Degenerate integrodifferential equations of Volterra type in Banach space

Il contributo è un articolo scientifico originale, non pubblicato altrove in nessuna forma. Introduction: "This paper is concerned with the following degenerate integro-differential equation of parabolic type: $$\gathered \frac{d}{dt}(M(t)u(t))+L(t)u(t)+\int_0^t K(t,s)u(s)ds=f(t),\\0\leq t\leq T, M(t)u(t)|_{t=0}=M(0)u_0.\endgathered\tag1$$ This type of equation without the integral terms is discussed in great detail in Chapters III and IV of the book \ref[A. Favini and A. Yagi, Degenerate differential equations in Banach spaces, Dekker, New York, 1999; MR1654663 (99i:34079)] based on the theory of analytic semigroups generated by multi-valued linear operators. In Section 1, making intensive use of the results of [op. cit. (Chapter IV)], we show the existence and uniqueness of a solution to the non-autonomous equation (1) described above. We use the idea of M. G. Crandall and J. A. Nohel \ref[Israel J. Math. 29 (1978), no. 4, 313--328; MR0477910 (57 \#17410) (Proposition 1)] to deal with the integral term. Section 2 is devoted to the autonomous case based on the results of Chapter III of [A. Favini and A. Yagi, op. cit.]. In the non-autonomous case rather restrictive assumptions are required for certain constants $\alpha, \beta $ which appear in the hypothesis for the operators $M(t)$ and $L(t)$. In the autonomous case this restriction is considerably relaxed. Finally in Section 3 we consider the case in which the assumption (P) of [A. Favini and A. Yagi, op. cit. (p. 92)] is satisfied with $\alpha =\beta=1$. In this case using the method of J. Prüss \ref[J. Integral Equations 5 (1983), no. 3, 211--236; MR0702432 (85d:45026)], we show the existence and uniqueness of a function satisfying the integro-differential equation (1) with the integral term understood in the improper sense under a weaker assumption on the initial data."