A completely integrable quasilinear partial differential equation of infinite dimension (Q5945050)

The local initial value problem for the first order partial differential equations of the form \[ \partial _1 u(t,x)+ \partial _2 u(t,x)\lambda (t,x, u(t,x))=f(t,x, u(t,x)) \] is considered. Here \(t\) and \(x\) move in a real Banach space. The values of the unknown function \(u\) also are elements of some Banach space. The symbols \(\partial_1, \partial_2\) denote the partial derivatives in the sense of Fréchet. Under the assumption that the functions \(\lambda\) and \(f\) are smooth and satisfy the conditions of complete inegrability, existence and uniqueness conditions of local solutions are established.