An abstract nonlinear Cauchy-Kovalevska theorem

Abstract:

A nonlinear version of Ovcyannikov’s theorem is proved. If $F(u,t)$ is an analytic function of $t$ real or complex and of $u$ varying in a scale of Banach spaces, valued in a scale of Banach spaces, the Cauchy problem ${u_t} = F(u,t),u(0) = {u_0}$, has a unique analytic solution. This is an abstract version of the Cauchy-Kovalevska theorem which can be applied to equations other than partial-differential, e.g. to certain differential-convolution or, more generally, differential-pseudodifferential equations.