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 $a$ a

, we show moreover that

u $u$ u

enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion

f left parenthesis u right parenthesis equals 1 $f(u)=1$ f ( u ) = 1

, the Lane–Emden equation 

f left parenthesis u right parenthesis equals uq $f(u)=u^q$ f ( u ) = u q

,

q element of left parenthesis 0 comma 1 right parenthesis $q\in(0,1)$ q ∈ ( 0 , 1 )

, the eigenfunction

f left parenthesis u right parenthesis equals u $f(u)=u$ f ( u ) = u

, the logarithmic equation 

f left parenthesis u right parenthesis equals ulog left parenthesis u 2 right parenthesis $f(u)=u\log(u^2)$ f ( u ) = u log ( u 2 )

, and the saturable nonlinearity

f left parenthesis u right parenthesis equals u 21 plus u $f(u)=\frac{u^2}{1+u}$ f ( u ) = u 2 1 + u

. The logistic equation 

f left parenthesis x comma u right parenthesis equals a left parenthesis x right parenthesis u minus u 2 $f(x,u)=a(x)u-u^2$ f ( x , u ) = a ( x ) u − u 2

can be treated as well.

Some exact results provide a different approach to the existing literature, as well as several generalizations. Moreover, some quantitative results are also valid and new in the elliptic setting

.