Regularity properties of solutions of some abstract parabolic equations

Consider the equation (i) (da/dt)—A(t)u(t)=f(t) where fort ∈ [a, b],A(t) is a densely defined and closed linear operator in a Banach spaceX. Assume the existence of bounded projectionsE i(t),i=1, 2, such thatA(t) E 1(t) and —A(t)E 2(t) are infinitesimal generators of analytic semigroups andA(t) is completely reduced by the direct sum decompositionX = Σ b = 1/2 ⊕E i (t)X. We show that any solutionu(t) of (i) is inC ∞(a, b) and satisfies the inequalities (1.2) provided thatf(t) andA(t) are infinitely differentiable in [a, b] in a suitable sense. In caseA(t) andf(t) are in a Gevrey class determined by the constants {M n} we have (1. 3). Applications are given to the study of solution of (i) where fort ∈ [a, b]A(t) is the unbounded operator inH 0,p (G) associated with an elliptic boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ > 0.