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

Abstract

In the article, the Cauchy problem of the form

(*) $\partial_{2}u(x,t)=f(u(x,t),\partial_{1}^{p}u(x,\alpha(t)t),x,t),\ u(x,0)=0$

or of the form

$(\dagger)$ $\partial_{2}u(x,t)=f(u(x,t), \partial_{1}^{p}u(\alpha(x,t)x,t),x,t),\ u(x,0)=0$

is studied. In (*) and $(\dagger)$ $u(x,t)$ 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 $(u,v,x)$. $\alpha \left(t\right)$ in (*) and $\alpha (x,t)$ in $(\dagger)$ are called shrinkings, since they satisfy the conditions sup$|\alpha (t\left)\right|<1$ and sup$|\alpha (x,t\left)\right|<1$, respectively.