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On the abstract nonautonomous parabolic Cauchy problem in the case of constant domains. (English) Zbl 0579.34001
Let E be a Banach space, \(\{A(t)\}_{t\in [0,T]}^ a \)family of closed linear operators on E, C([0,T],E) the space of continuous functions [0,T]\(\to E\), and consider the linear non-autonomous Cauchy problem (P): \(u'(t)-A(t)u(t)=f(t),\) \(u(0)=x\), with \(x\in E\) and \(f\in C([0,T],E)\) prescribed. The authors study the case where \(\{\) A(t)\(\}\) is a family of infinitesimal generators of analytic semi-groups (parabolic case) whose domain does not depend on t but can be not dense in E (so generalizing results of previous authors who have considered the case D(A(t))\(\equiv D(A(0))\) dense in E). They obtain (heuristically) a representation integral formula for a solution of (P),and introduce several adequate Banach spaces of functions where they study in detail the operators and functions appearing in this formula.
Then, they study at some length, what happens (w.r.t. uniqueness, existence, regularity) with strict, classical and strong solutions of the problem under some hypotheses (some of the generalizing results already known). Finally, the theory is applied to classical non-autonomous boundary-initial value problems.
Reviewer: F.R.Dias Agudo

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
47B38 Linear operators on function spaces (general)
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