2 problems

Let $T>0$, let $m$ be the porous-medium exponent, and let $g:{\mathbb T}\to[0,1]$ be measurable. For each $\kappa>0$, let…

Let $\rho$ be a solution to the porous medium equation on $[-1,0] \times B_2(0)$. Assume that there is a nonempty open cone $C$ such that $\rho(t,x) \leq \rho(t,y)$ whenever…