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Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. II. (English) Zbl 0706.35060

[For part I, see ibid. 32, No.1, 107-124 (1989; Zbl 0693.35074).]

This paper continues the study of a linear evolution equation of parabolic type

du/dt+A(t)u=f(t),0<tT,u(0)=u 0 (E)

in a Banach space X in which A(t), 0tT, are the generators of infinitely differentiable semigroups on X. We interpolate two results presented in part I, in which the two extreme cases that the domains 𝒟(A(t)) of A(t) are independent of t and that 𝒟(A(t)) are completely variable with t were discussed. Now 𝒟(A(t)) are assumed to vary with t temperately in the sense that

A(t)(λ-A(t)) -1 (A(t) -1 -A(s) -1 ) (X) N|t-s| μ (|λ|+1) -ν

with some suitable exponents 0<μ, ν1. Under this condition, a fundamental solution (evolution operator) U(t,s), 0t,sT, on X for (E) is constructed. The strict solution u to (E) is given in the form

u(t)=U(t,0)u 0 + 0 t U(t,τ)f(τ)dτ,0tT·

Reviewer: A.Yagi
MSC:
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
47D06One-parameter semigroups and linear evolution equations
34G10Linear ODE in abstract spaces