How (or if) a comparison principle works in the case of equations singular at some point? For example, I am analyzing a partial differential equation

$$ \partial_{t}u=\partial_{rr}u+\frac{2}{r}\partial_{r}u-\frac{2}{r^{2}}u\quad u(0,t)=0,\quad \partial_{t}u(1,t)=0 $$

There is a family of stationary solutions $u_{a}(r)=a\cdot r$ that could possibly function as sub or super solutions with suitably chosen $a$. But, to my understanding, comparison principle only works when coefficients in the equation are well behaved, which is not the case here: at $r=0$ the coefficients are diverging. How can I rigorously apply the comparison principle in this case?

A somewhat related question is, what do we know about $\partial_{r}u$? In particular, can we show that $\partial_{r}u$ is bounded when $a_1r\le u(r)\le a_2r$?

Ultimately, I would like to use comparison principle to give a bound on $u$ and, in consequence, a bound on $\partial_{r}u$ with $u$ solving a quasilinear equation

$$ \partial_{t}u=\partial_{rr}u+\frac{2}{r}\partial_{r}u-\frac{2}{r^{2}}f(u)\quad u(0,t)=0. $$

Here $f$ is smooth and $f(u)=u+\mathcal O(u^{2})$.