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 $$u_t -\Delta u= a(x,t)f(u)\quad\hbox{in $\Omega\times(0,T)$}$$ with $\Omega\subset\mathbb{R}^N$ bounded, convex domain and $T\in(0,+\infty]$. Under suitable assumptions on $a$ and $f$, we show logarithmic or power concavity (in space, or in space-time) of the solution $u$; under some relaxed assumptions on $a$, we show moreover that $u$ enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion $f(u)=1$, the Lane-Emden equation $f(u)=u^q$, $q\in(0,1)$, the eigenfunction $f(u)=u$, the logarithmic equation $f(u)=u\log(u^2)$, and the saturable nonlinearity $f(u)=\frac{u^2}{1+u}$. The logistic equation $f(x,u)=a(x)u-u^2$ can be treated as well. Some exact results give a different approach, as well as generalizations, to [Ishige-Salani2013, Ishige-Salani2016]. Moreover, some quantitative results are valid also in the elliptic framework $-\Delta u=a(x)f(u)$ and refine [Bucur-Squassina2019, Gallo-Squassina2024].