Vol. 22, No. 3, 1967

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The higher order differentiability of solutions of abstract evolution equations

Ponnaluri Suryanarayana

Vol. 22 (1967), No. 3, 543–561

In this paper, the regularity of the solution of the initial value problem for the abstract evolution equation

du+ A (t)u = f (t),u(0) ∈ X, 0 ≦ t ≦ T

and the associated homogeneous equation

du+ A (t)u = 0,u(0) ∈ X, 0 ≦ t ≦ T

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.

Definition. u(t) 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 differentiable in the semiclosed interval (s,T],

(ii) u(t) D(A(t)), the domain of A(t), for each t (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 t [0,T] satisfies the following conditions.

(i) A(t) generates a semigroup exp(sA(t)) of operators analytic in the sector |arg s| < 𝜃,s0,0 < 𝜃 < π∕2,

(ii) For any complex number λ satisfying |arg λ| < π∕2 + 𝜃, 0 < 𝜃 < π∕2,(∂∕∂t)(λ + A(t))1 exists in the operator topology and that there exist constants N and ρ independent of t and λ with N > 0,0 ρ < 1 such that

∥ ∂-(λ+ A(t))− 1∥ ≦ N |λ|p−1

The main result proved in the paper can be stated as follows. If, in addition to the above assumptions, A(t)1 Cn+α[0,T] in the uniform operator topology, B(t), a bounded operator for each t [0,T] is of class Cn1+β[0,T], and f(t) Cn1+γ[0,T] in the strong topology, then the unique strict solution u(t) of

dt-+ (A (t)+ B (t))u = f(t),u (0) ∈ X.0 ≦ t ≦ T

belongs to the class Cn+δ[s0,T],s0 > 0 arbitrary, δ > 0 depending on α,β,γ and ρ. In this no assumption regarding the constancy of the domain D(A(t)) is made.

Mathematical Subject Classification
Primary: 34.95
Received: 11 March 1965
Published: 1 September 1967
Ponnaluri Suryanarayana