Quantitative and exact concavity principles for parabolic and elliptic equations

Abstract Goal of this paper is to study classes of Cauchy–Dirichlet problems which include parabolic equations of the type \begin{equation*}u_t -\Delta u= a(x,t)f(u)\quad\hbox{in}\ \Omega\times(0,T)\end{equation*} u t − Δ u = a ( x , t ) f ( u ) in Ω × ( 0 , T ) with upper Omega subset of double struck upper R n $\Omega\subset\mathbb{R}^n$ Ω ⊂ ℝ n bounded, convex domain and upper T element of left parenthesis 0 comma plus infinity right bracket $T\in(0,+\infty]$ T ∈ ( 0 , + ∞ ] . Under suitable assumptions on a $a$ a and f $f$ f , we show logarithmic or power concavity (in space, or in space-time) of the solution u $u$ u ; under some relaxed assumptions on a $a$