The infinite-dimensional Liouville equation

A theorem on the existence of a solution of the Cauchy problem is proved for the infinite-dimensional Liouville equation for the $\varphi^n(x)$ model of a scalar bosonic field and for the $\varphi^n(x(\sigma))$ model of a bosonic string field. The Liouville equation is regarded as an ordinary differential equation in a locally convex space of functions on an infinite-dimensional phase space.