
doi: 10.1007/bf02761404
The paper deals with the initial value problem \[ (P)\quad \dot u(t)=Au(t)+A_ 1u(t-r)+\int^{0}_{-r}a(s)A_ 2u(t+s)ds+f(t),\quad 00)\), where u satisfies (P) with \(f=0\), and \(u_ t(s)=u(t+s)\), \(s\in [-r,0]\). Of special concern is the spectrum of the infinitesimal generator of S(t). The asymptotic behaviour of solutions of (P) where \(f=0\) is next investigated in the discrete delay \((A_ 2=0)\), and distributed delay \((A_ 1=0)\) case, respectively. As an application of the theory, a parabolic second order integrodifferential equation with delay is finally considered.
parabolic second order, infinitesimal generator, delay, asymptotic behaviour, Hilbert space, Abstract integral equations, integral equations in abstract spaces, Asymptotics of solutions to integral equations, solution semigroup, spectrum, Integro-partial differential equations, Groups and semigroups of linear operators, initial value problem
parabolic second order, infinitesimal generator, delay, asymptotic behaviour, Hilbert space, Abstract integral equations, integral equations in abstract spaces, Asymptotics of solutions to integral equations, solution semigroup, spectrum, Integro-partial differential equations, Groups and semigroups of linear operators, initial value problem
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