Electron. J. Diff. Eqns., Vol. 2007(2007), No. 130, pp. 1-20.

Variation of constants formula for functional parabolic partial differential equations

Alexander Carrasco, Hugo Leiva

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 5, 2007.
Math Subject Classifications: 34G10, 35B40.
Key Words: Functional partial parabolic equations; variation of constants formula; strongly continuous semigroups.

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Alexander Carrasco
Universidad Centroccidental Lisandro Alvarado
Decanato de Ciencias, Departamento de Matemática
Barquisimeto 3001, Venezuela
email: acarrasco@ucla.edu.ve
Hugo Leiva
Universidad de los Andes
Department of Mathemátics
Mérida 5101, Venezuela
email: hleiva@ula.ve

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