Nonautonomous partial functional differential equations; existence and regularity

Abstract

The aim of this work is to establish several results on the existence and regularity of solutions for some nondensely nonautonomous partial functional differential equations with finite delay in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family under the conditions introduced by N. Tanaka.We show the local existence of the mild solutions which may blow up at the finite time. Secondly,we give sufficient conditions ensuring the existence of the strict solutions. Finally, we consider a reaction diffusion equation with delay to illustrate the obtained results.

Existence results on nonautonomous partial functional differential equations with state-dependent infinite delay

“…Its applications in the life sciences range from simple predator-prey models to complicated signal traduction pathways in biological cells, in physics from the motion of a pendulum to complex climate models, and beyond that to further fields as diverse as chemistry (reaction kinetics), economics, engineering, sociology, demography, and biosciences. Nonautonomous differential equations has received the great attention see for instance the works [22,26,28,40,42,47,51] and some recent works [9,[37][38][39]. For some applications, we refer the reader to the handbook by Peter E. Kloeden and Christian Pötzsche [44].…”

“…The literature related to partial nonautonomous functional differential equations with delay for which ρ(t, ψ) = t is very extensive and we refer the reader to the papers in [9,13,25,37,38,40,47] concerning this case. Recently Kpoumié et al in [9], investigate several results on the existence of solutions of the following nonautonomous equation :…”