}^{0} \|+c_{1}\Big( \int_{0}^{t}\|W_{s}^{n}\|_{L^{2}}ds\Big)\Big)^{2} \end{align*} So that $$\| W_{t}^{n}\|_{L^{2}}\leq c_{2}\|w_{n}^ {0} \|+c_{1} \Big(\int_{0}^{t}\| W_{s}^{n}\|_{L^{2}}ds\Big)$$ Therefore, applying Gronwall's lemma we obtain $$\| W_{t}^{n}\|_{L^{2}}\leq c_{2}e^{c_{1}t}\|w_{n}^{0} \|,\quad t\geq0.$$ On the other hand, we obtain the estimate \begin{align*} \| W^{n}(t)\|_{Z} & \leq \| T_{A_{n}}(t)w_{n}^{0}\| + \|\int_{0}^{t}T_{A_{n}}(t-s) L_{n}W^{n}(s+\cdot)ds\| \\ & \leq M_{1}\|w_{n}^{0} \|+ M_{1}^{2}\int_{0}^{t}\| W^{n} (s+\cdot)ds\|\\ & \leq M_{1}\|w_{n}^{0} \|+ M_{1}^{2}\int_{0}^{t}c_{1}e^{c_{2}t} \|w_{n}^{0} \|ds\\ & = \big( M_{1} + \frac{M_{1}^{2}c_{1}}{c_{2}}e^{c_{2}t}\big) \|w_{n}^{0}\| \\ & \leq ce^{c_{2} t}\|w_{n}^{0} \|, \end{align*} wherec= M_{1} + \frac{M_{1}^{2}c_{1}}{c_ {2}}$,$t\geq0. Finally, we get $$\int_{0}^{t}\| W^{n}(u)\|_{Z}du\leq ke^{c_{2} t}\|w_{n}^{0} \|,\quad k=\frac{c}{c_{2}}, \;t\geq0.$$ This completes the proof of the claim. \smallskip Now, we use the above inequalities: \begin{align*} \big\| T_{n}(t) \begin{pmatrix} w^{0}_{n} \\ w_{n}\end{pmatrix} \big\|^{2} & = \| W^{n}(t)\|_{Z}^{2} +\int_{-\tau}^{0}\| W^{n}(t + \tau)\|^{2}_{Z}d\tau \\ & = \| W^{n}(t)\|_{Z}^{2}+ \int_{t-\tau}^{t}\| W^{n}(u)\|^{2}_{Z}du \\ & \leq \| W^{n}(t)\|_{Z}^{2}+ \int_{0}^ {t}\| W^{n}(u)\|^{2}_{Z}du +\|w_{n}\|^{2}_{L^{2}}\\ & \leq \big(c_{2}^2e^{2c_{2}t} + k^2e^{2c_{2}t} \big)\|w_{n}^{0} \|^2 + \| w_{n}\|^{2}_{L^{2}}\\ & \leq g(t)2\big(\|w_{n}^{0}\|^2 + \| w_{n}\|^{2}_{L^{2}} \big), \quad n\geq 1,2,\dots . \end{align*} Hence, $$\| T_{n}(t) \| \leq g(t), \quad n\geq 1,2,\dots .$$ Therefore, applying Lemma \ref{LG}, we obtain thatT(t)$is bounded and$\{T(t)\}_{t\geq0}$is a strongly continuous semigroup on the Hilbert space$\mathbb{M}_{2}$, whose generator$\Lambda$is given by $$\Lambda W=\sum_{n=1}^{\infty}\Lambda_{n}Q_{n}W,\quad W\in D(\Lambda),$$ with $$D(\Lambda)=\big\{ W\in\mathbb{M}_{2}/\sum_{n=1}^{\infty}\|\Lambda_{n}Q_{n} W\|^{2}<\infty \big\}$$ and the spectrum$\sigma(\Lambda)$of$\Lambda$is given by $$\label{L66} \sigma(\Lambda) = \overline{\cup_{n=1}^{\infty} \sigma(\bar{\Lambda}_n)},$$ where$\bar{\Lambda}_{n} = \Lambda_{n}Q_{n} :\mathcal{R}(Q_{n}) \to \mathcal{R}(Q_{n})$. \end{proof} \begin{lemma} \label{lem4.2} Let$\Lambda$be the infinitesimal generator of the semi-group$\{T(t)\}_{t\geq0}. Then $$\Lambda\tilde{\varphi}(s) = \begin{pmatrix} -A\varphi(0) + L\phi(s) \\ \frac{\partial\phi(s)}{\partial s} \end{pmatrix}, \quad -\tau\leq s \leq 0,$$ \begin{align*} D(\Lambda) = \big\{&\begin{pmatrix} \phi_{0} \\ \phi(\cdot) \end{pmatrix} \in \mathbb{M}_{2} : \phi_{0} \in D(A), \phi \mbox{ is a.c., }\frac{\partial\phi(s)}{\partial s} \in L^{2}([-\tau,0];Z) \\ &\mbox{and } \phi(0)= \phi_{0} \big\}, \end{align*} and $$\sigma(\Lambda) = \overline{\cup_{n=1}^{\infty}\{ \lambda \in \mathbb{C} : \mbox{det}(\Lambda_{n}(\lambda)) = 0 \} }$$ \end{lemma} \begin{proof} Consider\begin{pmatrix} \phi_{0} \\ \phi(\cdot) \end{pmatrix}$in$\mathbb{M}_{2}. Then \begin{align*} \Lambda W&=\Lambda \begin{pmatrix} \phi_{0} \\ \phi(\cdot)\end{pmatrix} = \sum_{n=1}^{\infty} \Lambda_{n}Q_{n}W \\ & = \sum_{n=1}^{\infty}\Lambda_{n}\begin{pmatrix} P_{n} & 0 \\ 0 & \widetilde{P}_{n} \end{pmatrix} \begin{pmatrix} \phi_{0} \\ \phi(\cdot) \end{pmatrix} = \sum_{n=1}^{\infty}\Lambda_{n}\begin{pmatrix} P_{n}\phi_{0} \\ \widetilde{P_{n}}\phi(\cdot) \end{pmatrix} \\ & = \sum_{n=1}^{\infty}\begin{pmatrix} -\Lambda_{n}D\widetilde{P_{n}}\phi(0) + L_{n}\widetilde{P_{n}}\phi\\ \frac{\partial \widetilde{P}_{n}\phi(\cdot)}{\partial (s)} \end{pmatrix}\\ & = \begin{pmatrix} -\sum_{n=1}^{\infty}\Lambda_{n}DP_{n}\phi(0) + L\sum_ {n=1}^{\infty}\widetilde{P}_{n}\phi\\ \frac{\partial}{\partial s}\Big(\sum_{n=1}^{\infty} \widetilde{P}_{n}\phi(\cdot)\Big) \end{pmatrix} \\ & = \begin{pmatrix} -A\phi(0) + L\phi(\cdot) \\ \frac{\partial \phi(\cdot)}{\partial s} \end{pmatrix}. \end{align*} The other part of the lemma follows from (\ref{L66}) \end{proof} Therefore, the systems (\ref{ecuacion diferencial abstracta homogenea}) and (\ref{ecuacion diferencial abstracta no homogenea}) are equivalent to the following two systems of ordinary di-fferential equations in\mathbb{M}_{2}$respectively: $$\label{ecuacion ordinaria homogenea} \begin{gathered} \frac{dW(t)}{dt} = \Lambda W(t) ,\quad t>0 \\ W(0) = W_{0} = (\phi_{0},\phi(\cdot)) \end{gathered}$$ and $$\label{ecuacion ordinaria no homogenea} \begin{gathered} \frac{dW(t)}{dt} = \Lambda W(t) + \Phi(t),\quad t>0 \\ W(0) = W_{0} = (\phi_{0},\phi(\cdot)), \end{gathered}$$ where$\Lambda$is the infinitesimal generator of the semigroup$\{T(t)\}_{t\geq0}$and$\Phi(t) = (f^{e}(t),0)$. The steps we have taken to arrive here allow us to conclude the proof of the main result of this work: The Variation of Constants Formula for Functional Partial Parabolic Equations. This result is presented as the final Theorem of the this work. \begin{theorem} \label{thm4.3} The abstract Cauchy problem in the Hilbert space$\mathbb{M}_{2}$, \begin{gather*} \frac{dW(t)}{dt} = \Lambda W(t) + \Phi(t),\quad t>0 \\ W(0) = W_{0} \end{gather*} where$\Lambda$is the infinitesimal generator of the semigroup$\{T(t)\}_{t\geq0}$and$\Phi(t) = (f^{e}(t),0)$is a function taking values in$\mathbb{M}_{2}$, admits one and only one mild solution given by $$\label{variacion de parametro} W(t) = T(t)W_{0} + \int_{0}^{t}T(t-s)\Phi(s)ds$$ \end{theorem} \begin{corollary} \label{coro4.4} If$z(t)$is a solution of \eqref{ecuacion diferencial abstracta no homogenea}, then the function$W(t) := (z(t),z_{t}) $is solution of the equation \eqref{ecuacion ordinaria no homogenea} \end{corollary} \section{Conclusion} As one can see, this work can be generalized to a broad class of functional reaction diffusion equation in a Hilbert space$Z$of the form $$\label{ecuacion abstracta} \begin{gathered} \frac{dz(t)}{dt} = {\mathcal{A}}z(t) + Lz_{t} + F (t),\quad t>0\\ z(0) = \phi_{0} \\ z(s) = \phi(s),\quad s\in[-\tau,0), \end{gathered}$$ where $$\label{L4} {\mathcal{A}}z = \sum_{n=1}^{\infty} A_n P_n z, \quad z \in D({\mathcal{A}}),$$ where$L : L^{2}([-\tau,0];Z) \to Z$is linear and bounded$F :[-\tau, \infty) \to Z$is a suitable function. Some examples of this class are the following well known systems of partial differential equations with delay: The equation modelling a damped flexible beam: $$\begin{gathered} {\partial^2 z \over \partial t^2} =-{\partial3 z \over \partial3 x} +2\alpha {\partial3 z \over \partial t \partial2 x} + z(t-\tau,x) + f(t,x) \quad t \geq 0, \; 0 \leq x \leq 1 \\ z(t,1) = z(t,0) = {\partial2 z \over \partial2 x}(0,t)= {\partial2 z \over \partial2 x}(1,t)=0,\\ z(0,x) = \phi_{0}(x), \quad {\partial z \over \partial t}(0,x) = \psi_{0}(x), \quad 0 \leq x \leq 1 \\ z(s,x) = \phi(s,x), \quad {\partial z \over \partial t}(s,x) = \psi(s,x), \quad s \in [-\tau, 0), \; 0 \leq x \leq 1 \\ \end{gathered}$$ where$\alpha >0$,$f:\mathbb{R} \times [0,1] \to \mathbb{R}$is a smooth function,$\phi_{0}, \psi_{0} \in L^{2}[0,1]$and$\phi, \psi \in L^{2}([-\tau, 0]; L^{2}[0,1])$. The strongly damped wave equation with Dirichlet boundary conditions $$\label{W1} \begin{gathered} {\partial^2 w \over \partial t^2}+ \eta(-\Delta)^{1/2}{\partial w \over \partial t} + \gamma(- \Delta) w = Lw_t + f(t,x), \quad t \geq 0, \; x \in \Omega, \\ w(t,x) = 0, \quad t \geq 0, \; x \in \partial \Omega. \\ w(0,x) =\phi_{0}(x), \quad {\partial z \over \partial t}(0,x) =\psi_{0}(x), \quad x \in \Omega, \\ w(s,x) = \phi(s,x), \quad {\partial z \over \partial t}(s,x) = \psi(s,x), \quad s \in [-\tau, 0), \; x \in \Omega, \end{gathered}$$ where$\Omega$is a sufficiently smooth bounded domain in$\mathbb{R}^N$,$f:\mathbb{R} \times \Omega \to \mathbb{R}$is a smooth function,$\phi_{0}, \psi_{0} \in L^{2}(\Omega)$and$\phi, \psi \in L^{2}([-\tau, 0]; L^{2}(\Omega))$and$\tau\geq 0$is the maximum delay, which is supposed to be finite. We assume that the operators$L : L^{2}([-\tau,0];Z) \to Z$is linear and bounded and$Z = L^{2}(\Omega)$. The thermoelastic plate equation with Dirichlet boundary conditions $$\label{Th1} \begin{gathered} {\partial2 w \over \partial2 t}+ \Delta^{2}w + \alpha \Delta \theta = L_{1}w_t + f_{1}(t,x) \quad t \geq 0, \; x \in \Omega, \\ {\partial \theta \over \partial t}- \beta \Delta \theta - \alpha \Delta {\partial w \over \partial t} = L_{2}\theta_t + f_{2}(t,x) \quad t \geq 0, \; x\in \Omega, \\ \theta = w = \Delta w= 0, \quad t \geq 0, \; x \in \partial \Omega, \\ w(0,x) =\phi_{0}(x), \quad {\partial w \over \partial t}(0,x) = \psi_{0}(x), \quad \theta(0,x) =\xi_{0}(x) \quad x \in \Omega, \\ w(s,x) = \phi(s,x), \quad {\partial w \over \partial t}(s,x) = \psi(s,x), \quad \theta(0,x) =\xi(s,x), \quad s \in [- \tau, 0), \; x \in \Omega, \end{gathered}$$ where$\Omega$is a sufficiently smooth bounded domain in$\mathbb{R}^N$,$f_1, f_2 :\mathbb{R} \times \Omega \to \mathbb{R}$are smooth functions,$\phi_{0}, \psi_{0}, \xi_{0} \in L^{2}(\Omega)$and$\phi, \psi, \xi \in L^{2}([-\tau, 0]; L^{2}(\Omega))$and$\tau\geq 0$is the maximum delay, which is supposed to be finite. We assume that the operators$L_1, L_2 : L^{2}([-\tau,0];Z) \to Z$are linear and bounded and$Z = L^{2}(\Omega)$. \begin{thebibliography}{00} \bibitem {Alek} V. M. Alekseev, An estimate for perturbations of the solutions of ordinary diffe rential equations'' Westnik Moskov Unn. Ser. 1. Math. Merk. 2 (1961).28-36. \bibitem {Ba} H. T. Banks, The representation of solutions of linear functional di-fferential equations'' J. Diff. Eqns., 5 (1969). \bibitem {BT} J. G. Borisovich and A. S. Turbabin, On the Cauchy problem for linear non-homegeneous di-fferential equations with retarded arguments'' Soviet Math. Dokl., 10 (1969), pp. 401-405 \bibitem {CP1} R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems'', Lecture Notes in Control and Information Sciences, Vol. 8. Springer Verlag, Berlin (1978). \bibitem {CP3} R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory'', Tex in Applied Mathematics, Vol. 21. Springer Verlag, New York (1995). \bibitem {IV} A. E. Ize and A. Ventura, Asymptotic behavior of a perturbed neutral funcional differential equation related to the solution of the unperturbed linear system", Pacific J. Math., Vol. 1. pp. 57-91 (1984) \bibitem {DH} D. Henry, Geometric theory of semilinear parabolic equations'' , Springer, New York (1981). \bibitem {HLe} H. Leiva, A Lemma on$C_{0}\$-Semigroups and Appications PDEs Systems'', Quaestions Mathematicae 26(2003), 247-265. \bibitem {LG} J. Lopez G. and R. Pardo San Gil, Coexistence in a Simple Food Chain with Diffusion'' J. Math. Biol. (1992) 30: 655-668. \bibitem {MSt} R. H. Marlin and H. L. Struble, Asymptotic Equivalence of Nonlinear Systems'', Journal of Differential Equations, Vol. 6, 578-596(1969). \bibitem {MS} R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence", J. reine angew. Math. 413 (1991),1-35. \bibitem {LO2} Luiz A. F. de Oliveira, On Reaction-Diffusion Systems'' E. Journal of Differential Equations, Vol. 1998(1998), N0. 24, pp. 1-10. \bibitem {SHA} G. A. Shanholt, A nonlinear variations of constants formula for functional differential equations'', Math Systems Theory 6, No. 5 (1973). \bibitem {TW} C. C. Travis and G. F. Webb (1974), Existence and Stability for Partial Functional Differential Equations'', Transation of the A.M.S, Vol 200. \bibitem {JW} J. Wu (1996), Theory and Applications of Partial Functional Differential Equatios", Springer Verlag, Applied Math. Science Vol 119. \end{thebibliography} \end{document}