Abstract
In the present paper, the well-posedness of the initial value problem for the delay differential equation d v ( t ) d t +Av(t)=B(t)v(t−ω)+f(t), t≥0; v(t)=g(t) (−ω≤t≤0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A)⊆D(B(t)) is studied. Two main theorems on well-posedness of this problem in fractional spaces E α are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.MSC:35G15.
Highlights
1 Introduction The stability of delay ordinary differential and difference equations and delay partial differential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works and insight has developed over the last three decades
The theory of stability and coercive stability of delay partial differential and difference equations with unbounded operators acting on delay terms has received less attention than delay ordinary differential and difference equations
It is well known that various initial-boundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form dv(t) dt
Summary
The stability of delay ordinary differential and difference equations and delay partial differential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works (see [ – ] and the references therein) and insight has developed over the last three decades. The coercive stability estimates in Hölder norms for the solutions of the mixed problem of the delay differential equations of the parabolic type are obtained. In Section , the coercive stability estimates in Hölder norms for the solutions of the initialboundary value problem for delay parabolic equations are obtained. We consider the initial value problem ( ) for delay differential equations of parabolic type in the space C(Eα) of all continuous functions v(t) defined on the segment [ , ∞) with values in a Banach space Eα. . This result completes the proof of Theorem Note that these abstract results are applicable to the study of stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. To the study of the coercive stability of initial-boundary value problem for delay parabolic equations are given In Section , applications of Theorem . to the study of the coercive stability of initial-boundary value problem for delay parabolic equations are given
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