Quantitative and exact concavity principles for parabolic and elliptic equations

Abstract

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 $a$