*(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

in a Banach space $X$ in which $A\left(t\right)$, $0\le t\le T$, 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 $\mathcal{D}\left(A\right(t\left)\right)$ of $A\left(t\right)$ are independent of $t$ and that $\mathcal{D}\left(A\right(t\left)\right)$ are completely variable with $t$ were discussed. Now $\mathcal{D}\left(A\right(t\left)\right)$ are assumed to vary with $t$ temperately in the sense that

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

##### MSC:

35G10 | Initial value problems for linear higher-order PDE |

35K25 | Higher order parabolic equations, general |

47D06 | One-parameter semigroups and linear evolution equations |

34G10 | Linear ODE in abstract spaces |