Singular evolution integro-differential equations with kernels defined on bounded intervals

Abstract

We study linear singular first-order integro-differential Cauchy problems in Banach spaces. The adjective “singular” means here that the integro-differential equation is not in normal form neither can it be reduced to such a form. We generalize some existence and uniqueness theorems proved in [5] for kernels defined on the entire half-line R ₊ to the case of kernels defined on bounded intervals removing the strict assumption that the kernel should be Laplace-transformable.

Particular attention is paid to single out the optimal regularity properties of solutions as well as to point out several explicit applications relative to singular partial integro-differential equations of parabolic and hyperbolic type.

Keywords:

• Abstract linear singular first-order integro-differential equations
• Existence and uniqueness results
• Maximal regularity of solutions
• Linear singular partial integro-differential equations of parabolic type
• 2000 Mathematics Subject Classifications: Primary 45Q05
• Secondary 45K05
• 34G10
• 35K20
• 35K35

Acknowledgements

The first two authors are members of G.N.A.M.P.A of the Italian Istituto Nazionale di Alta Matematic (I.N.d.A.M).

Work partially supported by the Project FIRB 2001 Analisi di equazioni a derivate parziali lineari e non lineari: aspetti metodologici, modellistici, applicazioni, of the Italian Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica and by University of Bologna Funds for selected research topics.

Notes

However, in Section 3 less regular kernels k will be considered.