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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 $$ du/dt + A(t)u = f(t),\quad 0<t\le T,\quad u(0)=u\sb 0 \tag E$$ in a Banach space $X$ in which $A(t)$, $0\le t\le T$, 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 ${\cal D}(A(t))$ of $A(t)$ are independent of $t$ and that ${\cal D}(A(t))$ are completely variable with $t$ were discussed. Now ${\cal D}(A(t))$ are assumed to vary with $t$ temperately in the sense that $$ \|A(t)(\lambda-A(t))^{-1}(A(t)^{-1}-A(s)^{-1})\|_{{\cal L}(X)} \le N |t-s|^{\mu}(|\lambda|+1)^{-\nu} $$ with some suitable exponents $0<\mu$, $\nu\le 1$. Under this condition, a fundamental solution (evolution operator) $U(t,s)$, $0\le t,s\le T$, on $X$ for (E) is constructed. The strict solution $u$ to (E) is given in the form $$ u(t)=U(t,0)u\sb 0+\int^{t}_{0}U(t,\tau)f(\tau)d\tau,\quad 0\le t\le T.$$
Reviewer: A.Yagi

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