  We study the existence of positive viscosity solutions to Trudinger's
equation for cylindrical domains $\Omega\times[0, T)$, where $\Omega\subset
\mathbb{R}^n,\;n\ge 2,$ is a bounded domain, $T>0$ and $2\leq p<\infty$. We
show existence for general domains $\Omega,$ when $n<p<\infty$. For $2\leq
p\leq n$, we prove existence for domains $\Omega$ that satisfy a uniform outer
ball condition. We achieve this by constructing suitable sub-solutions and
super-solutions and applying Perron's method.
