The unique existence of the solution of the Cauchy problem for PDE of the form $$ \pa_1 u(t, x)=f(t, x, u(t, x), \pa_2 u(t, x), \pa_2^p u(\alpha(t)t, x), \pa_2^q u(t, \beta(t, x)x)) $$ is proved. $t$ is in ${\mbb R}$, $x$ is in ${\mbb C}$ and $u(t,x)$ is in ${\mbb C}$. $p$ and $q$ are positive integers. $\pa_1$ and $\pa_2$ denote differentiations with respect to the $1^{st}$ and $2^{nd}$ variables, respectively. $f(t, x, u_{1}, \cdots, u_{4})$ is assumed to be holomorphic in $(x, u_{1}, \cdots, u_{4})$. $\alpha$ and $\beta$ are called {\it shrinking functions}. It is assumed that $\sup |\alpha(t)|<1$ and $\sup |\beta(t,x)|<1$.