Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 130, pp. 1-20.

Title: Variation of constants formula for functional  parabolic
       partial differential equations

Authors: Alexander Carrasco (Univ. Centroccidental, Barquisimeto, Venezuela)
         Hugo Leiva         (Univ. de Los Andes, Merida, Venezuela)

Abstract: 
 This paper presents a variation of constants formula for
 the  system of functional parabolic partial differential equations
 $$\displaylines{
 \frac{\partial u(t,x)}{\partial t}
 = D\Delta u+Lu_t+f(t,x), \quad t>0,\; u\in \mathbb{R}^n  \cr
 \frac{\partial u(t,x)}{\partial \eta} = 0, \quad t>0, \; x\in \partial\Omega
 \cr
 u(0,x) =  \phi(x)  \cr
 u(s,x) = \phi(s,x), \quad s\in[-\tau,0),\; x\in\Omega\,.
 }$$
 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.

Submitted July 2, 2007. Published October 05, 2007.
Math Subject Classifications: 34G10, 35B40.
Key Words: Functional partial parabolic equations;
    variation of constants formula; strongly continuous semigroups.