## On abstract parabolic fundamental solutions.(English)Zbl 0616.47032

We construct the evolution operator associated with the abstract evolution equation of parabolic type $$\dot u+A(t)u=f(t)$$ in a Banach space E if the domain D(A(t)) of A(t) is not constant. The main novel idea is the use of ”extrapolation spaces”, defined by means of A(t), on which the extended operators have constant domains. This leads to results which are effectively applicable to the study of quasilinear parabolic systems with nonlinear boundary conditions [cf. H. Amann: Semigroups and nonlinear evolution equations. Lin. Algebra Appl. 84, 3-32 (1986), for extensions and a survey of further applications of this device].

### MSC:

 47D03 Groups and semigroups of linear operators 47E05 General theory of ordinary differential operators 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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