Row-finite systems of ordinary differential equations in a scale of Banach spaces

Motivated by the study of dynamics of interacting spins for infinite particle systems, we consider an infinite family of first order differential equations in a Euclidean space, parameterized by elements $x$ of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the $x$-equation depends on a finite but in general unbounded number $n_x$ of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of $n_x$, we show the existence of the solutions with infinite life-time, and prove that they live in an increasing scale of Banach spaces. For this, we obtain uniform estimates for solutions to approximating finite systems using a version of Ovsyannikov's method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.