Two point problems and analyticity of solutions of abstract parabolic equations

Fort ∈ [a, b], letA(t) be 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. Existence and uniqueness results are obtained for weak and strict solutions of two-point problems of the type (du/dt)−A(t) u(t) =f(t),E 1(α)u (α)=u α,E 2 (β)u (β)=u β. Here [α, β) χ- [a, b],E 1 (α) andE 2 (β) are spectral projections associated withA(α) andA(β) respectively, andA(α)E 1 (α) and =A (β)E 2 (β) are infinitesimal generators of analytic semigroups. WhenA(t) andf(t) are analytic in a convex, complex neighborhoodO of [a, b] we show that for someθ i ,i=1,2, any solution ofdu/dt =A(t)u (t)=f(t) in [a, b] is analytic and satisfies the above equation in the setO∩{t; t ≠ a, t ≠ b, | arg (t −a)