×

Regular solutions for time dependent abstract integrodifferential equations with singular kernel. (English) Zbl 0638.45007

The problems of existence, uniqueness and regularity of solutions to the linear integrodifferential equation \[ u'(t)=A(t)u(t)+ \int^{t}_{0} B(t,s)u(s)ds+f(t),\quad 0<t\leq T \] depending on the regularity of the initial value \(u(0)=u_ 0\) are considered. For each \(t\in [0,T]\) the operator \(A(t): F\subset E\to E\) generates an analytic semigroup \(e^{sA(t)}\) in E and \(B(t,s)\) \((0\leq s\leq t\leq T)\) is a linear operator from F to E. A(t) and f(t) are Hölder continuous with exponent \(\alpha\in (0,1)\) and, for each \(y\in F\) \(B(t,s)y\) is \(\alpha\)-Hölder continuous, with respect to t and \(L^ p\), with respect to s, for some \(p>1\). As an example of applications a linear parabolic Volterra equation is given.
Reviewer: I.Foltynska

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acquistapace, P., Existence and maximal time regularity for linear parabolic integro-differential equations, (Proceedings, Integrodifferential Evolution Equations and Applications. Proceedings, Integrodifferential Evolution Equations and Applications, Trento, 1984. Proceedings, Integrodifferential Evolution Equations and Applications. Proceedings, Integrodifferential Evolution Equations and Applications, Trento, 1984, J. Integral Equations, 10 (1985)), 5-43 · Zbl 0584.45012
[2] Acquistapace, P.; Terreni, B., Existence and sharp regularity for linear parabolic nonautonomous integro-differential equations, Israel J. Math., 53, 257-303 (1986) · Zbl 0603.45019
[3] Butzer, P. L.; Berens, H., Semigroups of Operators and Approximation (1967), Springer: Springer Berlin · Zbl 0164.43702
[4] Da Prato, G.; Sinestrari, E., Hölder regularity for nonautonomous abstract parabolic equations, Israel J. Math., 42, 1-19 (1982) · Zbl 0495.47031
[5] Lunardi, A., Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations, Math. Nachr., 121, 295-318 (1985) · Zbl 0568.47035
[6] Lunardi, A.; Sinestrari, E., \(C^α\)-regularity for nonautonomous linear integrodifferential equations, J. Differential Equations, 63, 88-116 (1986) · Zbl 0596.45019
[7] Prüss, J., On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations, 5, 211-236 (1983) · Zbl 0518.45008
[8] Sinestrari, E., On the abstract Cauchy problem in spaces of continuous functions, J. Math. Anal. Appl., 107, 16-66 (1985) · Zbl 0589.47042
[9] Stewart, H. B., Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc., 199, 141-162 (1974) · Zbl 0264.35043
[10] Taibleson, M. H., On the theory of Lipschitz spaces of distributions on euclidean \(n\)-space. 1. Principal properties, J. Math. Mech., 13, 407-479 (1964) · Zbl 0132.09402
[11] Yosida, K., Functional Analysis (1968), Springer: Springer Berlin · Zbl 0217.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.