Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer analysis of a model?

For example,in https://arxiv.org/pdf/2006.03894.pdf - the authors consider the following problem: let us consider a bounded domain $\Omega \subset \mathbb{R}^{n}$ and a positive number $T$, then in $\Omega \times (0,T)$ the locally bounded solutions of the inhomogeneous porus medium equation

$u_{t} - div(m\lvert u \rvert^{m-1}\nabla u) = f \in L^{q,r}, m>1$

are locally Hölder continuous, with exponent

$\gamma = min \Big(\frac{\alpha^{-}_{0}}{m}, \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \Big)$

where $\iota^{-}$ means that we can select all $s \in (0,\iota)$ and $L^{q,r}$ is a Lebesgue space with mixed norm (see the paper above for more informations)

So, why find this $\alpha$ which is a sharp exponent is so important and what applications we can obtain from the quantitative Holder regularity perspective?