Quantitative and exact concavity principles for parabolic and elliptic equations

Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type $$u_t -Δu= a(x,t)f(u)\quad\hbox{in $Ω\times(0,T)$}$$ with $Ω\subset\mathbb{R}^N$ bounded, convex domain and $T\in(0,+\infty]$. Under suitable assumptions on $a$ and $f$, we show logarith