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 ∂u(t, x)/∂t = DΔu + Lut + ƒ(t, x), t > 0, u ∈ ℝn ∂u(t, x)/∂η = 0, t > 0, x ∈ ∂Ω u(0, x) = φ(x) u(s, x) = φ(s, x), s ∈ [-τ, 0), x ∈ Ω. Here Ω is a bounded domain in ℝn, the n x 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 ut(x)(s) = u(t + s, x) is used.