On abstract parabolic fundamental solutions. (English) Zbl 0616.47032

Amann, Herbert

J. Math. Soc. Japan 39, 93-116 (1987).

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

Keywords:

evolution operator; abstract evolution equation of parabolic type; extrapolation spaces; quasilinear parabolic systems with nonlinear boundary conditions