Variation of constants formula for functional parabolic partial differential equations

Abstract

This paper presents a variation of constants formula for the system of functional parabolic partial differential equations $$displaylines{ frac{partial u(t,x)}{partial t} = DDelta u+Lu_t+f(t,x), quad t>0,; uin mathbb{R}^n cr frac{partial u(t,x)}{partial eta} = 0, quad t>0, ; xin partialOmega cr u(0,x) = phi(x) cr u(s,x) = phi(s,x), quad sin[-au,0),; xinOmega,. }$$ Here $Omega$ is a bounded domain in $mathbb{R}^n$, the $nimes n$ 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.

Keywords

• Functional partial parabolic equations
• variation of constants formula
• strongly continuous semigroups