Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: Lp-theory

The aim of this paper consists in solving integrodifferential problem of type (1.1)–(1.2) that may degenerate both in space and time. More precisely, {Mp(t)}t∈[0,T]{Mp(t)}t∈[0,T] is a family of multiplication operators related to a scalar function m(t,x)m(t,x) that may vanish, while {Lp(t)}t∈[0,T]{Lp(t)}t∈[0,T] is the realization of a family of linear second-order differential operators, with smooth coefficients, {L(t)}t∈[0,T]{L(t)}t∈[0,T], {Lp(t)}{Lp(t)} being invertible for all t∈[0,T]t∈[0,T]. Moreover, {Bp(t,s)}t,s∈[0,T],s≤t{Bp(t,s)}t,s∈[0,T],s≤t is the realization of a family {B(t,s)}t∈[0,T],s≤t{B(t,s)}t∈[0,T],s≤t of linear second-order differential operators with smooth coefficients. Finally, the scalar functions a and b   are such that 1/a1/a and b/ab/a are Hölder-continuous with suitable Hölder exponents.