The higher order differentiability of solutions of abstract evolution equations

Abstract

In this paper, the regularity of the solution of the initial value problem for the abstract evolution equation (o.i) 4r + A (t)u = /(*) > % (°) e x > at and the associated homogeneous equation (0.2) 4£-+ A(t)u = 0 , u(0)eX, Q^tST at in a Banach space X is considered.Here u -u(t) and f(t) are functions from [0, T] to X and A(t) is a function on [0, T] to the set of (in general) unbounded linear operators acting in X a DEFINITION, nit) is called a strict solution of (0.1) or (0.2) in (s, T] if (i) u(t) is strongly continuous in the closed interval [s, T] and is strongly continuously diff erentiable in the semi closed interval (s, Γ], (ii) u(t)eD(A(t)), the domain of A(t), for each te(s, T],(iii) u(t) satisfies (0.1) resp.(0.2) in (s, T], u(s) coinciding with the given initial value at t -s.It is assumed that A(t) for each te[Q, T] satisfies the following conditions.(i) -A(t) generates a semigroup exp(-sA(t)) of operators analytic in the sector | arg s\ < θ, s Φ 0,0 < θ < π/2, (ii) For any complex number 1 satisfying | arg λ \ < π/2 + θ, 0 < 0 < π/2, (dldt)(λ + A(ί))" 1 exists in the operator topology and that there exist constants N and p independent of t and λ with N > 0 9 0 ^ p <1 such that The main result proved in the paper can be stated as follows.If, in addition to the above assumptions, Aψ)-1 e C n+α [0, T] in the uniform operator topology, B(t), a bounded operator for each ίe[0, Γ] is of class C n ~^[0 9 T], and f(t)e O-1+ ?[0,T] in the strong topology, then the unique strict solution u(t) of 4^ + (A(t) + B(t))u = /(t) , u(0) G X. 0 ^ t g Γ belongs to the class C n+8 [s θ9 T], s 0 > 0 arbitrary, δ > 0 depending on α, β 9 γ and /o.In this no assumption regarding the constancy of the domain D(A(t)) is made.

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF