Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

: In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a ( k , m ) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.


Introduction
In this paper, we establish elliptic type gradient estimates for positive solutions to a class of nonlinear parabolic equations involving the Witten Laplacian in the context of smooth metric measure spaces where both the metric and potential are time dependent. We also discuss some applications and consequences of these estimates, most notably to Liouville theorems and global estimates.
Gradient estimates have a central place in geometric analysis with a vast scope of applications (for some recent work, see [2-6, 9-11, 13-15, 17, 23, 26, 27, 29, 31, 32, 34] and the references therein). The estimates of interest in this paper fall under the category of Souplet-Zhang and Hamilton type estimates that were first formulated and proved for the heat equation on static manifolds in [11,23]. Such estimates along with the differential Harnack and Li-Yau type inequalities have since been extensively studied and applied to families of parabolic equations on manifolds with both static and evolving metrics (e.g., under the Ricci flow and its relatives). They are well known not only for their effectiveness in proving classical Harnack inequalities, Liouville theorems and sharp bounds on heat kernels to mention a few, but also more recently and in conjunction with the Perelman entropy method for establishing sharp geometric and functional inequalities in a variety of contexts and settings (see, e.g., [4, 10, 13, 15-18, 28, 34]). In this paper, we make a contribution to the subject by deriving such estimates in the context of smooth metric measure spaces equipped with time dependent metrics and potentials under suitable bounds on the generalized Bakry-Émery curvature tensor before presenting some applications, most notably here to Liouville theorems.
To this end, suppose that (M, g, e −f dv) is a smooth metric measure space, by which it is meant that (M, g) is a complete Riemannian manifold of dimension n (here n ≥ 2), f is a smooth potential on M and dμ = e −f dv is a weighted measure on M conformally equivalent to the Riemannian volume measure dv = dv g . (For the significance and role of smooth metric measure spaces in modern geometric analysis see [3,4,10,22,28], the sizeable recent literature, e.g., [8,15,16,19,24,30,33,34] as well as the discussion in the next section). A (k, m)-super Perelman-Ricci flow on M refers to the evolution of a one parameter family of smooth metricpotential pairs (g, f) on M subject to the relation Here k, m are real constants with m ≥ n (not necessarily an integer). The second-order symmetric tensor field Ric m f (g) is the Bakry-Émery m-Ricci curvature tensor given in this evolutionary context by (cf. also (2.2) below) with Ric(g) the usual Ricci tensor and ∇ g ∇ g f the Hessian of f . Note that the definition in (1.2) extends to m = n only for constant functions f hence giving Ric n f (g) ≡ Ric(g), and to m = ∞ by Ric f (g) = Ric ∞ f (g) ≡ Ric(g) + ∇ g ∇ g f.
In this paper, we consider positive smooth solutions to the semilinear parabolic equation on M: ∂u ∂t − ∆ f u = F(u), t > 0, (1.3) where L = ∆ f denotes the Witten Laplacian (also known as Nelson's diffusion operator in stochastic mechanics, or else the weighted Laplacian). It acts on smooth functions v ∈ C ∞ (M) by ¹ and constitutes a natural extension of the Laplace-Beltrami operator to the context of smooth metric measure spaces. The Witten Laplacian has close links with Markov processes and probability theory, quantum field theory, Riemannian geometry and more modern areas in mathematical physics. Our goal here is to establish among other things local and global elliptic type gradient estimates for positive smooth solutions to (1.3) when both the metric and potential evolve under a (k, m)-super Perelman-Ricci flow (see Theorem 3.1) and discuss some implications of these estimates. Regarding the nonlinearity in (1.3), we take F = F(u) a sufficiently smooth function on the half-axis u > 0. Of particular interest, prompted by the surge of recent research work in the literature, is when F has a power-like growth and/or a logarithmic singularity at infinity, e.g., F(u) = Au p log u + Bu q with real p, q, or F(u) = Au|log u| p + Bu q with real p, q and p > 1 (see Sections 5 and 6 for more details). It turns out that a crucial role in the estimates is played here by the non-negative, F and u dependent quantity, We discuss some consequences of these estimates and other closely related issues. In particular, and as a byproduct, we establish some new relative evolutionary estimates that include a Hamilton-Zhang type global gradient estimate to the nonlinear context with dimension free constants. As the static case is a particular instance of the system, we also discuss consequences of the gradient estimate to that context. In particular, here we establish new Liouville type results for the stationary elliptic counterpart of (1.3): ∆ f u + F(u) = 0 subject to the non-negativity of the Bakry-Émery curvature tensors Ric m f ≥ 0 in Section 5 and global curvature free gradient estimates for the full parabolic equation (1.3) in Section 6 (see in particular Theorem 6.1 and the subsequent discussion). The non-negativity and/or monotonicity of related entropy like quantities is a question of great contemporary interest [1,7,8,17,22,26,28,34].

Preliminaries
Let (M, g) be a complete n-dimensional Riemannian manifold. Given f ∈ C ∞ (M), put dμ = e −f dv g where dv g is the Riemannian volume measure. The triple (M, g, dμ) is then called a smooth metric measure space, or equivalently a weighted manifold (see [3,4,10,19] for nice and exquisite introductions to the subject).² The f -Laplacian (1.4) is the natural substitute for the Laplace-Beltrami operator in this context (in fact, as is easily seen, the two coincide when the potential f is constant). It is a symmetric diffusion operator with respect to the invariant measure dμ = e −f dv g [borrowing the terminology from the theory of Dirichlet forms and Itô's SDE theory prompted by the second integral in (2.1)], and as one readily verifies for u, v ∈ C ∞ 0 (M): Next regarding the curvature properties of (M, g, dμ), the theory of carré du champ for Markov diffusion operators and the curvature dimension condition CD(k, m) as developed by Bakry-Émery (see [3,4]) naturally lead to the generalized m-Ricci curvature tensor Ric m f = Ric m f (g) associated with the f -Laplacian defined by When m = n, to make sense of (2.2) one only allows constant functions f as admissible potentials, in which case ∆ f = ∆ and Ric n f (g) ≡ Ric(g). By contrast when m = ∞, one simply defines Upon switching the inequality sign in (1.1) to equality, we get the (k, m) Perelman-Ricci flow ((2.4) below). Under the additional assumption that the weighted measure dμ = e −f dv remains static in time, it can be seen that Here Tr stands for the metric trace on (0, 2)-tensors, and the combined system resulting from the flow and the stationary volume measure constraint takes the form In particular, it follows from (2.4) and (2.5) that the potential f satisfies the evolution equation ∂f ∂t The case (k, m) = (0, ∞) in system (2.4)-(2.5) is what was introduced by Perelman in [22] as the L 2 -gradient flow of the functional P(g, f) = ∫ M (R + |∇ f| 2 )e −f dv g subject to the stationary volume measure constraint considered above. Here R = R(g) is the scalar curvature, Ric m f (g) = Ric(g) + ∇ 2 f is the modified Ricci tensor, and f is the potential satisfying the so-called conjugate or adjoint heat equation, The above discussion illustrates another aspect of how the problems considered here link with the larger and more recent body of research on the subject and the remarkable work on the Poincaré conjecture [22]. For further reading and related work with volume measure constraints, see [8,12,15,17,21,27,34] and the references therein.
Notation. For any pair of points x, y on M we designate by d = d g(t) (x, y) or d(x, y; t) the Riemannian distance between x, y with respect to the metric g = g(t). Fixing a reference point x 0 on M, we denote by ϱ = ϱ(x, t) the geodesic radial variable measuring the distance between x and x 0 at time t. For R, T > 0 we introduce the compact set Throughout, we assume R ≥ 2 and we make use of the notation s + = max(s, 0) and s − = max(−s, 0).

A Hamilton-Souplet-Zhang type gradient estimate
The main result here is a local elliptic gradient estimate on positive smooth solutions to equation ( Here Moreover, in the case m < ∞ one can remove the term √ζ/R in (3.1).

Remark 3.2. From the bounds Ric
The lower bound Ric m f (g) ≥ −k m g here is required for the Wei-Wylie weighted Laplacian comparison theorem (step (v) in the proof) and the bound ∂g/∂t ≥ −2hg for controlling the time derivative of the geodesic distance, namely d t = ∂[d g(t) (x, x 0 )]/∂t (in step (vi)). As is evident from (2.2), a lower bound on Ric f (g) is a weaker assumption than one on Ric m f (g) for m < ∞ by virtue of Ric m f ≥ kg implying Ric f ≥ kg but not vice versa. with Ric m f (g) ≥ −k m g; see Section 5.
If u is a positive bounded solution to (1.3) and the lower bounds Ric m f (g) ≥ −k m g and ∂g/∂t ≥ −2hg in Theorem 3.1 are global on M × [0, T], then, by passing to R ↗ ∞, we have the following global counterpart of (3.1).

Corollary 3.4. Under the assumptions of Theorem 3.1 and the global bounds
Note that the boundedness of u does not necessarily imply that the right-hand side of (3.2) is always finite; however, this is not an issue for the estimate itself (see Section 5 for more on this). The proof of Theorem 3.1 and all the necessary tools span Section 4. In Sections 5 and 6, we give some interesting consequences of the local gradient estimate (3.1) that includes global elliptic and parabolic estimates for positive solutions, a curvature free estimate for the case when M is closed and a number of important Liouville type results on the elliptic (non-evolutionary) counterpart of (1.3).
4 Proof of the main estimate and Theorem 3.1

Parabolic lemmas
Before proceeding onto the proof of Theorem 3.1, we present a chain of lemmas that will be used in the course of the argument. First we establish some parabolic estimates under the flow on auxiliary functions involved in the proof. Note that hereafter for convenience we abbreviate the metric inner product by writing We also write h t = ∂h/∂t and as before ∆ f h = ∆h − ⟨∇ f, ∇h⟩.
Moreover, it is easily seen that Hence referring to (1.3) and (1.4) and putting the above pieces together, we have and so the conclusion follows immediately. (4.1).

Lemma 4.2. Suppose g = g(t) and f = f(t) are of class C 2 and let h be a positive solution to
Proof. By using the description W = |∇h| 2 /(1 + h) 2 and the conclusion of Lemma 4.1, let us proceed by explicitly calculating the left-hand side of (4.2). Towards this end, we first note that where we have made use of the identity (|∇h| 2 ) t = −g t (∇h, ∇h) + 2∇h∇h t . Similarly, for the gradient and the Laplacian of W we have which then gives Now putting together the individual descriptions of W t and ∆ f W from (4.3) and (4.4), respectively, and taking into account the relevant cancellations, we have or, after doing some algebra, Hence by using the weighted Bochner-Weitzenböck formula (2.3), we can write or, upon rearranging terms, Now by substituting and reverting to W in the terms involving |∇h| 2 in the last two lines, this results in which is the desired conclusion. Note that here we have made use of the relation This therefore completes the proof.
Proof. This is a straightforward consequence of (4.2) by using the flow inequality (1.1) and 2 The proof of Theorem 3.1 uses Lemma 4.3 and a space-time localization using suitable cut-off functions whose main properties appear in Lemma 4.4 below. (For more on the method, its background and underlying ideas, see [2,6,14,23,30] and the references therein. For more discussions on the use of cut-off techniques and localization in deriving various estimates in PDEs, see also [25,26].) Now fix R, T > 0 and then τ ∈ (0, T]. Denoting by ϱ(x, t) = d g(t) (x, x 0 ) the geodesic radial variable at time t with respect to the reference point x for a smooth cut-off function supported in the compact set B R,T ⊂ M × [0, T]. The functionψ =ψ(ϱ, t) appearing on the right-hand side of (4.5) is one granted by and described in the following statement (see [2,6,23,31]).

Proof of Theorem 3.1
Let ψ be as described in (4.5) withψ as in Lemma 4.4. Note that here we have fixed R ≥ 2, T > 0 and 0 < τ ≤ T. We will show that (3.1) holds for all (x, τ) with d(x, x 0 ; τ) ≤ R 2 , and then the arbitrariness of τ will grant the assertion for all (x, t) in B R/2,T with t ̸ = 0. Now by proceeding forward, a straightforward calculation gives ∆ f (ψW) = ψ∆ f W + 2∇ψ∇W + W∆ f ψ. So when combined with (ψW) t = ψW t + Wψ t , we can write At this stage we refer to the conclusion in Lemma 4.3 and, by substitution and making note of the nonnegativity of ψ, we write Next by utilizing the two basic vector identities ψ∇h∇W = ∇h∇(ψW) − (∇h∇ψ)W and ∇ψ∇W = (∇ψ/ψ)∇(ψW) − ( |∇ψ| 2 ψ )W and upon substituting in (4.6), we have Assume now that the localized function ψW attains its maximum value on the compact set B R,T at (x 1 , t 1 ). We suppose without loss of generality that x 1 is not on the cut-locus of M by Calabi's argument [14]. We also assume that (ψW)(x 1 , t 1 ) > 0, as otherwise the desired estimate becomes trivial with W(x, τ) ≤ 0 whenever d(x, x 0 ; τ) ≤ R 2 . Thus in particular t 1 > 0 by (iii) and at (x 1 , t 1 ) we have ∆ f (ψW) ≤ 0, (ψW) t ≥ 0 and ∇(ψW) = 0. Therefore, (4.7) after taking into account all the necessary cancellations implies that . After multiplying through by h 2 /2(1 + h), this can be rewritten as The plan is now to exploit the above inequality and the maximal characterization of (x 1 , t 1 ) to establish the required estimate at the space-time point (x, τ). Towards this end, we proceed by considering two cases depending as to whether d(x 1 , x 0 ; t 1 ) ≥ 1 or d(x 1 , x 0 ; t 1 ) ≤ 1. Let us consider the first case. Here we apply the properties (i)-(iv) of ψ as listed in Lemma 4.4, and the Cauchy-Schwarz and Young inequalities, respectively, to obtain suitable upper bounds for each of the six terms on the right-hand side of (4.8).
(i) For the first term, by recalling the definition of W, noting 0 < (2h + 1)/(1 + h) ≤ 2 and using basic inequalities, we can write (ii) For the second term, noting 1/(1 + h) ≤ 1 and proceeding in a similar way, we have (iii) For the third term, noting 0 ≤ ψ ≤ 1 and taking positive parts, we can write (iv) For the fourth term we can write (v) For the fifth term we treat the cases m = ∞ and n ≤ m < ∞ separately. Before that however, we note that in view of the relation Ric 2) and Remark 3.2), the estimate obtained in the first case under a lower bound on Ric f (g) remains true also in the second case. However, as will be seen, one can take advantage of the lower bound on Ric m f (g) in the second case to obtain a slightly less involved estimate (compare (4.9) with (4.10) below). Proceeding now onto the first case, we Using Ric f (g) ≥ −(n − 1)kg and the weighted Laplacian comparison theorem [30, Theorem 3.1], we have ∆ f ϱ ≤ ζ + (R − 1)(n − 1)k for ϱ ≥ 1. From ψ being radial withψ ϱ ≤ 0 it then follows that and so −∆ f ψ ≤ |ψ ϱϱ | + [ζ + (R − 1)(n − 1)k]|ψ ϱ |.

By using the bound
(here we are making use of v coth v ≤ 1 + v and the monotonicity of coth v for v > 0) and noting thatψ ϱ ≡ 0 for 0 ≤ ϱ ≤ R 2 , it then follows that Therefore, we can write Thus referring to (4.5) and applying the properties ofψ as listed in Lemma 4.4, we deduce that Now with the aid of this estimate of ψ t , we can proceed onto the last term and write Completing the estimate of the individual terms on the right-hand side of (4.8), by inserting these estimates back and after basic considerations and adjusting the constants upon noting R ≥ 2, we arrive at the following bound at (x 1 , t 1 ): Referring to the expression on the right-hand side in (4.11), note that in the case resulting from (4.10) in case two in (v), we can remove the term ζ 2 R 2 . Now invoking the maximal characterization of (x 1 , t 1 ), we have for all (x, t) ∈ B R,T the inequalities and Since ψ(x, τ) = 1 in d(x, x 0 ; τ) ≤ R 2 and by the definition W = |∇h| 2 /(1 + h) 2 , upon recalling h = √u and |∇h| = |∇√u|, this gives We now move to the second case and consider d(x 1 , x 0 ; t 1 ) ≤ 1. As by Lemma 4.4 (ii), ψ is a constant function in the space direction (when d(x, x 0 ; t) ≤ R 2 , t ∈ [0, T], with R ≥ 2). By referring to (4.8), all the terms involving spatial derivatives of ψ vanish or simplify (including ψ t =ψ t ), and so we have at the point (x 1 , t 1 ), , the above and the arbitrariness of τ > 0 easily lead to a special case of (3.1). We have now established the estimate in both cases, and so the proof is complete.

Applications to special nonlinearities F and Liouville theorems
In this section, we present some corollaries of Theorem 3.1 to the static case (see Remark 3.3). Here (M, g, e −f dv) is a smooth metric measure space where the metric g and the potential f are time independent whilst Ric m f (g) ≥ −k m g in B R with k m ≥ 0 and R ≥ 2. For clarification, B R ⊂ M is the geodesic ball with center x 0 and radius R > 0. We consider positive bounded solutions to the elliptic equation ∆ f u + F(u) = 0. Since u is a time independent solution of (1.3), setting t ↗ ∞ in (3.1) and noting that the constants do not depend on t gives the corresponding estimate for x ∈ B R/2 : As g t ≡ 0, we have set h = 0 and so K = k whilst Of particular interest here are Liouville type theorems; the underlying idea upon referring to (5.2) being that if Ric m f (g) ≥ 0 (i.e., k = 0) and R F (u) ≡ 0, then combined with the above estimate these together imply that ∇u = 0 and so u is a constant. Below we examine this more closely in the context of certain power-like and logarithmic type nonlinearities F that arise frequently in the literature and include the following: (with arbitrary real exponents p, q). (iv) F(u) = Au p log u + Bu q (with arbitrary real exponents p, q).

The case F(u) = Au| log u| α with α > 1
The function F here is continuously differentiable, and so, by referring to the description of R F (u) (see (1.5)), a basic calculation gives and so as readily seen Proof. We have A + = 0 and [A|log u| α−2 log u] + = 0 by referring to the above calculation R F (u) = 0. Since by assumption k = 0, it then follows from (5.2) that ∇u ≡ 0, and this leads immediately to the desired conclusion.

The case F(u) = Au(log u) d with integer d ≥ 1
By comparing with Section 5.1 above, the modulus sign in F has been removed from the term log u; however, it is evident that due to the sign changing nature of the logarithm some restriction has to be imposed on the exponent (here an integer d ≥ 1) as speaking of arbitrary powers of log u for u > 0 is in general meaningless. Now a short calculation gives and so referring to (1.5), we can write

The case F(u) = Au p + Bu q with real exponents p, q
A direct calculation in the spirit of those in the previous sections here gives and so from (1.5) and by considering positive parts, we have Note that regardless of the upper bound on u the condition p ≥ [1 Proof. From the discussion preceding the theorem, the assumptions on the exponents p, q the solution u and the coefficients A, B give R F ≡ 0. Hence from the global estimate (5.2) with k = 0, it follows that ∇u ≡ 0. This immediately gives the conclusion.

The case F(u) = Au p log u + Bu q
Here again by a direct differentiation, we have So referring to (1.5), we can write Let us now set s(p) = [(2p − 1) + 2h(p − 1)]/[2(1 + h)]. Then from the above we have In what follows, we look more closely at the sign of the term log(eu s(p) ), or equivalently that of s(p) log u + 1. Towards this end, consider the function , h ≥ 0.
As for the range 1 2 < p < 1, the function ζ p has a singularity at h = (2p − 1)/[2(1 − p)] on the halfaxis h > 0 and is in particular unbounded. Hereafter, we restrict attention to p outside this interval, i.e., p ≤ 1 2 or p ≥ 1. Now it is easily seen that ζ p < 0, and so ζ p is monotonically decreasing on h ≥ 0. Moreover, by direct evaluation, ζ p (0) = 2/(1 − 2p) and lim ζ p As regarding the term B, we can argue exactly as in Section 5.3. By putting the above together, we have proved the following statement.
together with B ≥ 0 and q ≤ 1 2 , or B ≤ 0 and q ≥ 1. If these conditions hold, then u must be a constant.

A global gradient estimate for solutions to (1.3)
We have already seen some implications of the local and global estimates to Liouville type results on the elliptic counterpart of (1.3) in the previous sections. In this section, we continue by establishing estimates on the full parabolic equation that are of a global nature whereas before the metric and potential evolved under a (k, m)-super Perelman-Ricci flow. for all x ∈ M and 0 < t ≤ T.

(6.4)
Now the function γ(t) = t/(2kt + 1) with k, t ≥ 0 is seen to be non-negative, smooth and to satisfy γ(0) = 0 and γ + 2kγ ≤ 1. The conclusion thus follows by an application of the weak maximum principle by virtue of M being compact. Remark 6.2. Note that in this argument we did not use the fact that u is positive, but only uF(u) ≤ 0 to hold along u.
Specializing to the nonlinearities F introduced and studied in Section 5, we have the following corollaries of Theorem 6.1 for the positive solutions u to equation (1.3): (i) For the equation ∂u/∂t = ∆ f u + Au|log u| α with α > 1 we have from (6.4) that P t ≤ ∆ f P + Au 2 |log u| α + 2tA|∇u| 2 [|log u| α + α|log u| α−2 log u].
Therefore, if A ≤ 0 and u ≥ 1, then P t ≤ ∆ f P, and so the global estimate (6.1) holds. Hence if d ≥ 2 is even, then it is exactly as in (i). If d ≥ 1 is odd, and if A ≥ 0 and 0 < u ≤ e −d , or if A ≤ 0 and u ≥ 1, then in either case the global estimate (6.1) holds. (iii) For the equation ∂u/∂t = ∆ f u + Au p + Bu q we have again from (6.4) that P t ≤ ∆ f P + Au p+1 + Bu q+1 + 2t|∇u| 2 (Apu p−1 + Bqu q−1 ).