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Nonlinear analytic semiflows. (English) Zbl 0723.34047
Extending the approach of G. daPrato and G. Grisvard [Ann. Mat. Pura Appl., IV. Ser. 120, 329-396 (1979; this Zbl 0471.35036)] the author presents a local existence and regularity theory for abstract parabolic initial value problems of the type \[ x'(t)=f(x(t)),\quad t>0,\quad x(0)=x_ 0\in {\mathcal O} \] where f: \({\mathcal O}\to E_ 0\) is a smooth map, \({\mathcal O}\) is an open subset of \(E_ 1\) and \(E=(E_ 0,E_ 1)\) is a pair of Banach spaces, \(E_ 1\) continuously and densely embedded into \(E_ 0\). The main hypothesis is that the linearizations df(x) for all \(x\in {\mathcal O}\) belong to a certain class of generators of analytical semigroups in \(E_ 0\) with domain \(E_ 1\). This theory is extended to quasilinear problems \[ x'(t)=A(x(t))x(t)+f(x(t)),\quad t>0,\quad x(0)=x_ 0\in {\mathcal O}, \] where \({\mathcal O}\) is now an open subset in a suitable intermediate space \(E_{\theta}\) between \(E_ 1\) and \(E_ 0\) and for all \(x\in {\mathcal O}\) now A(x) belongs to a certain class \({\mathcal M}_{\theta}(E)\) of generators of analytic semigroups in \(E_ 0\) with domain \(E_ 1\). The author proves also some results on the classes \({\mathcal M}_{\theta}(E)\) and presents for a given generator of a holomorphic semigroup the construction of a pair E such that A belongs to \({\mathcal M}_{\theta}(E)\).

34G20 Nonlinear differential equations in abstract spaces
47N20 Applications of operator theory to differential and integral equations
35K55 Nonlinear parabolic equations
Zbl 0471.35036
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