Gradient estimates for a weighted nonlinear parabolic equation and applications

Abstract This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.


Introduction
, otherwise known as a smooth metric measure space. Here ∈ β and ( ) p x t , and ( ) q x t , are smooth functions at least C 1 in x and C 0 in t. Suppose ( ) p x t , and ( ) q x t , are zero, then (1.1) is the heat equation: whose minimal positive solution is known as the weighted heat kernel. The major aim of this paper is to establish improved elliptic gradient estimates of elliptic type on smooth solutions and then derive Liouville-type theorems, Harnack inequalities and heat kernel estimates, as applications of the obtained gradient estimates. In recent years, gradient estimates for both elliptic and parabolic equations have become fundamental tools in geometric analysis. In their celebrated work [1], Li and Yau established parabolic gradient estimates on solutions to the linear heat equation on Riemannian manifolds having Ricci curvature bounded from below. They applied their results to get Harnack inequalities and various estimates on the heat kernel.
Since then there has been a lot of work to improve, extend or generalize these results, for instance, see [2][3][4][5][6][7][8][9][10][11][12] and references therein. Historically, the technique of gradient estimates emanated from the study by Yau [13] (see also [14,15]), in which a gradient estimate for harmonic functions was first established using the maximum principle. This estimate was applied to obtain a Liouville theorem. A Liouville theorem says that a bounded positive solution to the harmonic equation is constant. Global elliptic gradient estimate was derived by Hamilton [9] for the heat equation on a closed manifold. Localized elliptic-type estimate was proved by Souplet and Zhang [11] for the heat equation by adding a logarithmic correction term. In the same spirit, we deal with local gradient estimates on weighted manifolds with Bakry-Émery tensor bounded from below. Thus, it is in order to give background information about this space. is called a weighted Riemannian manifold (smooth metric measure space). The space is endowed with weighted Ricci tensor and weighted Laplacian.

Basics of weighted Riemannian manifold
The m-weighted Ricci tensor otherwise known as the Bakry-Émery tensor in the literature is defined for some constant > m 0 as Ric is the Ricci tensor of ( ) M g , and ∇ 2 is the Hessian operator with respect to g. Clearly, = Ric Ric . Note that the ∞-Bakry-Émery tensor defines special solutions to the Hamilton Ricci flow [16]. These solutions are called gradient Ricci solitons = λg Ric f , for ∈ λ . For a detailed discussion on Ricci solitons, interested readers can see [17] for a survey on Ricci solitons.
The weighted Laplacian is defined as where Δ and ∇ are the usual Laplace-Beltrami and gradient operators, respectively.
The weighted Bochner formula for any ∈ ( ) ∞ u C M , gives a relation between the weighted Laplacian and weighted Ricci tensor. The heat kernel on weighted manifold is the minimal positive solution to the heat equation , , for each ∈ y M, satisfying the initial condition is the weighted delta function given by Various Liouville-type theorems have been obtained for harmonic functions on weighted manifold, see for instance [18][19][20][21][22] and Cao and Zhou [23] and Wu and Wu [24,25] for related results. Specifically, we mention that Brighton [26] derived some elliptic-type gradient estimates for weighted harmonic functions when ≥ Ric 0 f , using a power of u rather than u log in Yau [15]. Brighton's approach was refined by Munteanu and Wang [27] to show that positive weighted harmonic function of sub-exponential growth must be constant on the condition ≥ Ric 0 f .

Motivations
The present work is an extension of results in [28], in which it was stated that equations of the form (1.1) arise in geometry and physics. For instance, the static form of (1.1) is equivalent to the weighted Yamabe equation [29]. Hopefully, elliptic gradient estimates for Eq. (1.1) can be applied to get information that will be useful in solving the Yamabe problem on weighted manifolds [30]. Another motivation is coming from physical applications. Suppose f is a constant function on M and take either ( ) p x t , or ( ) q x t , to be zero, we have in particular which is a simple ecological model for population dynamics, where ( ) u x t , is the population density at time t. Recently, Zhu [31] derived local elliptic-type gradient estimates for (1.4) on ( ) M g , N with Ricci tensor bounded below, where > β 0. He used his estimates to obtain some Liouville-type theorems for positive solutions, thereby generalizing result [11] according to the study by Souplet and Zhang. The present author in [28] has extended Zhu's results to weighted manifold for an arbitrary constant ∈ β . Later, Wu [30] obtained similar estimates on weighted manifolds and used those estimates to determine sufficient conditions on parameter which guarantee nonexistence of positive solutions to some cases of (1.1) viz a viz existence of Yamabe-type problems on weighted manifolds.
According to [27,32], one encounters some sort of difficulties in obtaining estimates of Li-Yau type for positive solutions to (1.2) when Ric f is bounded below, even when growth condition is imposed on f. However, Wu [22] has recently proved local and global elliptic gradient estimates for positive solutions to (1.2) without any assumption on f. Thereby, extending Souplet-Zhang's result [11] to (1.2) with only Ric f bounded from below. At present, therefore, we combine the approaches in [28] and [22,30] to derive elliptic gradient estimates for positive solutions of (1.1). Here, the estimates obtained do not require any condition on f.

Main result
and the distance function from x 0 to x with respect to g is denoted by is the geodesic ball centered at x 0 with > R 0, ∈ t 0 and > T 0. Denote by ∥⋅∥ = |⋅| sup R T , , the norm with respect to g and , , 0, . The main result concerning the localized elliptic gradient estimates is stated below.
is a positive solution to (1.1) in R T , , > T 0, ≥ R 2 for some constant D. Then there exists a constant ( ) C N such that Note that if ≡ q 0 estimates (1.5) and (1.6) are equal to the estimates in Theorem 1.1 of [28]. Note also that (1.6) includes (1.5). It is therefore obvious that Theorem 1.1 is a generalization of Theorem 1.1 in [28]. See [33] for similar results when p and q are some constants.
The aforementioned estimates have numerous applications. We only mention three of them here: namely, we derive Harnack-type inequalities in Section 4.1, Liouville-type theorems in Section 4.2 and weighted Heat kernel bound in Section 5. In Section 2, we prove a lemma which is fundamental to the proof of main theorem and then introduce a cutoff function depending on space and time. We then apply the lemma and properties of the cutoff function to prove Theorem 1.1 in Section 3.

Basic lemma
Consider (1.1) where p and q are continuous differentiable functions and ∈ β .
The following Lemma is key to the proof of Theorem 1.1.
Proof. By the weighted Bochner formula (1.3) we have Using the fact that by the hypothesis, m can be sent to ∞ and By direct computation, we have The last lemma and some localization techniques introduced in [11] will be applied to prove Theorem 1.1. As in [11,22,28], we use a cutoff function satisfying the following properties [28, Lemma 2.2]. The properties will be used to obtain the required bounds in R T , .
with the following properties: (ii) ψ is radially decreasing and In the next section, we use Lemma 2.1 and properties of ψ together with the maximum principle in a local space-time supported set. We follow the usual methodology applied in [28]. to be a cut-function with support in R T , and satisfies properties of (i)-(iv) stated in Lemma 2.2. We want to obtain some estimates and do some analysis at a space-time point where ψG reaches maximum. We shall show that inequalities (1.5) and (1.6) hold at the point ( )

Gradient estimates
and then obtain the conclusion of the theorem at once since τ is arbitrarily chosen.
A straightforward computation yields Now substituting (2.2) into (3.1) and rearranging give where the following identity has been used We suppose ψG attains its maximum at the point ( ) , . By Calabi's argument [1], we assume that x 1 is not in the cut locus of M. Then ( )( ) ψG x t , 1 1 is assumed to be positive, otherwise ≤ G 0 and the result then holds trivially whenever ( Now, consider the case where ∉ ( ) x B x , 1 0 . First, we obtain upper bounds on each term at the right hand side of (3.3) at ( ) x t , 1 1 and then do the analysis. Let = ( ) > C C N 0 be a constant whose values may vary from line to line. With repeated use of Young's inequality as in [11,28] , the third term on the right hand side of (3.3) Using similar arguments to those in [28], the 4th, 5th, 6th and 7th terms of (3.3) are, respectively, estimated as follows: , the 8th term on the right hand side of (3.3) can be estimated as follows.
The 9th term is estimated as follows. Note that < ≤ Putting (3.4)-(3.12) back into the right hand side of (3.3) and rearranging to get  (3.14) as defined before. For the 8th term on the right hand side of (3.3) is estimated as follows: . Then    analogous to (3.13) ( ≥ β 1). Following similar steps to those in the case of ≥ β 1 we arrive at (1.6). The proof is complete for the case ∉ ( , ≥ R 2, due to the assumption. Hence, by (3.3) we have for ≥ β 1 that where Λ is as defined in the theorem. We now compute B . Then from the last inequality we have Hence, some elementary computations imply (4.1), and this completes the proof. □

Liouville-type theorems
This section presents some Liouville-type theorems for various cases of (1).  near infinity.
Note that if ( ) ≡ ≡ ( ) p x qx 0 (1.1) reduces to (1.2). Any solution which exists for negative time in all space is referred to as the ancient solution.
Proof of Theorem 4.2. We only consider case ≥ β 1 for the proof of (a) since the case < β 1 is similar. Suppose near infinity. For a fixed point ( and using the function holds, where ( ) x y d , denotes the geodesic distance from x to y.
In [22], this type of estimate was proved via a global Hamilton-type gradient estimate, which was derived using Bernstein-Shi-type estimate and the weighted Laplacian comparison theorem.  [24,25]), we have , , , depending only on N and This completes the proof. □

Concluding remark
We have adopted the classical approach of Souplet and Zhang [11] to obtain elliptic gradient estimates on positive solutions to nonlinear parabolic equations on weighted manifolds with lower boundedness condition on the Bakry-Émery tensor. Our result is an improvement of some known results. The gradient estimates derived have been used to establish Harnack-type inequalities (Section 4.1) and Liouville-type theorems (Section 4.2). The weighted heat kernel estimates obtained in Section 5 is interesting on its own. Furthermore, the gradient estimates obtained can be used to study the existence and nonexistence of Yamabe-type problems on weighted manifolds. In this direction, Wu [30, Theorem 5.1] has used some parabolic gradient estimates to prove that