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.