Finkelshtein, Dmitri Around Ovsyannikov’s method. (English) Zbl 1340.35177 Methods Funct. Anal. Topol. 21, No. 2, 134-150 (2015). 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. Reviewer: Anatoly N. Kochubei (Kyïv) Cited in 6 Documents MSC: 35K90 Abstract parabolic equations 47D06 One-parameter semigroups and linear evolution equations 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) Keywords:evolution equations; Ovsyannikov’s method; birth-and-death stochastic dynamics Citations:Zbl 0205.39202; Zbl 1317.82031 × Cite Format Result Cite Review PDF Full Text: arXiv