On the Cauchy problem for non linear PDEs in the Gevrey class with shrinkings

Abstract

In the article, the Cauchy problem of the form
(*) ∂₂u(x, t)=f(u(x, t), ∂₁^pu(x, α(t)t), x, t), \ u(x, 0)=0
or of the form
(†) ∂₂u(x, t)=f(u(x, t), ∂₁^pu(α(x, t)x, t), x, t), \ u(x, 0)=0
is studied. In (*) and (†) u(x, t) denotes a real valued unknown function of the real variables x and t.\ p denotes a fixed positive integer. It is assumed that f(u, v, x, t) is continuous in (u, v, x, t) and Gevrey in (u, v, x).\ α(t) in (*) and α(x, t) in (†) are called shrinkings, since they satisfy the conditions \displaystyle \sup|α(t)|<1 and \displaystyle \sup|α(x, t)|<1, respectively.