1. Introduction

In the 1970s and 1980s, significant progress was made in understanding the convexity and concavity properties of solutions to certain elliptic differential equations on convex domains and Dirichlet boundary conditions. In the pivotal papers by Brascamp and Lieb [Reference Brascamp and Lieb12] and Makar-Limanov [Reference Makar-Limanov58], the authors showed that solutions of the eigenfunction problem and of the torsion problem are concave up to a composition, respectively, with the logarithm function or the square root function. The seminal papers by Korevaar [Reference Korevaar51] and Kennington [Reference Kennington49] gave a new boost to this research by introducing general concavity maximum principles to deal with general classes of equations. We refer to [Reference Almousa, Assettini, Gallo and Squassina2, Reference Gallo and Squassina32] and references therein for a general overview of the topic. Let us mention, among various mathematical applications, that in [Reference Singer, Wong, Yau and Yau64] the authors exploit the $\log$-concavity of the eigenfunction to show notable results about the eigenvalue gap, i.e. the difference between the first and the second Dirichlet eigenvalue.

In the framework of parabolic equations and Cauchy–Dirichlet problems

\begin{equation*} \begin{cases} u_t - \Delta u = b(x,u,t) & \quad \hbox{in}\ \Omega \times (0,T),\\ u=0 & \quad \hbox{on}\ \partial \Omega \times [0,T), \\ u=u_0& \quad \hbox{on}\ \overline{\Omega} \times \{0\}, \end{cases} \end{equation*}

$T \in (0, +\infty]$, several questions arise. A first problem is to study very particular examples of $b$ and discuss if, under suitable assumptions on $u_0$, the solution $u(x,t)$ becomes logarithmic or power concave (i.e. $\log(u)$ or $u^\alpha$ for some $\alpha \gt 0$ is concave) after some amount of time ($t \gt 0$ or possibly $t\gg 0$): these kinds of results are strongly related to the particular shape of $b$ which allows, for instance, explicit representation of the solution, and have been investigated, for example, in [Reference Ishige39, Reference Lee and Vázquez54, Reference Lee and Vázquez55]. We highlight that these results allow, under suitable assumptions, to get interesting properties also in the elliptic case, by considering the limit of $u(x,t)$ as $t\to +\infty$, which coincides with the solution of the corresponding stationary problem.

A second question is the following: what is the optimal (in some sense) concavity on $u_0$ which is preserved by the heat flow? In this problem, the logarithmic concavity plays a very special role. This and similar questions have been addressed in a series of papers [Reference Ishige and Salani41, Reference Ishige, Salani and Takatsu44–Reference Ishige, Salani and Takatsu46]. See also [Reference Chau and Weinkove17, Reference Daskalopoulos, Hamilton and Lee26] for similar questions regarding power concavity and porous medium flow.

In this paper, we are interested in a different question: under which assumptions on $b$ and $u_0$ can we obtain that $u(x,t)$ enjoys some concavity properties for each $t \gt 0$? To the authors’ knowledge, only a few papers deal with this problem.

A first result can be found in Korevaar [Reference Korevaar51], where the author adapts the elliptic concavity maximum principles to study classes of parabolic equations and show that, for example, the parabolic eigenfunctions $u(\cdot, t)$ are spatially logarithmic concave for each $t \in [0,T]$ if $u_0$ is so. Let us mention that in [Reference Andrews and Clutterbuck6] a stronger notion of $\log$-concavity is proved and it is exploited to prove a sharp result on the eigenvalue gap; it is notable to observe that the technique adopted in [Reference Andrews and Clutterbuck6] relies on the introduction and analysis of some suitable multivariable function in a spirit relatable to the one introduced in [Reference Kennington49, Reference Korevaar51] and used in the present paper. Some partial generalizations of the results in [Reference Korevaar51] can be found in [Reference Porru and Serra62] (see also [Reference Greco and Kawohl35]), by allowing the inclusion of other interesting models. Other results are contained in [Reference Ishige, Salani and Takatsu44] (see e.g. Proposition 4.2 therein) by means of spatial log-concave envelopes as viscosity subsolutions of the problem and corresponding comparison principles; similar techniques go back to [Reference Alvarez, Lasry and Lions4] (see also [Reference Colesanti, Qin and Salani23]).

A second result, i.e. a parabolic generalization for equations in the spirit of the torsion and Lane–Emden problems, which allows us to obtain power concavity (stronger than logarithmic one), appears first in [Reference Kennington48]: here the author studies the case $a=1$ and $f(u)=(1-u)^q$, $q \in (0,1)$, obtaining that a suitable transformation of $u$ is concave and thus $u$ is quasi-concave (i.e. the superlevel sets of $u$ are convex). To gain the claim, the author exploits some concavity maximum principle again, and he strongly uses that $f(0) \neq 0$ coupled with some comparison principle; moreover, he relies on the information at infinity given by the stationary problem. Other results have been more recently obtained in [Reference Ishige and Salani42, Reference Ishige and Salani43] (see also [Reference Ishige, Liu and Salani40]): here the authors rely on the use of parabolic concave envelopes as viscosity subsolutions. In such a way, the authors are able to study the concavity of nondecreasing solutions with controlled behaviour on the boundary, in the case of nonnegative nonlinearities satisfying a priori some comparison principle, set in smooth domains: for example, they cover the cases $f=1$, or $a=1$ and $f(u)=u^q$, $q \in (0,1)$. Moreover, the authors study a rescaled concavity in space and time, namely they show that $(x,t) \mapsto u(x,t^{\beta})$ is power concave for suitable values of $\beta$.

The aim of this paper is twofold. First, we aim to recover and slightly generalize the abovementioned results, in a unified way, by means of parabolic concavity maximum principles. In particular, to recover the results previously obtained through different techniques, we need a finer analysis, especially regarding the boundary behaviour of the solution (including $t \to +\infty$). A second goal is to derive quantitative estimates of concavity maximum principles, which will require a careful study of the objects in play. For the sake of clarity in presentation, we describe first some applications of the exact results (which are indeed a consequence of the quantitative estimates).

Notation. We say that $u$ is $\alpha$-concave if $\alpha \gt 0$ and $u^{\alpha}$ is concave, or $\alpha=0$ and $\log(u)$ is concave (see (3.4)). Moreover, a solution of a PDE is meant positive when $u \gt 0$ in $\Omega\times (0,T]$. For the sake of simplicity, to say that $u\colon \overline{\Omega} \times [0,T] \to \mathbb{R}$ verifies $u(\cdot, t) \in C^k(\overline{\Omega})$ for each $t \in (0,T]$ (or $t \gt 0$ if $T=+\infty$) and $u(x, \star) \in C^h([0,T])$ for each $x \in \Omega$, we will simply write $u \in C^k_x(\overline{\Omega}) \cap C^h_t([0,T])$. Finally, in what follows, we use the convention, for the sake of simplicity, $1/\gamma=+\infty$ when $\gamma=0$.

We refer to the beginning of Section 3 for other notations.

We start with some examples which also cover results in [Reference Greco and Kawohl35, Reference Korevaar51, Reference Porru and Serra62] and [Reference Ishige, Salani and Takatsu44].

Theorem 1.1 (Weighted eigenfunctions and logarithmic equations)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, smooth, strongly convex domain. Let $T \gt 0$, $a\colon\Omega\times(0,T]\to \mathbb{R}$, $u_0\in C^1(\overline{\Omega})$ with $u_0=0$ on $\partial\Omega$ and $u \in C^2_x(\overline{\Omega}) \cap C^1(\overline{\Omega}\times[0,T])$ be a positive solution of

\begin{equation*} \begin{cases} u_t - \Delta u = a(x,t) f(u) &\quad \hbox{in}\ \Omega \times (0,T], \\ u=0 & \quad\text{on }\partial \Omega\times [0,T], \\ D_x u \cdot \nu \gt 0 & \quad\text{on }\partial \Omega\times [0,T], \\ u =u_0 & \quad \text{on } \overline{\Omega} \times \{0\}, \end{cases} \end{equation*}

where $\nu$ is the interior normal to $\partial \Omega$. Assume moreover that $u_0 \gt 0$ is $\log$-concave in $\Omega$ and one of the following:

1. i) $f(s)=s$, and $a(\cdot, t)$ is concave for each $t \in (0,T]$;

2. ii) $f(s)=s \log^q(1+s)$ or $f(s)=\frac{s^{q+1}}{1+s^q}$, $q \in (0,1)$, and $a(\cdot,t)$ is $\frac{1}{1-q}$-concave for each $t \in (0,T]$;

3. iii) $f(s)=s \log(1+s)$, $f(s)=s \log(s)$ or $f(s)=\frac{s^2}{1+s}$, and $a$ is a positive constant.

Then $\log(u(\cdot, t))$ is concave for each $t \in (0,T)$. Moreover, if $a \in C^{2,1}(\Omega\times(0,T])$ and $u \in C^{2,1}(\Omega\times(0,T])$, then $\log(u(\cdot, t))$ is strictly concave for each $t \in (0,T)$; as a consequence, for each $t \in (0,T)$ the function $u(\cdot, t)$ has a unique, nondegenerate, critical point.

Theorem 1.1 is a particular case of the more general Corollary 5.3; we highlight that the method allows a dependence of $a$ both in $x$ and $t$, and to choose $a$ sign-changing in point (i) (which allows us to cover a broader range of applicative problems, see also Theorem 1.2 below). Moreover, we highlight the information on strict concavity that we obtain. A generalization of Theorem 1.1 in the sense of perturbed concavity is presented in Theorem 1.5 below and Corollary 5.6. We refer to Remark 2.1 for some technical comments on Theorem 1.1.

As an additional example, we may consider logistic models of population dynamics in heterogeneous environments with lethal boundary. That is, the equation is governed by a logistic law $b(x,s) = a(x) s - s^2$ where $\{a \gt 0\}$ is the favourable habitat while $\{a \lt 0\}$ is the hostile one, see [Reference Belgacem and Cosner7, Reference Cantrell and Cosner15, Reference Oruganti, Shi and Shivaji60] (see also Remark 5.10 for bang-bang weights).

Theorem 1.2 (Logistic model)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, smooth, strongly convex domain. Let $T \gt 0$, $a\colon\Omega\times(0,T]\to \mathbb{R}$, $u_0\in C^1(\overline{\Omega})$ with $u_0=0$ on $\partial\Omega$ and $u \in C^2_x(\overline{\Omega}) \cap C^1(\overline{\Omega}\times[0,T])$ be a positive solution of

\begin{equation*} \begin{cases} u_t - \Delta u = a(x,t) u - u^2 &\quad \hbox{in}\ \Omega \times (0,T], \\ u=0 & \quad\text{on }\partial \Omega\times [0,T], \\ D_x u \cdot \nu \gt 0 & \quad\text{on }\partial \Omega\times [0,T], \\ u =u_0 & \quad \text{on } \overline{\Omega} \times \{0\}, \end{cases} \end{equation*}

where $\nu$ is the interior normal to $\partial \Omega$. Assume moreover that $u_0 \gt 0$ is $\log$-concave in $\Omega$ and $a(\cdot,t)$ is concave in $\Omega$ for each $t\in (0,T]$. Then $\log(u(\cdot, t))$ is concave for each $t \in (0,T)$. Moreover, if $a\in C^{2,1}(\Omega\times(0,T])$ and $u \in C^{2,1}(\Omega \times (0,T])$, then $\log(u(\cdot, t))$ is strictly concave for each $t \in (0,T)$.

Compared with [Reference Greco and Kawohl35, Reference Korevaar51, Reference Porru and Serra62], Theorems 1.1 and 1.2 point out some additional models and contain some minor generalizations (e.g. the presence of the weight $a(x,t)$), which are not substantial but allow us to give a more complete picture. We continue by presenting some examples, inspired by [Reference Kennington48], and which cover models presented in [Reference Ishige and Salani42, Reference Ishige and Salani43].

Theorem 1.3 (Weighted Lane–Emden)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, convex domain which satisfies the interior sphere property. Let $q \in [0,1)$ and $\gamma \in [0,1]$. Let $u\in C^2_x(\Omega)\cap C^1_x(\overline{\Omega}) \cap C^1_t((0,+\infty)) \cap C(\overline{\Omega}\times[0,+\infty))$ be a positive solution of

\begin{equation*} \begin{cases} u_t-\Delta u=a(x) t^{\gamma} u^q &\quad\text{in }\Omega\times (0,+\infty), \\ u=0 & \quad\text{on }\partial \Omega\times [0,+\infty), \\ u =0 & \quad \text{on } \overline{\Omega} \times \{0\}. \end{cases} \end{equation*}

Suppose $a\colon \Omega \to \mathbb{R}$ to be measurable and that there exists $m \gt 0$ such that $a(x)\geq m$ for all $x\in \Omega$. If $\beta \in [1, \frac{1}{\gamma}) \cap [1,2]$ and $a$ is $\theta$-concave with $\theta \geq \frac{1}{1-\beta \gamma}$, then $u(\cdot, \star^{\beta})$ is $\frac{(1-q)\theta}{2\theta +\beta \gamma \theta+1}$-concave. If $\beta \in [1, \min\{\frac{1}{\gamma},2\}]$ and $a$ is constant, then $u(\cdot, \star^{\beta})$ is $\frac{1-q}{2+\beta \gamma }$-concave.

We notice that, if $q=0$, the condition $a(x) \geq m \gt 0$ above can be relaxed into $a(x) \geq 0$, see Proposition 6.5; we highlight in Theorem 1.3 the possibility of the nonlinearity to vanish where $u=0$, which is a nontrivial improvement compared to [Reference Kennington48]. Theorem 1.3 is a particular case of the more general Corollary 5.18. We highlight that, whereas [Reference Ishige and Salani43, Theorem 3] and [Reference Ishige and Salani42, Theorem 4.1] rely on a priori assumptions on the solution $u$, in Theorem 5.16 and Corollary 5.18 we focus instead on furnishing operative conditions to check on the nonlinearity. Notice that Theorem 1.3 somehow unifies [Reference Ishige and Salani43, Theorem 5 and Theorem 6] (see also [Reference Ishige and Salani42, Theorem 5.3 and Corollary 6.1]), even if such generalization is not deducible from the arguments in [Reference Ishige and Salani42, Reference Ishige and Salani43].

A quantitative version of Theorem 1.3 can be found in Theorem 1.6 below and Corollary 5.17. We refer to Remark 2.2 for more technical comments on Theorem 1.3.

To mention another consequence of our general setting, we propose here the concavity of solutions for nonhomogeneous nonlinearities. Let us mention that our approach allows us to recover also the quasiconcavity obtained by [Reference Kennington48] (see Proposition 5.19 and the subsequent remark), and to treat gradient-type equations (see e.g. Remark 5.5).

Theorem 1.4 (Sum of powers)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, convex domain which satisfies the interior sphere property. Let $q\in (\frac{1}{3},1)$ and $p\in (\frac{3q-1}{2},q)$. Let $u\in C^2_x(\Omega)\cap C^1_x(\overline{\Omega}) \cap C^1_t((0,+\infty)) \cap C(\overline{\Omega}\times [0,+\infty))$ be a positive solution of

\begin{equation*} \begin{cases} u_t-\Delta u= a(x) u^p+u^q &\quad\text{in }\Omega\times (0,+\infty), \\ u=0 & \quad\text{on }\partial \Omega\times [0,+\infty), \\ u =0 & \quad \text{on } \overline{\Omega} \times \{0\}. \end{cases} \end{equation*}

Suppose $a\colon \Omega \to \mathbb{R}$ to be measurable and that there exists $m \gt 0$ such that $a(x)\geq m$ for all $x\in \Omega$; assume moreover $a$ to be $\theta$-concave with $\theta \in [\frac{1-q}{2(q-p)}, +\infty]$. Then we have that $u(\cdot, \star^{2})$ is $\frac{1-q}{2}$-concave.

We postpone the discussion of the technical difficulties of Theorems 1.1–1.4 (and of the corresponding generalizations Corollaries 5.3 and 5.18), see Remark 2.4.

As aforementioned, the second and main goal of this paper is to deal with quantitative versions of the concavity maximum principles. Recently, quantitative versions of symmetry results have been considered, for example, in [Reference Amato, Barbato, Masiello and Paoli5, Reference Ciraolo, Cozzi, Perugini and Pollastro21, Reference Dipierro, da Silva, Poggesi and Valdinoci29, Reference Gatti33]; we also mention [Reference Ciraolo, Figalli and Maggi22, Reference Deng, Sun and Wei27] where the error from bubble functions is quantified in Sobolev critical equations. In the realm of concavity properties, instead, similar results have been recently investigated in the elliptic case in [Reference Bucur and Squassina13] (see also [Reference Almousa, Bucur, Cornale and Squassina3, Reference Gallo and Squassina32]); such estimates are made in terms of some parameter of the weight $a(x)$: more precisely, in these papers the authors show that, if $a$ is close to a constant function, then some power of $u$ is close to a concave function, up to some explicit error. We also mention the recent [Reference Hamel and Nadirashvili36] where a concavity result is obtained for problems with small perturbations, by exploiting the strong $\log$-concavity of the eigenfunction.

In this paper, we deal with estimations of the (space) concavity function of a solution $v$ (see (3.5))

\begin{equation*} \mathcal{C}^*_v(x_1,x_3,\lambda, t) :=v(x_2, t)-\lambda v(x_3, t)-(1-\lambda)v(x_1,t), \end{equation*}

or the (spacetime) concavity function

\begin{equation*}\mathcal{C}_v(x_1,x_3,t_1,t_3,\lambda):= v(x_2,t_2)-\lambda v(x_3,t_3)-(1-\lambda)v(x_1,t_1),\end{equation*}

where $x_1, x_3 \in \Omega $, $t_1, t_3 \in [0,+\infty)$, $\lambda \in [0,1]$ and $x_2=\lambda x_3+(1-\lambda)x_1$ and $t_2=\lambda t_3+(1-\lambda)t_1$; notice that $\mathcal{C}_v \geq 0$ if and only if $v$ is concave. Our goal is, as already anticipated, to generalize the abovementioned results also to the parabolic setting, which seems to be new in the literature. As a consequence of our general results, we get in particular the following statement. Here $f^- := \max\{0,-f\}$ stands for the negative part of a function $f$, while ${\rm osc}(a):= \sup(a) - \inf(a)$ stands for the oscillation of $a$.

Theorem 1.5 (Quantitative concavity, I)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, smooth, strongly convex domain. Let $T \gt 0$, $a\colon \Omega \times (0,T] \to \mathbb{R}$ bounded, $u_0\in C^1(\overline{\Omega})$ with $u_0=0$ on $\partial\Omega$ and $u \in C^2_x(\Omega) \cap C^1(\overline{\Omega}\times[0,T])$ be a positive solution of

\begin{equation*} \begin{cases} u_t - \Delta u = a(x,t) u &\quad \hbox{in}\ \Omega \times (0,T], \\ u=0 & \quad\text{on }\partial \Omega\times [0,T], \\ D_x u \cdot \nu \gt 0 & \quad\text{on }\partial \Omega\times [0,T], \\ u =u_0 & \quad \text{on } \overline{\Omega} \times \{0\}, \end{cases} \end{equation*}

where $\nu$ is the interior normal to $\partial \Omega$. Assume moreover that $u_0 \gt 0$ is $\log$-concave in $\Omega$. Then

\begin{equation*}\inf_{\Omega \times \Omega \times [0,1] \times [0,T]} \mathcal{C}^*_{\log(u)} \geq -e T \sup_{\Omega \times \Omega \times [0,1] \times (0,T]} \left(\mathcal{C}^*_{a(\cdot, \star)}\right)^-.\end{equation*}

In Theorem 1.5, one may also consider other nonlinearities, like the ones in Theorem 1.1 (see Corollary 5.6). Moreover, if $u_0$ is not assumed to be $\log$-concave, then an additional error related to $\mathcal{C}_{\log(u_0)}$ appears (see Remark 5.2).

Theorem 1.6 (Quantitative concavity, II)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, convex domain which satisfies the interior sphere property. Let $q \in [0,1)$. Let $u\in C^2_x(\Omega)\cap C^1_x(\overline{\Omega}) \cap C^1_t((0,+\infty)) \cap C(\overline{\Omega}\times [0,+\infty))$ be a positive solution of

\begin{equation*} \begin{cases} u_t-\Delta u=a(x) u^q &\quad\text{in }\Omega\times (0,+\infty), \\ u=0 & \quad\text{on }\partial \Omega\times [0,+\infty), \\ u =0 & \quad \text{on } \overline{\Omega} \times \{0\}. \end{cases} \end{equation*}

Suppose $a\colon\Omega \to \mathbb{R}$ to be measurable and $0 \lt m \leq a \leq M$. Then $u$ is increasing in time and

(1.1)\begin{equation} \min_{\overline{\Omega}\times \overline{\Omega}\times[0,+\infty]\times [0,+\infty]\times[0,1]}\mathcal{C}_{u^{\frac{1-q}{2}}} \geq - \|u(\cdot, \infty)\|_{L^{\infty}(\Omega)}^{\frac{1-q}{2}} \frac{{\rm osc}(a^2)}{m^2}, \end{equation}

where $u(\cdot, \infty)$ is the solution of the corresponding stationary problem.

If moreover $\theta \geq 1$ and $m^\theta\geq M^{\theta}/2$, then

(1.2)\begin{equation} \min_{\overline{\Omega}\times \overline{\Omega}\times[0,+\infty]\times [0,+\infty]\times[0,1]}\mathcal{C}_{u^{\frac{\theta(1-q)}{2\theta+1}}} \geq -\frac{\theta }{\theta+1} \frac{1}{m} \|u(\cdot, \infty)\|_{L^{\infty}(\Omega)}^{\frac{\theta(1-q)}{2\theta+1}} \sup\limits_{\Omega \times \Omega\times[0,1]} (\mathcal{C}^-_{a^{\theta}})^{1/\theta}. \end{equation}

We refer to Remark 2.3 for some additional comments. We highlight that estimate (1.1) in Theorem 1.6 is slightly finer if compared with the ones obtained in the elliptic setting in [Reference Bucur and Squassina13, Reference Gallo and Squassina32]. Moreover, the estimate (1.2) does not involve the oscillation of $a$ and is totally new, even for the elliptic setting. Indeed, some results of this paper can be easily adapted to the elliptic framework, allowing us to improve and generalize some previous results [Reference Bucur and Squassina13, Reference Gallo and Squassina32]. By way of example, with the same techniques, we obtain the following statement. We leave the details to the interested reader.

Theorem 1.7 ($\theta$-concavity estimate)

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, convex domain which satisfies the interior sphere property, and let $q \in [0,1)$. Let $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ be a positive solution of

\begin{equation*}\begin{cases} -\Delta u = a(x) u^q & \quad \hbox{in}\ \Omega, \\ u=0 & \quad \hbox{on}\ \partial \Omega. \end{cases} \end{equation*}

Suppose $a\colon \Omega \to \mathbb{R}$ to be measurable and $0 \lt m\leq a \leq M$ and $1\leq\theta \leq\log(2)/\log(\frac{M}{m})$. Then

\begin{equation*} \min_{\overline{\Omega}\times \overline{\Omega}\times[0,1]}\mathcal{C}_{u^{\frac{\theta(1-q)}{1+2\theta}}} \geq -\frac{\theta}{1+\theta} \frac{1}{m}||u||_\infty^{\frac{\theta(1-q)}{1+2\theta}} \sup\limits_{\Omega \times \Omega\times[0,1]} (\mathcal{C}^-_{a^{\theta}})^{1/\theta}. \end{equation*}

As already mentioned, the main tool of this paper is composed by concavity and harmonic concavity maximum principles, which allow us to give an estimate on the concavity function $\mathcal{C}_v$ of a solution of a general PDE in terms of the concavity/harmonic concavity function of the nonlinear term (see Definition 3.7). The equations treated do not rely on the boundary conditions and can have a very general form (see Theorems 4.1, 4.4), and in particular include the equations satisfied by $v=\log(u)$ or $v=u^{\alpha}$, $\alpha \in (0,1)$, where $u$ is the solution of the equation of interest. We highlight that the concavity principle approach is flexible since it can be used, for example, to deal with weak solutions of quasilinear equations, see [Reference Borrelli, Mosconi and Squassina11, Reference Gallo and Squassina32, Reference Lin57] (see also Remark 5.10 for another nonsmooth setting), and thus possibly useful for treating $p$-Laplace parabolic equations.

We refer to Remark 2.4 for some technical comments on the proofs and corresponding difficulties and novelties.

The paper is organized as follows. In Section 2, we collect some remarks on the main theorems and on the proofs, while in Section 3, we recall some basic facts on parabolic equations, and we introduce the main assumptions that we will require on the nonlinearity. Moreover, we recall notations and properties about concavity functions; more technical results are instead collected in Appendix B. Section 4 is devoted to the proof of the main concavity principles, which take into account only the PDE and not the boundary/initial conditions; these theorems will lead both to exact and quantitative results. In particular, we show both concavity principles in space (in the presence of nonlinearities which are concave but not necessarily decreasing, and set in $(0,T]$) and in space-time (in the case of harmonic concave and decreasing nonlinearities set in $(0,+\infty)$); the latter will require the use of the harmonic concavity function. In Section 5, we take into account the boundary conditions and the initial condition, employing a suitable analysis of the solution and of the concavity function near the boundary of the parabolic cylinder; this analysis also exploits comparison principles and Hopf lemmas, which are collected in Appendix A. Some applications and examples follow, including the proofs of the main theorems presented in the introduction. When the nonlinearity does not depend on $u$, finally, we present some further results (both simplifications and improvements of the previous ones) in Section 6.