Eventual concavity properties of the heat flow

The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for nonnegative, bounded measurable initial functions with compact support.


Introduction
Let u be a nonnegative solution to the Cauchy problem for the heat equation where φ ∈ L and n ≥ 1.Here and in what follows we denote by L the set of nontrivial, nonnegative, bounded measurable functions in R n with compact support.The solution u is represented by u(x, t) = (e t∆ φ)(x) ≡ (4πt which leads to the eventual log-concavity property of u, that is, there exists T > 0 such that log u((1 − µ)x + µy, t) ≥ (1 − µ) log u(x, t) + µ log u(y, t) for x, y ∈ R n , µ ∈ (0, 1), and t ∈ [T, ∞) (see e.g., [24,Theoerem 5.1]).It was proved in [12,Theorem 3.3] that e t∆ φ eventually possesses stronger concavity properties than the logconcavity property (see also Theorem 1.1).The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations, and they have been studied for several nonlinear parabolic equations, e.g., a fully nonlinear variation of the heat equation (see [11]), the porous medium equation (see e.g., [1,3,17,24]), and the evolution of p-Laplace equation (see [23]).
In this paper we characterize the eventual concavity properties of the function where d ∈ R, using the notion of F -concavity (see Definition 1.1), and study geometric properties of the final state of U d [φ].F -concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity.Surprisingly, a variety of Fconcavity properties which U d [φ] eventually develops depends on d and n. (See Theorem 1.6 and Corollary 1.3.)

F -concavity and the heat flow
We follow [14] to introduce the notion of F -concavity. (2) Let F be admissible on I.For any f ∈ A(I), we say that for x, y ∈ R n and µ ∈ (0, 1).We denote by C[F ] the set of F -concave functions in R n .
(3) Let F 1 and F 2 be admissible on I.We say that F 1 -concavity is weaker (resp.strictly weaker) than F 2 -concavity in A(I), or equivalently that F 2 -concavity is stronger (resp.strictly stronger) than F 1 -concavity in We define the eventual F -concavity property of U d [φ] and the preservation of the F -concavity property by the heat flow in R n .
Definition 1.2 Let F be admissible on I = [0, a) with a ∈ (0, ∞]. (1) Let d ∈ R and φ ∈ L. We say that U d [φ] possesses the eventual F -concavity property if there exists T > 0 such that (2) We say that the F -concavity property is preserved by the heat flow in R n if, for any The preservation of concavity properties in parabolic problems has been studied in many papers since the pioneering work by Brascamp-Lieb [2] (see e.g., [3-10, 12-14, 16, 18-21, 24] and references therein).Among others, in [14], the author of this paper, Salani, and Takatsu gave a complete characterization of the F -concavity property preserved by the heat flow in R n , more generally, by the Dirichlet heat flow in convex domains in R n .The preservation of the F -concavity property by the heat flow causes the following question.
(Q) Assume that the F -concavity property is preserved by the heat flow in R n .Then does e t∆ φ possess the eventual F -concavity for φ ∈ L without the F -concavity property of the initial data φ?
One of the aims of this paper is to give an affirmative answer to question (Q) for the initial data φ in a subclass L A (see Section 1.3) of L.
We give some typical examples of F -concavity, and recall some results on the preservation of concavity properties by the heat flow in R n .

(E1) (Power concavity)
Let I = [0, ∞) and α ∈ R. Let Φ α be an admissible function on I defined by We often call 0-concavity log-concavity.Power concavity is a generic term of α-concavity, and it possesses the following properties. (a (c) Among power concavity properties, only the log-concavity property is preserved by the heat flow in R n .
Properties (a) and (c) follow from the Jensen inequality and [14, Theorem 1.5], respectively.Property (b) is a characteristic property of power concavity (see Lemma 2.3).
(a) If f is hot-concave in R n and λ ∈ (0, 1], then λf is hot-concave in R n . (b) The hot-concavity property is the strongest F -concavity properties in A(I) preserved by the heat flow in R n .
Theorem 1.1 Let φ ∈ L. For any β > 1/2, e t∆ φ possesses the eventual β-log-concavity concavity, that is, there exists T > 0 such that While Theorem 1.1 give a partial affirmative answer to question (Q), the following question (Q') naturally arises.
(Q') What kind of eventual F -concavity properties does the heat flow develops for φ ∈ L? In particular, does e t∆ φ possess the eventual 1/2-log-concavity concavity property and the eventual hot-concavity for φ ∈ L ?
In this paper we investigate the eventual F -concavity properties of U d [φ] and reveal the relation between the eventual F -concavity properties of U d [φ], the behavior of F (τ ) as τ → +0, and the parameter d.Furthermore, as an application of our study, we solve questions (Q) and (Q') for initial data φ in a suitable subclass L A of L.

Main results in the class L
In this section we state our results on the eventual F -concavity properties of U d [φ] for φ ∈ L.
In the case of d < n/2, we assume the following condition with a ∈ (0, ∞).
In the case of The function κ F is independent of a ∈ (0, ∞) (see Remark 1.1 ( 1)), and it has the following property: when F 1 and F 2 satisfies condition (F a ) with a ∈ (0, ∞), The first result of this paper concerns with a characterization of the eventual ( does not possess the eventual F -concavity property for any φ ∈ L. The case of κ * F = 1/2 is delicate, and it is treated in Section 1.3.Remark 1.1 We give some comments on the function κ F . (1) Let F satisfy condition (F a ) with a ∈ (0, ∞).For any k > 0, set ), where α ∈ R and Φ α is as in (E1).Since ), where β ∈ (0, ∞) and Ψ β is as in (E2).Then This implies that κ * F < 1/2 if and only if β > 1/2.
Next, we consider the case of d = n/2.Let F be admissible on and Theorem 1.3 Let d = n/2.Assume condition (F a ) with a ∈ (0, ∞).
(1) Assume that Then U d [φ] possesses the eventual F -concavity property for all φ ∈ L with M φ < a.
Theorem 1.4 Let F be admissible on [0, ∞) and d = n/2.Then U d [φ] possesses the eventual F -concavity property for all φ ∈ L if and only if F -concavity is weaker than log-concavity in A([0, ∞)).
Applying Theorems 1.2 and 1.3 to power log-concavity, we have: possesses the eventual β-log-concavity property for all φ ∈ L.
does not possess the eventual β-log-concavity property for any φ ∈ L. ( does not possess the eventual β-log-concavity property for some φ ∈ L with M φ < 1.
Next, we consider the case of ) is well-defined for all large enough t if and only if a = ∞.Theorem 1.5 Let F be admissible on [0, ∞), d > n/2, and φ ∈ L. Then U d [φ] possesses the eventual F -concavity property if and only if F -concavity is weaker than log-concavity in A([0, ∞)).
Theorems 1.4 and 1.5 completely characterize the eventual F -concavity properties which U d [φ] develops for all φ ∈ L in the case of a = ∞ and d ≥ n/2.In the next section we introduce a subclass L A of L, and study the eventual F -concavity properties which U d [φ] develops for all φ ∈ L A in the case of a ∈ (0, ∞) and d ≤ n/2.

Main results in the class L A
For any φ ∈ L, we say that φ ∈ L A if φ satisfies the following condition.
(A) There exists C > 0 such that R n y, ξ 2 e k y,ξ φ(y − z) dy ≤ Ck −2 R n e k y,ξ φ(y − z) dy for all k ≥ 1 and (ξ, z) ∈ S n−1 × R n satisfying ess sup y∈P φ y − z, ξ = 0.Here P φ is the positive set of φ, that is, P φ := {x ∈ R n : φ(x) > 0}.See Section 2.3 for sufficient conditions for φ ∈ L to satisfy condition (A).For example, if there exist C > 0 and a bounded smooth open set Ω in R n such that In this subsection we give a complete characterization of the eventual F -concavity properties which U d [φ] develops for all φ ∈ L A in the case of d ≤ n/2.The behavior of the function σ F defined by is crucial for the characterization.
Theorem 1.6 Assume condition (F a ) with a ∈ (0, ∞).Let σ F be as in (1.7). ( As a direct consequence of Theorem 1.6, we have the following results, which give our answers to questions (Q) and (Q') for φ ∈ L A .(2) Let d < n/2.Then U d [φ] possesses the eventual 1/2-log-concavity and the eventual hotconcavity property for all φ ∈ L A . ( does not possess the eventual 1/2-log-concavity property and the eventual hot-concavity property for some φ ∈ L A with M φ < 1. Corollary 1.3 Assume condition (F a ) with a ∈ (0, ∞).Then the heat fow e t∆ φ possesses the eventual F -concavity property for all φ ∈ L A if and only if In particular, there exists an admissible function F in I = [0, 1), satisfying condition (F a ) with a = 1, such that (1) F -concavity is strictly stronger than hot-concavity in A([0, 1)); (2) the F -concavity property is not preserved by the heat flow in R n ; (3) e t∆ φ possesses the eventual F -concavity property for all φ ∈ L A .
By Corollaries 1.1 and 1.2 we have the following list on whether U d [φ] possesses the eventual β-concavity property for all φ ∈ L A or not.
We explain our strategy of the proofs of our main results.In our analysis it is crucial to study geometric properties of U d [φ] outside parabolic cones B(0, L √ t), in particular, outside B(0, Lt), for large enough t, where L > 0. Let φ ∈ L. In the proofs of Theorem 1.2, Theorem 1.3, and Theorem 1.6 (2)-(4), we discuss the eventual F -concavity property of U d [φ] studying the sign of the function a −1 φ(y) dy. (1.12) and , where C > 0. In the case of κ * F = 1/2, thanks to the function κ F and the term |x| 2 /4t in (1.11), we study the sign of ∂ 2 F(v d )/∂ξ 2 systematically (see Proposition 3.1), and prove Theorems 1.2 and 1.3.On the other hand, in the case of κ * F = 1/2, we require more precise decay estimates of the derivatives of w in R n \B(0, Lt) for large enough t, where L > 0, than those of (1.14).Indeed, we improve decay estimate (1.14) under condition (A) (see Lemma 2.9).Furthermore, introducing the functions ν F and ψ d (see (2.3) and (4.9), respectively) and modifying the proof in the case of κ * F = 1/2, we study the sign of ∂ 2 F(v d )/∂ξ 2 in the case of κ * F = 1/2 (see Proposition 4.1) to prove Theorem 1.6.We also develop the arguments in [13] to prove Theorems 1.4 and 1.5.
The rest of this paper is organized as follows.Section 2 is divided into three subsections.In Section 2.1 we recall some properties of F -concavity and prove lemmas on the functions κ F and σ F .In Section 2.2 we obtain two lemmas on estimates of w.In Section 2.3 we obtain sufficient conditions for φ ∈ L to satisfy condition L A .In Section 3 we study the eventual F -concavity property of U d [φ] with d ≤ n/2 for φ ∈ L and prove Theorems 1.2 and 1.3.Furthermore, we prove Theorems 1.4 and 1.5 using a characterization of log-concavity.In Section 4 we study the eventual F -concavity property of U d [φ] with d ≤ n/2 under condition (A), and prove Theorem 1.6.We also prove Corollaries 1.2 and 1.3.

Preliminaries
Throughout this paper, for any x = (x 1 , . . ., x n ), y = (y 1 , . . ., y n ) ∈ R n , we denote by x, y the standard inner product of x and y, that is, x, y := n i=1 x i y i .Let e 1 , . . ., e n be the standard basis of R n .For any measurable set E in R n , we denote by χ E and |E| the characteristic function of E and the n-dimensional Lebesgue measure of E, respectively.We also use C to denote generic positive constants and may take different values within a calculation.

F -concavity
We recall some properties of F -concavity.Lemmas 2.1 and 2.2 concern with the strength of F -concavity.See [14, Lemmas 2.4 and 2.5].
We state two lemmas on necessary conditions for the F -concavity property to be preserved by the heat flow.Lemma 2.4 follows from [13,Lemma 4.1].
Lemma 2.7 Assume condition (F a ) with a = 1.Set for r ∈ (0, ∞).Let σ F be as in (1.7) and k > 0.Then, for any For any ǫ > 0, it follows that for large enough r with κ F (r) > 0. Since ǫ is arbitrary, we obtain lim sup Next, we assume (2.5).Then, for any ǫ > 0, we have for large enough r with κ F (r) > 0. Then for large enough r with κ F (r) > 0. This implies that for large enough r with κ F (r) > 0. Since ǫ is arbitrary, we obtain (2.4).Thus Lemma 2.7 follows.✷

Estimates of w
We obtain two lemmas on estimates of the function w given in (1.12).Lemma 2.8 (resp.Lemma 2.9) concerns with φ ∈ L (resp.φ ∈ L A ).
Lemma 2.8 Let φ ∈ L. Let w be as in (1.12).Then there exists C 1 > 0 such that for x ∈ R n and t > 0. On the other hand, since φ ≥ 0 and φ = 0 in R n , we find a measurable set E ⊂ P φ and ǫ > 0 such that This together with φ ∈ L implies that Let w be as in (1.12).Then there exists C 1 > 0 such that for x = (X, 0) with X > 0 and t > 1.Furthermore, there exists for x = (X, 0) with X ≥ 2t and t > 1, where i = 2, . . ., n.

Subclass L A
We give sufficient conditions for φ ∈ L to satisfy condition (A).
We can assume, without loss of generality, that ξ = e 1 , z = 0, where Π := {0} × R n−1 .For the proof, it suffices to prove that e ky 1 φ(y) dy for k ≥ 1. (2.12) Let α, δ > 0 be as in condition (A1).Thanks to the compactness of ∂P φ , by (2.11) we find finite points {x j } m j=1 ⊂ ∂P φ ∩ Π such that Furthermore, there exists ℓ > 0 such that On the other hand, it follows from condition (A1) that, for any j = 1, . . ., m, φχ B(x j ,δ) is α-concave in R n .Then we observe that φ α is concave in P φ ∩ B(x j , δ) and P φ ∩ B(x j , δ) is convex and open. (2.14) In particular, φ is continuous in P φ ∩ B(x j , δ).Then we find an (non-empty) open set Then we have This implies that On the other hand, for any fixed j = 1, . . ., m, we observe from x j ∈ Π that for y ∈ P φ ∩ B(x j , δ).These imply that e kz 1 φ(z) dz e ky 1 φ(y) dy.
This implies (2.12).Thus Lemma 2.10 follows.✷ As an application of Lemma 2.10, we see that Furthermore, we easily obtain the following properties.
(A2) Let φ ∈ L A and φ ∈ L. If there exists C ≥ 1 such that 3 Eventual F -concavity in L In this section we study the eventual F -concavity property of U d [φ] for φ ∈ L, and prove Theorems 1.2-1.5 and Corollary 1.1.We first prove Theorems 1.2 and 1.3.Let F satisfy condition (F a ) with a ∈ (0, ∞), d ≤ n/2, and φ ∈ L. Assume that M φ < a if d = n/2.Thanks to Remark 1.1 (1), we can assume, without loss of generality, that a = 1.Lemma 3.1 Assume condition (F a ) with a = 1.Let φ ∈ L. Let v d be as in (1.11).
Proof of Theorem 1.2.Let d < n/2, φ ∈ L, and a = 1.We first prove assertion (1).Assume that κ * F < 1/2.Then we find R > 0 and κ ′ > 0 such that This together with Lemma 3.1 (1) implies that for x ∈ R n and large enough t.Then, by Proposition 3.1 (2) we find T > 0 such that for large enough t.Since κ * F > 1/2, we find a sequence {r j } ⊂ R and κ ′′ ∈ R such that for j = 1, 2, . . . .By Lemma 3.1, applying by the intermediate value theorem, for any large enough T > 0, we find t T ∈ [T, ∞) such that v d (t T e 1 , t T ) = r j for some j = 1, 2, . . . .Then, taking large enough T if necessary, we see that This together with (3.5) implies that follows, and the proof of Theorem 1.2 is complete.
Next, we prove assertion (2).Assume that κ F (r * ) > 1/2 for some r * ∈ (0, ∞).Let φ ∈ L be such that M φ ∈ (e −r * /2 , 1).We can assume, without loss of generality, that φ satisfies (3.7).For any L > 0, by (1.11) and (3.8) we have uniformly for r ∈ (0, L).Then, for any large enough t > 0, applying the intermediate value theorem, we find r(t) > 0 such that for large enough t.Taking M φ close enough to 1 if necessary, we deduce that for large enough t.Since U d possesses the eventual F -concavity property, we have (3.17) On the other hand, it follows that for large enough t.This implies that for large enough t.Then for large enough t.Therefore, by (3.16), (3.17), and (3.18) we obtain Since r 0 , r 1 ∈ [0, ∞) and µ ∈ (0, 1) are arbitray, we deduce that the function is concave for any k > 0. Therefore we deduce from Lemma 2.5 that F -concavity is weaker than log-concavity in A([0, ∞)).
Proof of Theorem 1.4.We apply the same argument as in the proof of Theorem 1.5 with for t > 0 and r ∈ [0, ∞).
Assume that U n/2 [φ] possesses the eventual F -concavity property for all φ ∈ L. Let r 0 , Then, for any φ ∈ L, we see that for large enough t.This together with (3.19) implies that Similarly to the proof of Theorem 1.5, applying Lemma 2.5, we see that F -concavity is weaker than log-concavity in A([0, ∞)).Furthermore, we observe from the eventual log-concavity property of U d [φ] that U d [φ] possesses the eventual F -concavity property for all φ ∈ L if F -concavity is weaker than log-concavity in A([0, ∞)).Thus Theorem 1.4 follows.✷ 4 Eventual F -concavity with d ≤ n/2 in L A In this section, under condition (F a ) with a ∈ (0, ∞), we study the eventual F -concavity property of U d [φ] with φ ∈ L A to prove Theorem 1.6.Similarly to Section 3, we can assume, without loss of generality, that a = 1.We use the same notation as in Section 3.
Proof Theorem 1.6 (1).Assume condition (F a ) with a = 1.Furthermore, assume that the F -concavity property is preserved by the heat flow in R n .Then, by Lemma 2.4 we see that, for any k ∈ (0, 1), the function [0, ∞) ∋ r → F (ke −r 2 ) is concave, that is, for r ∈ (0, ∞), where s := − log k + r 2 .This implies that for large enough s.Then for large enough s.This implies the desired conclusion.The proof is complete.✷

Eventual F -concavity for L A
We assume condition (F a ) with a = 1 and Taking small enough ǫ if necessary, we have Then we find T > 0 and m > 0 such that Therefore, in the case of κ * F = 1/2, we observe that The main purpose of this subsection is to prove the following proposition.
Step 2. We complete the proof of Lemma 4.
Step 2. Assume that (1.8) does not hold.Then κ * F ≥ 1/2.Let ℓ be large enough, and set Then Lemma 2.10 implies that φ ℓ ∈ L A .We prove that U d [φ ℓ ](t) does not possess the eventual F -concavity property.By Theorem 1.2 (2) it suffices to consider the case of κ * F = 1/2.Let X > 0, and set X t := X/2t.It follows that for large enough t.Taking large enough L > 0, we have for large enough t and X t ≥ L. Since for X t ≥ L and large enough t, by (4.19) we have Step 1. Assume that (1.9) holds.It follows from Lemma 2.7 that (4.4) holds.Proposition 4.2 implies that U d [φ] possesses the eventual F -concavity property.

Corollary 1. 2 ( 1 )
Let F be admissible on I = [0, a) with a ∈ (0, ∞] and d < n/2.If the F -concavity property is preserved by the heat flow in R n , then U d [φ] possesses the eventual F -concavity property for all φ ∈ L A .