I wondering if thare is a bound $C>0$ such that $\|\Delta c\|_{\infty}\leq C\|c\|_{\infty}$ (or any bound that only depend of $c(0)$) where $c$ is in $C^{\infty}$ real valued non-negative function such that $\|c(t)\|_{i\infty}$ is decrecient. Here is important to know that $c(0)$ is knew but $c(t)$ fot $t>0$ does not, so the constant $C>0$ should not to depend of $c$.
My aproach consist in the fact that "roughly speaking, the Laplace operator gives you the difference between the value of a function at a point and the average value at neighboring points." [https://math.stackexchange.com/questions/35532/nice-way-of-thinking-about-the-laplace-operator-but-whats-the-proof], so I can conclude that since the essential supremum of $c(t)$ is decreasing then $\Delta c(t)$ is decreasing (I don't know if this is true, I until right now does not able to prove or give a counterexample), if the last aseveration will be true then I would can to bound with $\Delta c(0)$ taking $t\to 0$ and taking into account that $c\in C^{\infty}$ .
