The heat equation and the shrinking

This article concerns the Cauchy problem for the partial differential equation $$ partial_1 u(t,x)-apartial_2^2 u(t,x) =f(t,x,partial_2^p u(mu(t)t,x),partial_2^q u(t, u(t)x)),. $$ Here $t$ and $x$ are real variables, $p$ and $q$ are positive integers greater than 1, and the shrinking factors $mu(t)$, $ u(t)$ are positive-valued functions such that their suprema are less than 1.