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Existence and sharp regularity results for linear parabolic non- autonomous integro-differential equations. (English) Zbl 0603.45019
The authors solve the linear abstract integro-differential equation of parabolic type \[ du(t)/dt-A(t)u(t)- \int^{t}_{0}B(t,s)u(s)ds=f(t),\quad t\in [0,T],\quad u(0)=u_ 0 \] in a Banach space. It is assumed that the operator A(t) which is not necessarily densely defined generates an analytic semigroup for each fixed t and B(t,s) is an operator whose domain contains that of A(s) for each \(0\leq s<t\leq T.\)
Based on their theory of parabolic evolution equations the authors establish the existence, uniqueness and maximal regularity of the strict solution for any Hölder continuous inhomogeneous term and initial value satisfying a certain compatibility condition.
The main result is applied to second order parabolic integro-partial differential equations in which B(t,s) is a differential operator of second order in case of one space variable and one of first order in case of several space variables. The theory of interpolation spaces by the authors as well as some of other authors such as Da Prato, Grisvard, Lunardi, Sinestrari also plays an important role.
Reviewer: H.Tanabe

45N05 Abstract integral equations, integral equations in abstract spaces
45K05 Integro-partial differential equations
Full Text: DOI
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