Hamilton's gradient estimates and Liouville theorems for porous medium equations
Guangyue Huang1,
Ruiwei Xu1 &
Fanqi Zeng2
Journal of Inequalities and Applications volume 2016, Article number: 37 (2016) Cite this article
Let \((M^{n}, g)\) be an n-dimensional Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the porous medium equation
$$u_{t}=\Delta\bigl(u^{p}\bigr),\quad 1< p< 1+\frac{1}{\sqrt{n-1}}, $$
posed on \((M^{n}, g)\) with the Ricci curvature bounded from below. Moreover, we also obtain a Liouville type theorem. In particular, the results obtained in this paper generalize those in (Zhu in J. Math. Anal. Appl. 402:201-206, 2013).
In this paper we study the porous medium equation
$$ u_{t}=\Delta\bigl(u^{p}\bigr) $$
with \(p>1\), which is a nonlinear extension of the classical heat equation. As is typical of nonlinear problems, the mathematical theory of the porous medium equation is based on a priori estimates. In 1979, Aronson and Bénilan obtained a celebrated second-order differential inequality of the form [2]
$$\sum_{i}\frac{\partial}{\partial x^{i}} \biggl(pu^{p-2} \frac{\partial u}{\partial x^{i}} \biggr)\geq-\frac{k}{t},\quad k:=\frac{n}{n(p-1)+2}, $$
which applies to all positive smooth solutions of (1.1) defined on the whole Euclidean space on the condition that \(p>1-\frac{2}{n}\). For various values of \(p>1\), it has appeared in different applications to model diffusive phenomena; see [2–4] and the references therein.
In [5], Hamilton proved the following results.
Theorem A
(Hamilton [5])
Let \((M^{n}, g)\) be an n-dimensional compact Riemannian manifold with \(\operatorname{Ric}(M^{n})\geq-K\), where K is a non-negative constant. Suppose that u is a positive solution to the heat equation
$$ u_{t}=\Delta u $$
with \(u< M\) for all \((x,t)\in M^{n}\times(0,\infty)\). Then
$$ \frac{|\nabla u|^{2}}{u^{2}}\leq \biggl(\frac{1}{t}+2K \biggr)\log \frac{M}{u}. $$
Hamilton's estimate tells us that when the temperature is bounded we can compare the temperature of two different points at the same time. For the study of gradient estimates of the equation (1.1), see [6–10] and the references therein. In [1], Zhu applied similar techniques that were used for the heat equation, he derived the following Hamilton type estimate for equation (1.1).
Theorem B
(Zhu [1])
Let \((M^{n}, g)\) be an n-dimensional Riemannian manifold with \(\operatorname{ Ric}(M^{n})\geq-K\), where K is a non-negative constant. Suppose that u is a positive solution to the porous medium equation (1.1) in \(Q_{R,T}:=B_{x_{0}}(R)\times[t_{0}-T, t_{0}]\subset M^{n}\times(-\infty, \infty)\). Let \(v=\frac{p}{p-1}u^{p-1}\). Then for \(1< p<1+\frac{1}{\sqrt{2n}+1}\) and \(v\leq M\),
$$ v^{\frac{1}{4}\frac{2-p}{p-1}}|\nabla v|\leq CM^{1+\frac{1}{4}\frac{2-p}{p-1}} \biggl( \frac{1}{R}+\frac{1}{\sqrt {T}}+\sqrt{K} \biggr) $$
in \(Q_{R,T}\), where \(C=C(p,n)\) is a constant depending only on p and n.
In this paper, by introducing a new parameter and using a lemma in [11], we generalize Theorem B as follows.
Let \((M^{n}, g)\) be an n-dimensional Riemannian manifold with \(\operatorname{Ric}(M^{n})\geq-K\), where K is a non-negative constant. Suppose that u is a positive solution to the porous medium equation (1.1) in \(Q_{R,T}:=B_{x_{0}}(R)\times[t_{0}-T, t_{0}]\subset M^{n}\times(-\infty, \infty)\). Let \(v=\frac{p}{p-1}u^{p-1}\). Then for \(1< p<1+\frac{1}{\sqrt{n-1}}\) and \(v\leq M\),
$$ v^{\frac{1}{2(p-1)}}|\nabla v|\leq CM^{1+\frac{1}{2(p-1)}} \biggl( \frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{K} \biggr) $$
As an application, we get the following Liouville type theorem.
Let \((M^{n}, g)\) be an n-dimensional complete noncompact Riemannian manifold with non-negative Ricci curvature. Let u be a positive ancient solution to the porous medium equation (1.1) with \(1< p<1+\frac{1}{\sqrt{n-1}}\) such that \(u(x, t)=o ([d(x)+\sqrt{|t|}]^{\frac{2}{2p-1}} )\) near infinity. Then u must be a constant.
Note that \(\frac{1}{\sqrt{2n}+1}<\frac{1}{\sqrt{n-1}}\). Therefore, the results obtained in this paper generalize those of Zhu in [1].
Proof of Theorem 1.1
Let \(v=\frac{p}{p-1}u^{p-1}\). From (1.1), by simple calculations, it is easy to see that
$$ v_{t}=(p-1)v\Delta v+|\nabla v|^{2}. $$
We define
$$w=\frac{|\nabla v|^{2}}{v^{\beta}}, $$
where β is a constant to be determined. Then we have
$$\begin{aligned}& w_{t}= \frac{2v_{i}v_{it}}{v^{\beta}}-\beta\frac{v_{i}^{2}v_{t}}{v^{\beta+1}} \\& \hphantom{w_{t}}= 2\frac{v_{i}[(p-1)v\Delta v+|\nabla v|^{2}]_{i}}{v^{\beta}}-\beta\frac{v_{i}^{2}[(p-1)v\Delta v+|\nabla v|^{2}]}{v^{\beta+1}} \\& \hphantom{w_{t}}= 2(p-1)\frac{v_{i}^{2}v_{jj}}{v^{\beta}}+2(p-1)\frac{v_{i}v_{jji}}{v^{\beta-1}} +4\frac{v_{ij}v_{i}v_{j}}{v^{\beta}} -\beta(p-1)\frac{v_{i}^{2}v_{jj}}{v^{\beta}}-\beta\frac{v_{i}^{2}v_{j}^{2} }{v^{\beta+1}}, \end{aligned}$$
$$\begin{aligned}& w_{j}=\frac{2v_{i}v_{ij}}{v^{\beta}}-\beta\frac{v_{i}^{2}v_{j}}{v^{\beta+1}}, \end{aligned}$$
and hence
$$ w_{jj}=\frac{2v_{ij}^{2}}{v^{\beta}}+\frac{2v_{i}v_{ijj}}{v^{\beta}} -4\beta \frac{v_{ij}v_{i}v_{j}}{v^{\beta+1}}-\beta\frac{v_{i}^{2}v_{jj}}{v^{\beta+1}} +\beta(\beta+1)\frac{v_{i}^{2}v_{j}^{2}}{v^{\beta+2}}. $$
By virtue of (2.2) and (2.4),
$$\begin{aligned} (p-1)v\Delta w-w_{t} =&2(p-1)\frac{v_{ij}^{2}}{v^{\beta-1}}+2(p-1) \frac {v_{i}v_{ijj}}{v^{\beta-1}} -4\beta(p-1)\frac{v_{i}v_{j}v_{ij}}{v^{\beta}} \\ &{}-\beta(p-1)\frac{v_{i}^{2}v_{jj}}{v^{\beta}}+\beta(\beta+1) (p-1)\frac {v_{i}^{2}v_{j}^{2}}{v^{\beta+1}} \\ &{}-2(p-1)\frac{v_{i}^{2}v_{jj}}{v^{\beta}}-2(p-1)\frac{v_{i}v_{jji}}{v^{\beta -1}}-4\frac{v_{ij}v_{i}v_{j}}{ v^{\beta}} \\ &{}+\beta(p-1) \frac{v_{i}^{2}v_{jj}}{v^{\beta}}+\beta\frac{v_{i}^{2}v_{j}^{2}}{v^{\beta+1}} \\ =&2(p-1)\frac{v_{ij}^{2}}{v^{\beta-1}}-2(p-1)\frac{v_{i}^{2}v_{jj}}{v^{\beta}} +2(p-1)\frac{R_{ij}v_{i}v_{j}}{v^{\beta-1}} \\ &{}-4\bigl[1+\beta(p-1)\bigr]\frac{v_{ij}v_{i}v_{j}}{v^{\beta}} +\beta\bigl[(\beta+1) (p-1)+1 \bigr]\frac{v_{i}^{2}v_{j}^{2}}{v^{\beta+1}}, \end{aligned}$$
where, in the second equality, we use the Ricci formula: \(v_{ijj}-v_{jji}=R_{ij}v_{j}\).
In order to prove Theorem 1.1, we need the following lemma (cf. Lemma A.1 in [11]).
Let \(A=(a_{ij})\) be a nonzero \(n\times n\) symmetric matrix. Then for \(a, b\in\mathbb{R}\),
$$ \max_{A\in\mathcal{S}(n);|e|=1} \biggl(\frac{aA+b (\operatorname{ tr}A) I_{n}}{|A|}(e,e) \biggr)^{2}=(a+b)^{2}+(n-1)b^{2}, $$
where \(I_{n}\) is an identity matrix.
Notice that
$$ \nabla v\nabla w =2\frac{v_{ij}v_{i}v_{j}}{v^{\beta}}-\beta\frac{v_{i}^{2}v_{j}^{2}}{v^{\beta+1}}. $$
It follows from (2.5) that, for any constant ε,
$$\begin{aligned}& (p-1) v\Delta w-w_{t}+\varepsilon\nabla v\nabla w \\& \quad= 2(p-1)\frac{v_{ij}^{2}}{v^{\beta-1}}-2(p-1)\frac{v_{i}^{2}v_{jj}}{v^{\beta}} +2(p-1)\frac{R_{ij}v_{i}v_{j}}{v^{\beta-1}} \\& \quad\quad{}+\bigl[2\varepsilon-4\bigl(1+\beta(p-1)\bigr)\bigr]\frac{v_{ij}v_{i}v_{j}}{v^{\beta}} +\beta \bigl[(\beta+1) (p-1)+1-\varepsilon\bigr]\frac{v_{i}^{2}v_{j}^{2}}{v^{\beta+1}} \\& \quad= 2(p-1)\frac{|A|^{2}}{v^{\beta-1}}-2(p-1)\frac{\operatorname{tr}A}{|A|}w|A| +2(p-1)v\operatorname{Ric}(e,e)w \\& \quad\quad{}+\bigl[2\varepsilon-4\bigl(1+\beta(p-1)\bigr)\bigr]\frac{A(e,e)}{|A|}w|A| + \beta\bigl[(\beta+1) (p-1)+1-\varepsilon\bigr]v^{\beta-1}w^{2} \\& \quad= 2(p-1)\frac{|A|^{2}}{v^{\beta-1}}+ \biggl(\bigl[ 2\varepsilon-4\bigl(1+\beta(p-1)\bigr) \bigr]\frac{A(e,e)}{|A|}-2(p-1)\frac{\operatorname{tr}A}{|A|} \biggr)w|A| \\& \quad\quad{}+2(p-1)v\operatorname{Ric}(e,e)w+\beta\bigl[(\beta+1) (p-1)+1-\varepsilon \bigr]v^{\beta-1}w^{2} \\& \quad= 2(p-1) \biggl[\frac{|A|}{v^{\frac{\beta-1}{2}}}+\frac{1}{4(p-1)} \biggl(\bigl[ 2 \varepsilon-4\bigl(1+\beta(p-1)\bigr)\bigr]\frac{A(e,e)}{|A|}-2(p-1) \frac{\operatorname{tr}A}{|A|} \biggr)wv^{\frac{\beta-1}{2}} \biggr]^{2} \\& \quad\quad{}-\frac{1}{8(p-1)} \biggl(\bigl[ 2\varepsilon-4\bigl(1+\beta(p-1)\bigr)\bigr] \frac{A(e,e)}{|A|}-2(p-1)\frac{\operatorname{tr}A}{|A|} \biggr)^{2}v^{\beta-1}w^{2} \\& \quad\quad{}+2(p-1)v\operatorname{Ric}(e,e)w+\beta\bigl[(\beta+1) (p-1)+1-\varepsilon \bigr]v^{\beta-1}w^{2} \\& \quad\geq -\frac{1}{8(p-1)} \biggl|\bigl[ 2\varepsilon-4\bigl(1+\beta(p-1)\bigr)\bigr] \frac{A(e,e)}{|A|}-2(p-1)\frac{(\operatorname{tr}A)I_{n}(e,e)}{|A|} \biggr|^{2}v^{\beta-1}w^{2} \\& \quad\quad{}+2(p-1)v\operatorname{Ric}(e,e)w+\beta\bigl[(\beta+1) (p-1)+1-\varepsilon \bigr]v^{\beta-1}w^{2}, \end{aligned}$$
where \(A_{ij}=(v_{ij})\) and \(e=\nabla v/|\nabla v|\). By virtue of Lemma 2.1, we have
$$\begin{aligned}& (p-1) v\Delta w-w_{t}+\varepsilon\nabla v\nabla w \\& \quad\geq -\frac{1}{8(p-1)} \biggl|\bigl[ 2\varepsilon-4\bigl(1+\beta(p-1)\bigr)\bigr] \frac{A(e,e)}{|A|}-2(p-1)\frac{(\operatorname{tr}A)I_{n}(e,e)}{|A|} \biggr|^{2}v^{\beta-1}w^{2} \\& \quad\quad{}+2(p-1)v\operatorname{Ric}(e,e)w+\beta\bigl[(\beta+1) (p-1)+1-\varepsilon \bigr]v^{\beta-1}w^{2} \\& \quad\geq -\frac{1}{8(p-1)} \bigl(\bigl[2\varepsilon-4\bigl(1+\beta (p-1) \bigr)-2(p-1)\bigr]^{2}+4(n-1) (p-1)^{2} \\& \quad\quad{}-8(p-1)\beta\bigl[(\beta+1) (p-1)+1-\varepsilon\bigr] \bigr)v^{\beta -1}w^{2}+2(p-1)v\operatorname{Ric}(e,e)w \\& \quad= -\frac{1}{8(p-1)}f(\beta,\varepsilon)v^{\beta-1}w^{2}+2(p-1)v\operatorname{Ric}(e,e)w, \end{aligned}$$
$$f(\beta,\varepsilon)=\bigl[2\varepsilon-4\bigl(1+\beta(p-1)\bigr)-2(p-1) \bigr]^{2}+4(n-1) (p-1)^{2} -8(p-1)\beta\bigl[(\beta+1) (p-1)+1-\varepsilon\bigr]. $$
For the purpose of showing that the coefficient of \(v^{\beta-1}w^{2}\) is positive, we minimize the function \(f(\beta,\varepsilon)\) by letting
$$\varepsilon=p $$
$$\beta=-\frac{p-\varepsilon+2}{2(p-1)}=-\frac{1}{p-1}, $$
such that
$$f=-4+4(n-1) (p-1)^{2}. $$
Then (2.9) becomes
$$\begin{aligned} (p-1)v\Delta w-w_{t}+p\nabla v\nabla w \geq& \frac{1-(n-1)(p-1)^{2}}{2(p-1)}v^{\beta-1}w^{2}-2(p-1)Kvw \\ =&\alpha v^{\beta-1}w^{2}-2(p-1)Kvw, \end{aligned}$$
where \(\alpha=\frac{1-(n-1)(p-1)^{2}}{2(p-1)}\).
We first recall the well-known smooth cutoff function ψ which originated with Li and Yau [12], satisfying the following:
The cutoff function ψ satisfies \(\psi=\psi(d(x,x_{0}),t)\equiv\psi(r,t)\) and \(\psi(r,t)=1\) in \(Q_{R/2,T/2}\) with \(0\leq\psi\leq1\).
The function ψ is decreasing as a radial function in the spatial variables.
\(|\partial_{r}\psi|/\psi^{a}\leq C_{a}/R\) and \(|\partial_{r}^{2}\psi|/\psi^{a}\leq C_{a}/R^{2}\) for \(a\in(0,1)\).
\(|\partial_{t}\psi|/\psi^{\frac{1}{2}}\leq C/T\).
By virtue of (2.10), we have
$$\begin{aligned}& \bigl[(p-1)v\Delta-\partial_{t}\bigr](\psi w) \\& \quad= \psi\bigl[(p-1)v\Delta-\partial_{t}\bigr]w+(p-1)v w\Delta\psi-w \psi_{t} \\& \quad\quad{}+2(p-1)v\frac{1}{\psi}\nabla\psi\nabla(\psi w)-2(p-1)vw\frac{|\nabla \psi|^{2}}{\psi} \\& \quad\geq \alpha v^{\beta-1}\psi w^{2}-2(p-1)Kv\psi w-p\nabla v\nabla( \psi w) \\& \quad\quad{}+pw\nabla v\nabla\psi+(p-1)v w\Delta\psi-w\psi_{t} \\& \quad\quad{}+2(p-1)v\frac{1}{\psi}\nabla\psi\nabla(\psi w)-2(p-1)vw\frac{|\nabla\psi|^{2}}{\psi}. \end{aligned}$$
Next we will apply the maximum principle to ψw in a closed set. Assume ψw achieves its maximum at the point \((x_{1},t_{1})\) and assume \((\psi w)(x_{1},t_{1})>0\) (otherwise the proof is trivial), which implies \(t_{1}>0\). Then at the point \((x_{1},t_{1})\)
$$(\Delta-\partial_{t}) (\psi w)\leq0, \quad\nabla(\psi w)=0, $$
and (2.11) becomes
$$\begin{aligned} \alpha v^{\beta-1}\psi w^{2} \leq&-pw\nabla v\nabla \psi+2(p-1)vw\frac{|\nabla\psi|^{2}}{\psi} \\ &{}-(p-1)v w\Delta\psi+w\psi_{t}+2(p-1)Kv\psi w. \end{aligned}$$
$$\begin{aligned} 2\psi w^{2} \leq&-p\gamma v^{1-\beta} w\nabla v \nabla\psi+2(p-1)\gamma v^{2-\beta}w\frac{|\nabla\psi|^{2}}{\psi} \\ &{}-(p-1)\gamma v^{2-\beta} w\Delta\psi+\gamma v^{1-\beta} w \psi_{t}+2(p-1)\gamma Kv^{2-\beta}\psi w, \end{aligned}$$
where \(\alpha=\frac{2}{\gamma}\) and \(\gamma=\frac{1-(n-1)(p-1)^{2}}{4(p-1)}\).
It have been shown in [1] (see equations (2.6)-(2.10) in [1]) that
$$\begin{aligned} -p\gamma v^{1-\beta} w\nabla v\nabla\psi\leq \frac{1}{4}\psi w^{2}+CM^{4-2\beta}\frac{1}{R^{4}}, \end{aligned}$$
where we used the fact that \(0< v\leq M\) and \(\beta\leq2\),
$$\begin{aligned}& 2(p-1)\gamma v^{2-\beta}w\frac{|\nabla\psi|^{2}}{\psi}\leq \frac{1}{4}\psi w^{2}+CM^{4-2\beta}\frac{1}{R^{4}}, \end{aligned}$$
$$\begin{aligned}& -(p-1)\gamma v^{2-\beta} w\Delta\psi\leq\frac{1}{4} \psi w^{2}+CM^{4-2\beta} \biggl(\frac{1}{R^{4}}+K \frac{1}{R^{2}} \biggr), \end{aligned}$$
$$\begin{aligned}& \gamma v^{1-\beta} w\psi_{t}\leq\frac{1}{4} \psi w^{2}+CM^{4-2\beta}\frac{1}{T^{2}}, \end{aligned}$$
$$\begin{aligned}& 2(p-1)\gamma Kv^{2-\beta}\psi w\leq\frac{1}{4}\psi w^{2}+CM^{4-2\beta}K^{2}. \end{aligned}$$
Substituting (2.14)-(2.18) into (2.13), we obtain
$$ 2\psi w^{2}\leq \frac{5}{4}\psi w^{2}+CM^{4-2\beta} \biggl(\frac{1}{R^{4}}+\frac{1}{T^{2}}+K^{2} \biggr), $$
which gives at the point \((x_{1},t_{1})\)
$$ \psi w^{2}\leq CM^{4-2\beta} \biggl( \frac{1}{R^{4}}+\frac{1}{T^{2}}+K^{2} \biggr). $$
Therefore, for all \((x,t)\in Q_{R,T}\),
$$\begin{aligned} \bigl(\psi^{2} w^{2}\bigr) (x,t) \leq&\bigl( \psi^{2} w^{2}\bigr) (x_{1},t_{1}) \leq \bigl(\psi w^{2}\bigr) (x_{1},t_{1}) \\ \leq&CM^{4-2\beta} \biggl(\frac{1}{R^{4}}+\frac{1}{T^{2}}+K^{2} \biggr). \end{aligned}$$
Notice that \(\psi=1\) in \(Q_{R/2,T/2}\) and \(w=\frac{|\nabla v|^{2}}{v^{\beta}}\). Hence, we have
$$ \frac{|\nabla v|}{v^{\frac{\beta}{2}}}(x,t)\leq CM^{1-\frac{\beta}{2}}\biggl(\sqrt{K} + \frac{1}{R}+\frac{1}{\sqrt{T}}\biggr). $$
One concludes the proof of Theorem 1.1 by letting \(\beta=-\frac{1}{p-1}\).
Simple proof of Corollary 1.2
Suppose u is a positive ancient solution to the porous medium equation (1.1) such that \(v(x, t)=o ( [d(x_{0},x)+\sqrt{|t|} ]^{\frac{2}{2p-1}} ) \) near infinity, where \(v=\frac{p}{p-1}u^{p-1}\). Fixing \((x_{0},t_{0})\) in space-time and using Theorem 1.1 for u on the cube \(B(x_{0}, R) \times[t_{0} - R^{2} , t_{0}]\), we obtain
$$v(x_{0},t_{0})^{\frac{1}{2(p-1)}}\bigl|\nabla v(x_{0},t_{0})\bigr| \leq \frac{C}{R}\cdot o(R). $$
Letting \(R\rightarrow\infty\), it follows that \(|\nabla v(x_{0},t_{0})|=0\). Since \((x_{0},t_{0})\) is arbitrary, we see that v is a constant. Hence, u is also constant from \(v=\frac{p}{p-1}u^{p-1}\). Thus ends the proof of Corollary 1.2.
Zhu, XB: Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds. J. Math. Anal. Appl. 402, 201-206 (2013)
Article MATH MathSciNet Google Scholar
Aronson, DG, Bénilan, P: Régularité des solutions de l'équation des milieux poreux dans \({\mathbb {R}}^{n}\). C. R. Acad. Sci. Paris Sér. A-B 288, 103-105 (1979)
Li, S, Li, X-D: Perelman's entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials. arXiv:1303.6019
Lu, P, Ni, L, Vázquez, J, Villani, C: Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl. 91, 1-19 (2009)
Hamilton, RS: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1, 113-126 (1993)
Huang, G, Huang, Z, Li, H: Gradient estimates for the porous medium equations on Riemannian manifolds. J. Geom. Anal. 23(4), 1851-1875 (2013)
Huang, G, Li, H: Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian. Pac. J. Math. 268, 47-78 (2014)
Zhu, XB: Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds. Proc. Am. Math. Soc. 139, 1637-1644 (2011)
Article MATH Google Scholar
Wang, YZ, Chen, WY: Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds. Math. Methods Appl. Sci. 37, 2772-2781 (2014)
Souplet, P, Zhang, QS: Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38, 1045-1053 (2006)
Xu, XJ: Gradient estimates for \(u_{t}=\Delta F(u)\) on manifolds and some Liouville-type theorems. J. Differ. Equ. 252, 1403-1420 (2012)
Li, P, Yau, ST: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153-201 (1986)
Article MathSciNet Google Scholar
The authors' research was supported by NSFC (No. 11371018, 11171091,11401179) and partially supported by IRTSTHN (14IRTSTHN023) and Henan Provincial Education department (No. 14B110017, 12B110014).
Department of Mathematics, Henan Normal University, Xinxiang, 453007, China
Guangyue Huang & Ruiwei Xu
Department of Mathematics, Tongji University, Shanghai, 200092, China
Fanqi Zeng
Guangyue Huang
Ruiwei Xu
Correspondence to Ruiwei Xu.
Huang and Xu participated in gradient estimates in this paper. Zeng carried out the Laplacian operator and the applications of inequalities studies. All authors read and approved the final manuscript.
Huang, G., Xu, R. & Zeng, F. Hamilton's gradient estimates and Liouville theorems for porous medium equations. J Inequal Appl 2016, 37 (2016). https://doi.org/10.1186/s13660-016-0986-3
porous medium equation
Hamilton's gradient estimate
Liouville type theorem
Recent Advances in Inequalities and Applications