Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds

In this paper, we study the gradient estimates for positive solutions to the following parabolic Lichnerowicz equations

on complete noncompact Riemannian manifolds, where h, p, q, A, B are real constants and p>1, q>0.

MSC:58J05, 58J35.

Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we study the following nonlinear parabolic equation

where h, p, q, A, B are real constants and p>1, q>0.

Gradient estimates play an important role in the study of PDE, especially the Laplace equation and heat equation. Li [1] derived the gradient estimates and Harnack inequalities for positive solutions of nonlinear equations (\mathrm{△}-\frac{\partial }{\partial t})u(x,t)+h(x,t)u^{\alpha }(x,t)=0 and Au+b\mathrm{\nabla }u+h{u}^{\alpha }=0 on Riemannian manifolds. The author in [1] also obtained a theorem of Liouville-type for positive solutions of the nonlinear elliptic equation. Later, Yang [2] gave the gradient estimates for the solution to the elliptic equation with singular nonlinearity

where \alpha >0, c are two real constants. More precisely, the author [2] obtained the following result.

Theorem 1.1 (Yang [2])

Let M be a noncompact complete Riemannian manifold of dimension n without boundary. Let B_p(2R) be a geodesic ball of radius 2R around p\in M. We denote -K(2R), with K(2R)\ge 0, to be a lower bound of the Ricci curvature on B_p(2R), i.e., Ric(\xi ,\xi )\ge -K(2R){|\xi |}^2 for all tangent field ξ on B_p(2R). Suppose that u(x) is a positive smooth solution of the equation (1.2) with \alpha >0, c being two real constants. Then we have:

1. (i)

   If c>0, then u(x) satisfies the estimate

   \frac{{|\mathrm{\nabla }u|}^2}{u^2}+c{u}^{-(\alpha +1)}\le \frac{n(2n+1){ϵ}^2}{R^2}+\frac{n(n-1){ϵ}^2}{R}\sqrt{K(2R)}+\frac{n\nu }{R^2}+2nK(2R)

on B_p(R), where ϵ>0 and \nu >0 are some universal constants independent of geometry of M.

1. (ii)

   If , then u(x) satisfies the estimate

   \begin{array}{rcl}\frac{{|\mathrm{\nabla }u|}^2}{u^2}+c{u}^{-(\alpha +1)}& \le & (n(\alpha +1)(\alpha +2)+\sqrt{n}(\alpha +1))|c|{\left(\underset{B_p(2R)}{inf}u\right)}^{-\alpha -1}\\ +\frac{n\nu }{R^2}+(2n+\frac{\sqrt{n}}{\alpha +1})K(2R)\\ +\frac{n{ϵ}^2}{R^2}(2n+1)+\frac{\sqrt{n}}{2(\alpha +1)}+(n-1)\sqrt{K(2R)R}\end{array}

on B_p(R), where ϵ>0 and \nu >0 are some universal constants independent of geometry of M.

For some interesting gradient estimates in this direction, we can refer to [3–7].

Recently, Song and Zhao [8] studied a generalized elliptic Lichnerowicz equation

on compact manifold (M,g). The authors in [8] got the local gradient estimate for the positive solutions of (1.3). Moreover, they considered the following parabolic Lichnerowicz equation

on manifold (M,g) and obtained the Harnack differential inequality.

Theorem 1.2 (Song and Zhao [8])

Let M be a compact Riemannian manifold without boundary, Ric(M)\ge 0. Let c(t)\in C^1(0,\mathrm{\infty }). Assume that u(x,t) is any positive solution of (1.4) on M with A(x)\equiv A, B(x)\equiv B, and h(x)\equiv h. Denote \phi =lnu, suppose that A\le 0, B\ge 0, c(t)\ge 0, c^{\prime }(t)\ge 0. If {|\mathrm{\nabla }\phi |}^2-\frac{1}{p}{\phi }_t+\frac{1}{p}\tilde{H}-c(t)\le 0 with \tilde{H}=A{e}^{(p-1)\phi }+\frac{B}{e^{(q+1)\phi }}-h at t=0, then we have

While the author considered the gradient estimates on compact Riemannian manifolds in Theorem 1.2, it is natural to study this problem on complete noncompact manifolds. Motivated by the work above, we present our main results as follows.

Theorem 1.3 Let (M,g) be a complete noncompact n-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant -K=:-K(2R), where R>0 and K(2R)>0 in the metric ball B_{2R}(p) around p\in M. Assume that u is a positive solution of (1.1) with u\le M_1 for all (x,t)\in B_R(p)\times (0,+\mathrm{\infty }). Then

1. (1)

   if A\le 0, B\ge 0, we have

   \begin{array}{c}\beta \frac{{|\mathrm{\nabla }u|}^2}{u^2}+h+A{u}^{p-1}+B{u}^{-(q+1)}-\frac{u_t}{u}\hfill \\ \le \frac{n}{2(1-\delta )\beta }(\frac{n{c}_1^2}{4\delta \beta (1-\beta )R^2}-\frac{M_1^{p-1}A(p-1)(p-\beta )}{4{(1-\beta )}^2}+H+\frac{1}{t});\hfill \end{array}
2. (2)

   if A>0, B>0, we get

   \begin{array}{c}\beta \frac{{|\mathrm{\nabla }u|}^2}{u^2}+h+A{u}^{p-1}+B{u}^{-(q+1)}-\frac{u_t}{u}\hfill \\ \le \frac{n}{2(1-\delta )\beta }(\frac{n{c}_1^2}{4\delta \beta (1-\beta )R^2}+M_1^{p-1}A(p-1)+H+\frac{1}{t}),\hfill \end{array}

where H=\frac{(n-1)(1+\sqrt{K}R)c+c_2+2c_1^2}{R^2}, c, c_1, c_2, δ are positive constants with and \beta =e^{-2Kt}.

Let R\to \mathrm{\infty }, we can get the following global gradient estimates for the nonlinear parabolic equation (1.1).

Corollary 1.4 Let (M,g) be a complete noncompact n-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant -K=:-K(M), where K>0. Assume that u is a positive solution of (1.1) with u\le M_1 for all (x,t)\in M\times (0,+\mathrm{\infty }). Then

1. (1)

   if A\le 0, B\ge 0, we have

   \beta \frac{{|\mathrm{\nabla }u|}^2}{u^2}+h+A{u}^{p-1}+B{u}^{-(q+1)}-\frac{u_t}{u}\le \frac{n}{2(1-\delta )\beta }(-\frac{M_1^{p-1}A(p-1)(p-\beta )}{4{(1-\beta )}^2}+\frac{1}{t});
2. (2)

   if A>0, B>0, we get

   \beta \frac{{|\mathrm{\nabla }u|}^2}{u^2}+h+A{u}^{p-1}+B{u}^{-(q+1)}-\frac{u_t}{u}\le \frac{n}{2(1-\delta )\beta }(M_1^{p-1}A(p-1)+\frac{1}{t}),

δ are positive constants with and \beta =e^{-2Kt}.

Let \delta \to 0, A=0 in Corollary 1.4, we get a Li-Yau-type gradient estimate.

Corollary 1.5 Let (M,g) be a complete noncompact n-dimensional Riemannian manifold with Ric(M)\ge 0. Assume that u(x,t) is a positive solution to the equation

on complete noncompact manifolds, where h, q, B are real constants and q>0. Then we have

As an application, we have the following Harnack inequality.

Theorem 1.6 Let (M,g) be a complete noncompact n-dimensional Riemannian manifold with Ric(M)\ge 0. Assume that u(x,t) is a positive solution to the equation

on complete noncompact manifolds, where h, q, B are real constants and q>0, B>0. Then for any points (x_1,t_1) and (x_2,t_2) on M\times [0,+\mathrm{\infty }) with , we have the following Harnack inequality:

where \varphi (x_1,x_2,t_1,t_2)=\frac{d^2(x_1,x_2)}{t_2-t_1}, D=h(t_1-t_2).

Assume that u is a positive solution to (1.1). Set w=lnu, then w satisfies the equation

Lemma 2.1 Let (M,g) be a complete noncompact n-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant -K=:-K(2R), where R>0 and K>0 in the metric ball B_{2R}(p) around p\in M. Let w be a positive solution of (2.1), then

where

and \beta =e^{-2Kt}.

Proof Define

where \beta =e^{-2Kt}. By the Bochner formula, we have

By a direct computation, we have

and

and we know

Therefore, by equalities (2.2) and (2.3), we obtain

This implies that,

and

Therefore, it follows that

which completes the proof of Lemma 2.1. □

We take a C^2 cut-off function \tilde{\phi } defined on [0,\mathrm{\infty }) such that \tilde{\phi }(r)=1 for r\in [0,1], \tilde{\phi }(r)=0 for r\in [2,\mathrm{\infty }), and 0\le \tilde{\phi }(r)\le 1. Furthermore, \tilde{\phi } satisfies

and

for some absolute constants c_1,c_2>0. Denote by r(x) the distance between x and p in M. Set

Using an argument of Cheng and Yau [9], we can assume that \phi (x)\in C^2(M) with support in B_p(2R). Direct calculation shows that on B_p(2R)

By the Laplacian comparison theorem in [10],

In inequality (2.5), if , then △φ can be controlled by -\frac{(n-1)(1+\sqrt{K}R)c_1+c_2}{R^2}, so in any case, \mathrm{△}\phi \ge -\frac{(n-1)(1+\sqrt{K}R)c+c_2}{R^2}, where c is some positive constant.

For T\ge 0, let (x,s) be a point in B_{2R}(p)\times [0,T], at which φF attains its maximum value P, and we assume that P is positive (otherwise the proof is trivial). At the point (x,s), we have

It follows that

This inequality, together with inequalities (2.4) and (2.5), yields

where

At (x,s), by Lemma 2.1, we have

it follows that

here we used

Following Davies [11] (see also Negrin [12]), we set

Then we have

Next, we consider the following two cases:

1. (1)

   if A\le 0, B\ge 0, then we have

   \frac{2\phi \beta {((\beta -1)s\mu -1)}^2{F}^2}{ns}\le \frac{2c_1}{R}{\phi }^{\frac{1}{2}}{\mu }^{\frac{1}{2}}{F}^{\frac{3}{2}}+HF-M_1^{p-1}A(p-1)(p-\beta )\mu s\phi F+\frac{\phi F}{s},

multiplying both sides of the inequality above by sφ, we have

So, it follows that

Since

we get

Now, (1) of Theorem 1.3 can be easily deduced from the inequality above;

1. (2)

   if A>0, B>0, then we have

   \frac{2\phi \beta {((\beta -1)s\mu -1)}^2{F}^2}{ns}\le \frac{2c_1}{R}{\phi }^{\frac{1}{2}}{\mu }^{\frac{1}{2}}{F}^{\frac{3}{2}}+M_1^{p-1}A(p-1)\phi F+HF+\frac{\phi F}{s},

multiplying both sides of the inequality above by sφ, we have

So, it follows that

Similarly, we can obtain (2) of Theorem 1.3.

Proof of Theorem 1.6 For any points (x_1,t_1) and (x_2,t_2) on M\times [0,+\mathrm{\infty }) with , we take a curve \gamma (t) parameterized with \gamma (t_1)=x_1 and \gamma (t_2)=x_2. One gets from Corollary 1.5 that

which means that

Therefore,

where \varphi (x_1,x_2,t_1,t_2)=\frac{d^2(x_1,x_2)}{t_2-t_1}, D=h(t_1-t_2). □

The author would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improved the quality of the paper. Moreover, the author would like to thank his supervisor Professor Kefeng Liu for his constant encouragement and help. This work is supported by the Postdoctoral Science Foundation of China (2013M531342) and the Fundamental Research Funds for the Central Universities (NS2012065).

Affiliations

1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China

   • Liang Zhao

Corresponding author

Correspondence to Liang Zhao.

Competing interests

The author declares that they have no competing interests.

Author’s contributions

The author completed the paper. The author read and approved the final manuscript.

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