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Around Ovsyannikov’s method. (English) Zbl 1340.35177

The author considers the Cauchy problem for evolution equations of the form \[ \frac{d}{dt}u(t)=Zu(t)+Au(t),\quad u(0)=u_0, \] where the operator \(Z\) is defined on an increasing scale of Banach spaces and satisfies conditions typical for the so-called Ovsyannikov’s method (see, for example, F. Treves [Notas Mat. No. 46, 238 p. (1968; Zbl 0205.39202)]. The operator \(A\) is a generator of the contraction semigroup acting in each space of the scale.
General existence and uniqueness results are obtained extending the method used earlier for a particular operator (D. Finkelshtein et al. [Math. Models Methods Appl. Sci. 25, No. 2, 343–370 (2015; Zbl 1317.82031)]. An application to a birth-and-death stochastic dynamics in the continuum is considered.

MSC:

35K90 Abstract parabolic equations
47D06 One-parameter semigroups and linear evolution equations
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)