Let $ (M, g, d\mu) $ be a weighted Riemannian manifold equipped with a reference measure $ d\mu = e^{-f}{\hbox{vol}} $, there is a canonical differential operator associated to the triple given by the weighted Laplacian

$ \Delta_fu\doteqdot{\rm div}_f(\nabla u), $

where the notation $ \doteqdot $ means definition, $ {\rm div}_f\doteqdot e^f{\rm div}(e^{-f}\cdot) $ is the weighted divergence operator. The weighted Laplacian $ \Delta_f $ is a self-adjoint operator on the Hilbert space $ L^2(M, d\mu) $. Recall the carré du champ operator of the weighted Laplacian defined by Bakry-Émery [1]

$ \Gamma(u, v)\doteqdot \frac12[\Delta_f(uv)-u\Delta_fv-v\Delta_fu], $

and the iterated carré du champ operator

$ \Gamma_2(u, v)\doteqdot \frac12[\Delta_f\Gamma(u, v)-\Gamma(u, \Delta_fv)-\Gamma(v, \Delta_fu)]. $

A weighted Riemannian manifold $ (M, g, d\mu) $ is said to satisfy the curvature dimensional condition $ CD(-K, m) $ if for every function $ u $,

$ \Gamma_2(u, u)\ge\frac1m(\Delta_fu)^{2}-K\Gamma(u, u). $

In fact, by the enhanced Bochner formula (see P.383 in Villani's book [2])

$ \begin{align*} \label{EnBochner} &\frac12\Delta_f|\nabla u|^2-\nabla u\cdot\nabla\Delta_fu = |\nabla\nabla u|^2+{\rm Ric}_f(\nabla u, \nabla u)\notag\\ = &\frac1m(\Delta_fu)^2+{\rm Ric}_f^m(\nabla u, \nabla u)+|\nabla\nabla u-\frac{\Delta u}ng|^2+(\frac1n-\frac1m)(\Delta u+\frac{n}{m-n}\nabla f\cdot\nabla u)^2, \end{align*} $

the curvature dimensional condition $ CD(-K, m) $ for $ m \in (-\infty, 0) \cup [n, \infty) $ is equivalent to $ m $-dimensional Bakry-Émery Ricci curvature $ {\rm Ric}^m_f $ is bounded below by $ -K $, i.e.,

$ \begin{equation} {\rm Ric}^m_f\doteqdot{\rm Ric}_f-\frac1{m-n}\nabla f\otimes\nabla f\doteqdot{\rm {Ric}}+\nabla^{2} f-\frac1{m-n}\nabla f\otimes\nabla f\ge-Kg. \end{equation} $

(1.1)

In this paper, we consider the weighted doubly nonlinear diffusion equation

$ \begin{equation} \partial_{t}u = \Delta_{p, f} u^{\gamma}, \end{equation} $

(1.2)

where $ \Delta_{p, f}\cdot\doteqdot e^{f}{\rm div}(e^{-f}|\nabla \cdot |^{p-2}\nabla \cdot) $ is the weighted $ p $-Laplacian, which appears in non-Newtonian fluids, turbulent flows in porous media, glaciology and other contexts. We are mainly concerned on the concavity of the Rényi type entropy power with respecting to the equation (1.2) on the weighted Riemannian manifolds.

In the classic paper [3], Shannon defined the entropy power $ N(X) $ for random vector $ X $ on $ \mathbb {R}^{n} $,

$ N(X)\doteqdot e^{\frac{2}{n}H(X)}, \quad H(X)\doteqdot -\int_{\mathbb{R}^{n}}u\log udx, $

where $ u $ is the probability density of $ X $ and $ H(X) $ is the information entropy. Moreover, he proved the entropy power inequality (EPI) for independent random vectors $ X $ and $ Y $,

$ \begin{equation} N(X+Y)\ge N(X)+N(Y). \end{equation} $

(1.3)

Later, Costa [4] proved EPI when one of a random vector is Gaussian and established an equivalence between EPI and the concavity of Shannon entropy power $ N(u) $ when $ u $ satisfies heat equation $ \partial_{t}u = \Delta u $, that is,

$ \frac{d^{2}}{dt^{2}}N(u(t))\leq0. $

Moreover, Villani [2] gave a short proof by using of the Bakry-Émery identities. In a recent paper, Savaré and Toscani [5] proved the concavity of the Rényi entropy power $ N_{\gamma}(u) $ along the porous medium equation $ \partial_{t}u = \Delta u^{\gamma} $ on $ \mathbb{R}^n $, where $ N_{\gamma}(u) $ is given by

$ \begin{align} N_{\gamma}(u)\doteqdot(\int_{\mathbb {R}^{n}}u^{\gamma}(x)dx)^{\alpha}, \quad \alpha = \frac{2}{n(\gamma-1)}+1. \end{align} $

(1.4)

In [6], the first author and coauthors studied the the concavities of $ p $-Shannon entropy power for positive solutions to $ p $-heat equations on Riemannian manifolds with nonnegative Ricci curvature.

On the other hand, Li and Li [11] proved the concavity of Shannon entropy power for the heat equation $ \partial_tu = \Delta_f u $ and the concavity of Rényi entropy power for the porous medium equation $ \partial_tu = \Delta_f u^{\gamma} $ with $ \gamma>1 $ on weighted Riemannian manifolds with $ CD(0, m) $ or $ CD(-K, m) $ condition, and also on $ (0, m) $ or $ (-K, m) $ super Ricci flows.

Inspired by works mentioned above, we study $ p $-Rényi type entropy power for the weighted doubly nonlinear diffusion equation (1.2) on the weighted Riemannian manifolds and prove its concavity under the curvature dimensional condition $ CD(0, m) $ and $ CD(-K, m) $.

Let us define the weighted $ p $-Rényi entropy $ R_p(u) $ and $ p $-Rényi entropy power $ N_p(u) $ on the weighted Riemannian manifolds,

$ \begin{equation} R_{p}(u)\doteqdot-\frac{1}{b}\log\int_{M}u^{b+1}d\mu\doteqdot-\frac{1}{b}\log(E_{p}(u)), \end{equation} $

(1.5)

$ \begin{equation} N_{p}(u)\doteqdot\exp({-\sigma b R_{p}(u)})\doteqdot(E_{p}(u))^{\sigma}, \end{equation} $

(1.6)

where $ b $ and $ \sigma $ are constants

$ b = \gamma-\frac1{p-1}, \quad \sigma = -(p-1)-\frac{p}{mb}, $

and $ E_p(u) $ is given by

$ \begin{align} E_p(u)\doteqdot \int_Mu^{b+1}d\mu. \end{align} $

(1.7)

When $ p = 2 $ and $ f $ = const., (1.6) reduces to (1.4).

Theorem 1.1  Let $ (M, g, d\mu) $ be a weighted closed Riemannian manifold. If $ u $ is a positive solution to (1.2) on $ (M, g, d\mu) $, then we have

$ \begin{align} &\frac{d^2}{dt^2}N_{p}(u) = \sigma(1-\sigma)b^{2}(E_{p}(u))^{\sigma-1}\int_M(\Delta_{p, f}v+I_{p}(u))^{2}u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M[||\nabla v|^{p-2}\nabla^{2} v-\frac{1}{n}(\Delta_{p}v) a_{ij}|_{A}^{2}+ |\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v)]u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M[(\frac{1}{n}-\frac{1}{m})(\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f)^{2}]u^{b+1}d\mu, \end{align} $

(1.8)

where $ a_{ij} $ is the inverse of $ A^{ij} = g^{ij}+\left(p-2\right)\frac{\nabla^i v\nabla^j v}{|\nabla v|^2} $ and $ I_p(u) $ is the weighted Fisher information in (3.3).

When $ m\geq n $, $ b\geq-\frac{1}{m} $, $ b\neq 0 $ and $ M $ satisfies the curvature-dimension condition $ CD(0, m) $, the weighted $ p $-Rényi entropy power is concave, that is,

$ \begin{equation*} \frac{d^2}{dt^2}N_{p}(u)\leq0. \end{equation*} $

By modified the definition of $ p $-Rényi entropy power, we can also obtain the concavity under the curvature dimensional condition $ CD(-K, m) $.

Theorem 1.2  If $ u $ is a positive solution to (1.2) on the weighted closed Riemannian manifolds with curvature dimensional condition $ CD(-K, m) $ for $ K\geq0 $, define $ p $-$ K $-Rényi entropy power $ N_{p, K} $ such that

$ \begin{equation} \frac{d}{dt}N_{p, K}(u) = \exp\{-\frac{p\gamma}{b+1}\kappa t\}\frac{d}{dt}N_{p}(u), \end{equation} $

(1.9)

where

$ \begin{equation} \kappa\doteqdot\frac{\int_M|\nabla v|^{2p-2}u^{b+1}d\mu}{\int_M|\nabla v|^{p}ud\mu}K, \end{equation} $

(1.10)

we obtain

$ \begin{align} &\frac{d^2}{dt^2}N_{p, K}(u) = \sigma(1-\sigma)b^{2}(E_{p}(u))^{\sigma-1}\exp\{-\frac{p\gamma}{b+1}\kappa t\}\int_M|\Delta_{p, f}v+I_{p}(u)|^{2}u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\exp\{-\frac{p\gamma}{b+1}\kappa t\}\int_M||\nabla v|^{p-2}\nabla^{2} v-\frac{1}{n}(\Delta_{p}v) a_{ij}|_{A}^{2}u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\exp\{-\frac{p\gamma}{b+1}\kappa t\}\int_M|\nabla v|^{2p-4}({\rm Ric}_{f}^{m}+Kg)(\nabla v, \nabla v)u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\exp\{-\frac{p\gamma}{b+1}\kappa t\}\int_M[(\frac{1}{n}-\frac{1}{m})(\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f)^{2}]u^{b+1}d\mu. \end{align} $

(1.11)

When $ m\ge n $, $ b\geq -\frac{1}{m} $, $ b\neq 0 $ and $ {\rm Ric}^m_f\ge-Kg $, we have

$ \begin{equation} \frac{d^{2}}{dt^{2}}N_{p, K}(u)\leq 0. \end{equation} $

(1.12)

This paper is organized as follows. In section 2, we will prove some identities as lemmas, and they are important tools for proving theorems. In section 3, we will finish the proofs of the main results.

According to [7], the pressure function is given by

$ \begin{equation} v = \frac{\gamma}{b}u^{b}, \quad b = \gamma-\frac{1}{p-1}, \end{equation} $

(2.1)

then $ v $ satisfies the following equation

$ \begin{equation} \partial_{t}v = bv\Delta_{p, f}v+|\nabla v|^{p}. \end{equation} $

(2.2)

Let us recall the weighted entropy functional

$ \begin{equation} E_{p}(u) = \int_{M} u^{b+1}d\mu, \end{equation} $

(2.3)

and the weighted $ p $-Rényi entropy

$ \begin{equation} R_{p}(u) = -\frac{1}{b}\log\int_{M}u^{b+1}d\mu = -\frac{1}{b}\log(E_{p}(u)), \end{equation} $

(2.4)

where $ d\mu = e^{-f}d $Vol. Hence, the weighted $ p $-Rényi entropy power is given by

$ \begin{equation} N_{p}(u) = \exp(-b\sigma R_{p}(u)) = (E_{p}(u))^{\sigma}, \quad -\frac{1}{\sigma} = \frac{mb}{(p-1)mb+p}. \end{equation} $

(2.5)

In order to obtain our results, we need to prove following lemmas. Motivated by the works of [8], we apply analogous methods in this paper.

Lemma 2.1   For any two functions $ h $ and $ g $, let us define the linearized operator of the weighted $ p $-Laplacian at point $ v $,

$ {L}_{p, f}(\psi)\doteqdot e^{f}{\rm div}(e^{-f}|\nabla v|^{p-2}A(\nabla \psi)), $

where $ A^{ij} = g^{ij}+(p-2)\frac{\nabla^i v\nabla^j v}{|\nabla v|^2} $, then we get

$ \begin{equation} {L}_{p, f}(hg) = g({L}_{p, f}h)+({L}_{p, f}g)h+2|\nabla v|^{p-2}\langle \nabla h, \nabla g\rangle_{A}, \end{equation} $

(2.6)

$ \begin{equation} \int_M({L}_{p, f}h)gd\mu = \int_M h({L}_{p, f}g)d\mu = -\int_M |\nabla v|^{p-2}\langle \nabla h, \nabla g\rangle_{A}d\mu, \end{equation} $

(2.7)

where $ \langle \nabla h, \nabla g\rangle_{A} = A(\nabla h)\cdot\nabla g = \nabla h\cdot A(\nabla g). $

proof  The proof is a direct result by the definition of $ {L}_{p, f} $ and integration by parts.

Lemma 2.2   We have the modified weighted $ p $-Bochner formula

$ \begin{equation} {L}_{p, f}|\nabla v|^{p} = p\Gamma_{2, A}(v)+p|\nabla v|^{p-2}\langle\nabla v, \nabla\Delta_{p, f}v\rangle, \end{equation} $

(2.8)

where

$ \begin{align} \Gamma_{2, A}(v)\doteqdot& |\nabla v|^{2p-4}(|\nabla\nabla v|^{2}_{A}+{\rm Ric}_f(\nabla v, \nabla v)) \end{align} $

(2.9)

$ \begin{align} = &\frac{1}{m}(\Delta_{p, f}v)^{2}+||\nabla v|^{p-2}\nabla\nabla v-\frac{1}{n}(\Delta_{p}v) a_{ij}|_{A}^{2} +|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v)\\ &+(\frac{1}{n}-\frac{1}{m})(\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f)^{2}. \end{align} $

(2.10)

proof  According to [9], we have

$ \begin{equation*} {L}_{p, f}|\nabla v|^{p} = p|\nabla v|^{2p-4}\left(|\nabla\nabla v|^{2}_{A}+{\rm Ric}_f(\nabla v, \nabla v)\right)+p|\nabla v|^{p-2}\langle\nabla v, \nabla\Delta_{p, f}v\rangle, \end{equation*} $

where $ |\nabla\nabla v|_A^2 = \frac{(p-2)^{2}}{4}\frac{\left|\nabla v\cdot\nabla|\nabla v|^{2}\right|^{2}}{|\nabla v|^{4}}+\frac{p-2}{2}\frac{\left|\nabla|\nabla v|^{2}\right|^2}{|\nabla v|^{2}}+|\nabla\nabla v|^2. $

Using the definition of $ m $-Bakry-Emery Ricci curvature in (1.1), we find

$ \begin{align*} \Gamma_{2, A}(v) = &|\nabla v|^{2p-4}|\nabla\nabla v|^{2}_{A}+|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v)+\frac{1}{m-n}(|\nabla v|^{p-2}\nabla v\cdot\nabla f)^{2}\\ = &||\nabla v|^{p-2}\nabla\nabla v-\frac1n(\Delta_pv)a_{ij}|^2_A+\frac1n(\Delta_pv)^2+|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v)\\ &+\frac{1}{m-n}(|\nabla v|^{p-2}\nabla v\cdot\nabla f)^{2}\\ = &\frac1m(\Delta_{p, f}v)^2+||\nabla v|^{p-2}\nabla\nabla v-\frac1n\Delta_pva_{ij}|_A^2+|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v)\\ &+(\frac1n-\frac1m)(\Delta_pv+\frac{n}{m-n} |\nabla v|^{p-2}\nabla v\cdot\nabla f)^2, \end{align*} $

which completes the proof of (2.8). In particular, when $ m>n $ or $ m<0 $, we have

$ \begin{equation} \Gamma_{2, A}(v)\geq\frac{1}{m}(\Delta_{p, f}v)^{2}+|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v). \end{equation} $

(2.11)

Lemma 2.3    If $ u $, $ v $ satisfies equations (1.2) and (2.2), we obtain that

$ \begin{equation} \label{pDelta{p, f}} \partial_{t}(\Delta_{p, f}v) = {L}_{p, f}(\partial_{t}v), \end{equation} $

(2.12)

$ \begin{equation} \partial_{t}(uv) = \frac{b}{p-1}v{L}_{p, f}(uv). \end{equation} $

(2.13)

proof  By the definitions of $ A $ and $ {L}_{p, f} $, then

$ \begin{align*} \partial_{t}(\Delta_{p, f}v) = &\partial_{t}[e^{f}{\rm div}(e^{-f}|\nabla v |^{p-2}\nabla v)] \\ = &e^{f}{\rm div}(e^{-f}[(p-2)|\nabla v |^{p-4}(\nabla v, \nabla \partial_{t} v)\nabla v+|\nabla v|^{p-2}\nabla \partial_{t} v])\\ = &e^{f}{\rm div}(e^{-f}|\nabla v |^{p-2}A(\nabla \partial_{t} v))\\ = &{L}_{p, f}(\partial_{t} v). \end{align*} $

By the definition of $ v $ in (2.1), we have $ u\nabla v = bv\nabla u $. Hence

$ \begin{align*} A(\nabla(uv)) = (1+\frac{1}{b})A(u\nabla v) = (p-1)(1+\frac{1}{b})u\nabla v \end{align*} $

and

$ \begin{align} {L}_{p, f}(uv) = &e^{f}{\rm div}\left(e^{-f}|\nabla v |^{p-2}A(\nabla (uv))\right)\\ = &(p-1)(1+\frac{1}{b})e^{f}{\rm div}\left(e^{-f}u|\nabla v |^{p-2}\nabla v\right)\\ = &(p-1)(1+\frac{1}{b})\partial_{t}u. \end{align} $

(2.14)

A direct calculation shows that

$ \begin{align} \partial_{t}(uv) = (1+b)v\partial_{t}u. \end{align} $

(2.15)

Combining (2.14) with (2.15), we can show (2.13).

Lemma 2.4   Let $ u $ be a positive solution to (1.2) and $ v $ satisfies (2.2), then

$ \begin{equation} \frac{d}{dt}E_{p}(u) = b\int_{M}(\Delta_{p, f}v)u^{b+1}d\mu = -\frac{b(b+1)}{\gamma}\int_{M}|\nabla v |^{p}ud\mu = -\frac{b(b+1)}{\gamma} \int_{M}u^{-\frac{1}{p-1}}|\nabla u^{\gamma} |^{p}ud\mu \end{equation} $

(2.16)

and

$ \begin{equation} \frac{d^{2}}{dt^{2}}E_{p}(u) = pb\int_{M} [\Gamma_{2, A}(v)+b(\Delta_{p, f}v)^{2} ]u^{b+1}d\mu , \end{equation} $

(2.17)

where $ \Gamma_{2, A} $ is defined in (2.10).

proof  By integrating by parts, we can obtain

$ \begin{align*} \frac{d}{dt}E_{p}(u) = &(b+1)\int_{M}u^{b}\partial_{t}ud\mu = \frac{b(b+1)}{\gamma}\int_{M}ve^{f}{\rm div} (e^{-f}u|\nabla v |^{p-2}\nabla v )d\mu\\ = &-\frac{b(b+1)}{\gamma}\int_{M}|\nabla v |^{p}ud\mu = b\int_{M}(\Delta_{p, f}v)u^{b+1}d\mu, \end{align*} $

where

$ u\nabla v = bv\nabla u, \quad buv = \gamma u^{b+1}. $

Using identities (2.2), (2.8), (2.12) and (2.13), we flnd that

$ \begin{align*} &\frac{d}{dt}\int_M \Delta_{p, f}v(uv)d\mu = \int_{M} [\partial_{t}(\Delta_{p, f}v)(uv)+(\Delta_{p, f}v)\partial_{t}(uv) ]d\mu\\ = &\int_M[{L}_{p, f}(\partial_{t}v)(uv)+\frac{b}{p-1}{L}_{p, f}(uv)(v\Delta_{p, f}v)]d\mu\\ = &\int_M[b{L}_{p, f}(v\Delta_{p, f})(uv)+{L}_{p, f}(|\nabla v|^{p})(uv)+\frac{b}{p-1}{L}_{p, f}(v\Delta_{p, f}v)(uv)]d\mu\\ = &p\int_M[\Gamma_{2, A}(v)+|\nabla v|^{p-2}\langle\nabla v, \nabla\Delta_{p, f}v\rangle]uvd\mu+\frac{pb}{p-1}\int_M{L}_{p, f}(v\Delta_{p, f}v)(uv)d\mu. \end{align*} $

According to the properties of the linearized operator $ {L}_{p, f} $ (2.6) and (2.7), we have

$ \begin{align*} \int_M{L}_{p, f}(v\Delta_{p, f}v)(uv)d\mu = &\int_M({L}_{p, f}(v))(\Delta_{p, f}v)(uv)d\mu\\ &+\int_M{L}_{p, f}(\Delta_{p, f}v)(uv^{2})d\mu+2\int_M|\nabla v|^{p-2}\langle\nabla v, \nabla\Delta_{p, f}v\rangle_{A}(uv)d\mu\\ = &(p-1)\int_M(\Delta_{p, f}v)^{2}(uv)d\mu-\int_M|\nabla v|^{p-2}\langle\nabla (uv^{2}), \nabla\Delta_{p, f}v\rangle_{A}d\mu\\ &+2\int_M|\nabla v|^{p-2}\langle\nabla v, \nabla\Delta_{p, f}v\rangle_{A}(uv)d\mu\\ = &(p-1)\int_M(\Delta_{p, f}v)^{2}(uv)d\mu-\frac{1}{b}\int_M|\nabla v|^{p-2}\langle\nabla v, \nabla\Delta_{p, f}v\rangle_{A}(uv)d\mu, \end{align*} $

where we use the fact

$ \nabla(uv^{2}) = (\frac{1}{b}+2)uv\nabla v. $

Hence, we get

$ \begin{align*} b\frac{d}{dt}\int_M(\Delta_{p, f}v)u^{b+1}d\mu = &\frac{b^{2}}{\gamma}\frac{d}{dt}\int_M(\Delta_{p, f}v)(uv)d\mu\\ = &\frac{b^{2}}{\gamma}\int_M[p\Gamma_{2, A}(v)+pb(\Delta_{p, f}v)^{2}]uvd\mu\\ = &pb\int_M[\Gamma_{2, A}(v)+b(\Delta_{p, f}v)^{2}]u^{b+1}d\mu, \end{align*} $

which is (2.17).

Proof of Theorem 1.1   Let $ \sigma $ be a constant, $ N_{p}(u) = (E_{p}(u))^{\sigma} $, a direct computation implies

$ \begin{align*} \frac{d}{dt}N_{p}(u) = \sigma(E_{p}(u))^{\sigma-1}\frac{d}{dt}E_{p}(u). \end{align*} $

By using of identities (2.16), (2.10) and (2.17), we get that

$ \begin{align} &\frac{d^2}{dt^2}N_{p}(u)\\ = &\sigma(E_{p}(u))^{\sigma-2} [E_{p}(u)\frac{d^2}{dt^2}(E_{p}(u))+(\sigma-1)(\frac{d}{dt}(E_{p}(u)))^2 ]\\ = &\sigma(E_{p}(u))^{\sigma-1}\int_M pb [\Gamma_{2, A}(v)+b(\Delta_{p, f}v)^{2} ]u^{b+1}d\mu +\sigma(\sigma-1)(E_{p}(u))^{\sigma-2} (\int_M(b\Delta_{p, f}v)u^{b+1}d\mu )^2\\ = &pb\sigma(E_{p}(u))^{\sigma-1}\int_M [(\frac{1}{m}+b)(\Delta_{p, f}v)^{2}+ ||\nabla v|^{p-2}\nabla^{2}v-\frac{1}{n}(\Delta_{p}v) a_{ij} |_{A}^{2}\\& +|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v) ]u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M [ (\frac{1}{n}-\frac{1}{m} ) (\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f )^{2} ]u^{b+1}d\mu\\ &+\sigma(\sigma-1)(E_{p}(u))^{\sigma-2} (\int_M(b\Delta_{p, f}v)u^{b+1}d\mu )^2. \end{align} $

(3.1)

In particular, when $ b>0 $, $ \sigma<0 $ or $ -\frac{1}{m}\leq b<0 $, $ \sigma>1 $ and $ {\rm Ric}_{f}^{m}\ge0 $, the Cauchy-Schwarz inequality yields

$ \begin{align*} \frac{d^2}{dt^2}N_{p}(u)\leq&\sigma(E_{b}(u))^{\sigma-1}\int_Mp(\frac{1}{mb}+1)(b\Delta_{p, f}v)^{2}u^{b+1}d\mu\\ &+(\sigma-1)(E_{p}(u))^{\sigma-2} (\int_M(b\Delta_{p, f}v)u^{b+1}d\mu )^2\\ \leq&\sigma(E_{p}(u))^{\sigma-1} [p(\frac{1}{mb}+1)+(\sigma-1) ]\int_M(b\Delta_{p, f}v)^{2}u^{b+1}d\mu. \end{align*} $

We can choose a proper constant $ \sigma $ such that

$ \begin{equation} p(\frac{1}{mb}+1)+(\sigma-1) = 0. \end{equation} $

(3.2)

Thus $ \frac{d^2}{dt^2}N_{p}(u)\le0 $, that is the weighted $ p $-Rényi entropy power is concave along the weighted doubly nonlinear diffusion equations (1.2) on $ (M, g, d\mu) $ with $ CD(0, m) $ condition.

In fact, we can obtain a precise form of the second order derivative of the weighted $ p $-Rényi entropy $ N_p(u) $. Let $ I_{p}(u) $ be the weighted Fisher information with respect to $ R_p(u) $,

$ \begin{align} I_{p}(u)\doteqdot\frac{d}{dt}R_{p}(u) = -\frac{1}{b}\frac{1}{E_{p}(u)}\frac{d}{dt}E_{p}(u) = -\frac{1}{\int_{M}u^{b+1}d\mu} \int_{M}(\Delta_{p, f}v)u^{b+1}d\mu. \end{align} $

(3.3)

Applying identities (3.1) and (3.2), we get

$ \begin{align*} &\frac{d^2}{dt^2}N_{p}(u) = \sigma(1-\sigma)b^{2}(E_{p}(u))^{\sigma-1}[\int_M(\Delta_{p, f}v)^{2}u^{b+1}d\mu -(E_{p}(u))^{-1}(\int_{M}(\Delta_{p, f}v)u^{b+1}d\mu)^{2}]\\ +&pb\sigma(E_{p}(u))^{\sigma-1}\int_M[||\nabla v|^{p-2}\nabla^{2}v-\frac{1}{n}(\Delta_{p}v) a_{ij}|_{A}^{2}+|\nabla v|^{2p-4}{\rm Ric}_{f}^{m}(\nabla v, \nabla v)]u^{b+1}d\mu\\ +&pb\sigma(E_{p}(u))^{\sigma-1}\int_M[(\frac{1}{n}-\frac{1}{m})(\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f)^{2}]u^{b+1}d\mu. \end{align*} $

Combining this with (3.3), one has

$ \begin{align*} &\frac{d^2}{dt^2}N_{p}(u) = \sigma(1-\sigma)b^{2}(E_{p}(u))^{\sigma-1}\int_M|\Delta_{p, f}v+I_{p}(u)|^{2}u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M[||\nabla v|^{p-2}\nabla^{2}v-\frac{1}{n}(\Delta_{p}v) a_{ij}|_{A}^{2}+|\nabla v|^{2p-4} {\rm Ric}_{f}^{m}(\nabla v, \nabla v)]u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M[(\frac{1}{n}-\frac{1}{m})(\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f)^{2}]u^{b+1}d\mu. \end{align*} $

Thus, we obtain an explicit expression about $ \frac{d^2}{dt^2}N_{p}(u) $. Moreover, if $ -\frac{1}{m}\leq b<0 $, $ \sigma>1 $ and $ b>0 $, $ \sigma<0 $, then $ \frac{d^2}{dt^2}N_{p}(u)\leq0 $.

Proof of Theorem 1.2  Motivated by [10], formula (1.8) can be rewritten as

$ \begin{align} &\frac{d^2}{dt^2}N_{p}(u) = \sigma(1-\sigma)b^{2}(E_{p}(u))^{\sigma-1}\int_M|\Delta_{p, f}v+I_{p}(u)|^{2}u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M [ ||\nabla v|^{p-2}\nabla^{2}v-\frac{1}{n}(\Delta_{p}v) a_{ij} |_{A}^{2}+|\nabla v|^{2p-4}({\rm Ric}_{f}^{m}+Kg)(\nabla v, \nabla v) ]u^{b+1}d\mu\\ &+pb\sigma(E_{p}(u))^{\sigma-1}\int_M [ (\frac{1}{n}-\frac{1}{m} ) (\Delta_{p}v+\frac{n}{m-n}|\nabla v|^{p-2}\nabla v \cdot\nabla f )^{2} ]u^{b+1}d\mu+\frac{p\gamma}{b+1}\kappa\frac{d}{dt}N_{p}(u), \end{align} $

(3.4)

where we use the definition of $ \kappa $ in (1.10) and variaitonal formula in (2.16), that is

$ \begin{align*} \frac{d}{dt}N_{p}(u) = \sigma(E_{p}(u))^{\sigma-1}\frac{d}{dt}E_{p}(u) = -\frac{\sigma b(b+1)}{\gamma}(E_{p}(u))^{\sigma-1}\int_{M}|\nabla v |^{p}ud\mu. \end{align*} $

Defining a functional $ N_{p, K} $ such that [11]

$ \begin{equation*} \frac{d}{dt}N_{p, K}(u)\doteqdot\exp \{-\frac{p\gamma}{b+1}\kappa t \}\frac{d}{dt}N_{p}(u). \end{equation*} $

A direct computation yields

$ \begin{equation} \frac{d^2}{dt^2}N_{p, K}(u) = \exp \{-\frac{p\gamma}{b+1}\kappa t \} (\frac{d^2}{dt^2}N_{p}(u)-\frac{p\gamma}{b+1}\kappa\frac{d}{dt}N_{p}(u) ). \end{equation} $

Hence, when $ (M, g, d\mu) $ satisfies the curvature dimensional condition $ CD(-K, m) $, i.e., $ {\rm Ric}_{f}^{m}\geq-Kg $ with $ K>0 $, $ m\geq n $, by formula (3.5), we obtain the concavity of $ N_{p, K} $, that is $ \frac{d^2}{dt^2}N_{p, K}(u)\leq 0. $ Furthermore, we obtain an explicit variational formula (1.11), which finishes the proof of Theorem 1.2.