On the viscosity solutions to Trudinger’s equation

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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.