Some new properties of biharmonic heat kernels

Filippo GAZZOLA^{∗} – Hans-Christoph GRUNAU^{†}

Abstract Contrary to the second order case, biharmonic heat kernels are sign-changing. A deep knowledge of their behaviour may however allow to prove positivity results for solutions of the Cauchy problem. We establish further properties of these kernels, we prove some Lorch-Szegö-type monotonicity results and we give some hints on how to obtain similar results for higher polyharmonic parabolic problems.

Keywords: biharmonic parabolic equations, heat kernels.

1 Introduction

Consider the following Cauchy problem for the biharmonic heat equation

\{ \begin{matrix}u_{t} + Δ^{2}u = 0 & in ℝ_{+}^{n+1} := ℝ^{n} × [0, ∞) \\ u (x, 0) = u_{0}(x) & in ℝ^{n} , \end{matrix}  (1)

where n ≥ 1 and u_{0} ∈ C^{0} ∩ L^{∞} (ℝ^{n}). It is well-known (see [5] and references therein) that (1) admits a unique global in time solution explicitly given by

u(x, t) = α_{n}t^{−n/4} ∫_{ℝ^{n}} u_{0}(x − y)f_{n} (\frac{|y|}{t^{1/4}} ) dy , (x, t) ∈ ℝ_{+}^{n+1} .

Here α_{n} > 0 denotes a suitable normalization constant and

f_{n}(η) = η^{1−n} ∫_{0}^{∞} e^{−s^{4}}(ηs)^{n/2}J_{(n−2)/2}(ηs) ds ,  (2)

where J_{ν} denotes the ν-th Bessel function of the first kind. Contrary to the second order heat equation, no general positivity preserving property holds for (1), namely the positivity of the initial datum u_{0} may not imply positivity (in space and time) for the solution u = u(x, t) of (1). However, a careful analysis of the kernels f_{n} in (2) enables us to obtain some restricted and somehow hidden positivity, see [5, 9]. This property is called eventual local positivity and reflects the fact that, for suitable initial data, the solution of (1) becomes positive on compact domains of ℝ^{n} for sufficiently large time t and the time depends on the compact set itself. Let us also mention that positivity for (1) with a source term (namely u_{t} + Δ^{2}u = f) has been studied in [2] for linear problems when f = f(x, t) and in [5] for nonlinear problems when f = |u|^{p−1}u for p > 1 + 4/n (the so-called super-Fujita case, see [6]). See also [8] for estimates, existence and decay of global solutions. We also refer [3, 4] for related and blow-up results in the case f = |u|^{p}. A better understanding of the behaviour of the kernels will certainly allow to reach stronger results on positivity of solutions to (1). This is precisely the first goal of the present paper. After recalling in Section 2 some known results, in Section 3 we establish some new features of the f_{n}-functions. These features enable us to reach the second purpose of this paper, namely Lorch-Szegö-type monotonicity results for the f-functions, see Section 4. This means that the sequence of moduli of certain weighted integrals of f_{n} between consecutive zeros of f_{n} is monotonically decreasing. Let us briefly explain how such monotonicity results may be used in order to obtain eventual local positivity. If the initial datum u_{0} is positive and behaves like |x|^{−β} (0 < β < n) at ∞ (see (9)), it was proved in [5] that one should know that

∗Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 32, 20133 Milano (Italy)

†Fakultät für Mathematik, Otto–von–Guericke–Universität, Postfach 4120, 39016 Magdeburg (Germany)

∫_{0}^{∞} η^{−β} [η^{n−1}f_{n}(η)] dη = ∫_{0}^{∞} η^{n−β−2} [ηf_{n}(η)] dη > 0.  (3)

It is this constant times t^{−β/4} which locally gives the asymptotic behaviour for t → ∞ of the solution of the Cauchy problem. Below we will prove a Lorch-Szegö-type monotonicity result for η ↦ ηf_{n}(η), (n ≥ 2), so that positivity of the above integral is immediate for β ∈ [(n − 2), n). In [5], inequality (3) (and more) was proved without referring explicitly to Lorch-Szegö-type results, but we think that our present approach gives a more natural interpretation of certain eventual local positivity features. Finally, in Section 5 we give hints on how to extend some properties of the kernels and their consequences to polyharmonic heat equations.

2 Preliminaries and notations

Let us first recall that thanks to Galaktionov-Pohožaev [7], we know that the f-functions have exponential decay at infinity. More precisely, for any integer n ≥ 1 there exist K = K_{n} > 0, μ = μ_{n} > 0 such that

|f_{n}(η)| ≤ K exp (−μη^{4/3})  for all η ≥ 0.  (4)

Then, we recall some properties of the f-functions proved in [5]. Firstly, a recursion formula holds:

f′_{n}(η) = −η f_{n+2}(η)  for all n ≥ 1.  (5)

Moreover, f_{n} satisfies the following third order differential equation (see [5, Theorem 6])

f′′′_{n}(η) + \frac{n − 1}{η}f′′_{n}(η) − \frac{n − 1}{η^{2}}f′_{n}(η) − \frac{η}{4}f_{n}(η) = 0 ,  (6)

which we shall also exploit in the following equivalent form

(Δf_{n})′ (η) = \frac{η}{4}f_{n}(η) .  (7)

Thanks to (5)-(6), in [5] the following result was proved

∫_{0}^{∞} η^{n−1−β}f_{n}(η) dη > 0  for all integer n ≥ 1 and all β ∈ [0, n).  (8)

In turn, (8) was used to prove the eventual local positivity property for (1) with initial data of the kind

u_{0}(x) = \frac{1}{g(x) + |x|^{β}}  where g ∈ C^{0}(ℝ^{n}, ℝ_{+}) satisfies lim_{|x|→∞} \frac{g(x)}{|x|^{β}} = 0  (9)

for some β ∈ [0, n). By eventual local positivity we mean that the solution of (1) is (locally) positive on compact domains of ℝ^{n} for (eventual) sufficiently large time t. The proof of (8) is quite lengthy and delicate.

It is also shown in [5, Theorem 7] that f_{n}(η) changes sign infinitely many times as η → +∞. For a fixed n, we denote by {ζ_{j}} the sequence of all the zeroes of f_{n}:

f_{n}(ζ_{j}) = 0 0 < ζ_{1} < ζ_{2} < ....

In some situations, we need to distinguish between “ascending” zeroes (where f′_{n} > 0) and “descending” zeroes (where f′_{n} < 0). To this end, we denote by P_{k} (resp. N_{k}) the successive intervals, where f_{n} is positive (resp. negative) so that we have

[0, ∞) = ⋃_{k=1}^{∞} (\overline{P_{k}} ∪ \overline{N_{k}}) .

Moreover, we write

z_{k}^{+} := sup P_{k} = inf N_{k},  z_{k}^{−} := sup N_{k} = inf P_{k+1}  (k ∈ ℕ)

so that ∪_{k}{z_{k}^{±}} ≡ ∪_{j}{ζ_{j}} are the zeroes of f_{n}. Let

μ_{k} ∈ P_{k} be such that f_{n}(μ_{k}) = max_{η∈P_{k}} f_{n}(η)
m_{k} ∈ N_{k} be such that f_{n}(m_{k}) = min_{η∈N_{k}} f_{n}(η) .

In particular, f′_{n}(μ_{k}) = f′_{n}(m_{k}) = 0 and we know that

0 = μ_{1} < z_{1}^{+} < m_{1} < z_{1}^{−} < μ_{2} < z_{2}^{+} < m_{2} < z_{2}^{−} < ...

3 Behaviour of the f-functions at some special points

3.1 Behaviour at zeroes

Proposition 1. Assume n ≥ 4. Then, for all j ≥ 1 we have ζ_{j}|f′_{n}(ζ_{j})| > ζ_{j+1}|f′_{n}(ζ_{j+1})|.

Proof. We use (7) and obtain, observing that f_{n}(ζ_{j}) = f_{n}(ζ_{j+1}) = 0:

−\frac{1}{2} (ζ_{j+1}^{2}f′_{n}(ζ_{j+1})^{2} − ζ_{j}^{2}f′_{n}(ζ_{j})^{2}) = −\frac{1}{2} ∫_{ζ_{j}}^{ζ_{j+1}} (η^{2}f′_{n}(η)^{2})′ dη

= − ∫_{ζ_{j}}^{ζ_{j+1}} (ηf′_{n}(η)^{2} + η^{2}f′_{n}(η)f′′_{n}(η)) dη = (n − 2) ∫_{ζ_{j}}^{ζ_{j+1}} ηf′_{n}(η)^{2} dη − ∫_{ζ_{j}}^{ζ_{j+1}} η^{2}f′_{n}(η)Δf_{n}(η) dη
= (n − 2) ∫_{ζ_{j}}^{ζ_{j+1}} ηf′_{n}(η)^{2} dη + 2 ∫_{ζ_{j}}^{ζ_{j+1}} ηf_{n}(η)Δf_{n}(η) dη + ∫_{ζ_{j}}^{ζ_{j+1}} η^{2}f_{n}(η)(Δf_{n})′(η) dη
= (n − 4) ∫_{ζ_{j}}^{ζ_{j+1}} ηf′_{n}(η)^{2} dη + 2(n − 2) ∫_{ζ_{j}}^{ζ_{j+1}} f_{n}(η)f′_{n}(η) dη + \frac{1}{4} ∫_{ζ_{j}}^{ζ_{j+1}} η^{3}f_{n}(η)^{2} dη
= (n − 4) ∫_{ζ_{j}}^{ζ_{j+1}} ηf′_{n}(η)^{2} dη + \frac{1}{4} ∫_{ζ_{j}}^{ζ_{j+1}} η^{3}f_{n}(η)^{2} dη > 0

since we assume that n ≥ 4.

In lower dimensions one has a slightly weaker statement:

Proposition 2. For any n ≥ 1 and any j ≥ 1 one has |f′_{n}(ζ_{j})| > |f′_{n}(ζ_{j+1})|.

Proof. The differential equation (7) directly shows:

0 < ∫_{ζ_{j}}^{ζ_{j+1}} \frac{η}{4}f_{n}(η)^{2} dη = ∫_{ζ_{j}}^{ζ_{j+1}} f_{n}(η)(Δf_{n})′(η) dη = − ∫_{ζ_{j}}^{ζ_{j+1}} f′_{n}(η)Δf_{n}(η) dη
= − ∫_{ζ_{j}}^{ζ_{j+1}} f′_{n}(η)f′′_{n}(η) dη − (n − 1) ∫_{ζ_{j}}^{ζ_{j+1}} \frac{f′_{n}(η)^{2}}{η} dη ≤ −\frac{1}{2}f′_{n}(ζ_{j+1})^{2} + \frac{1}{2}f′_{n}(ζ_{j})^{2}

and the statement follows.

3.2 Behaviour at critical points

We observe first that in successive local maxima and minima, the f-functions are decreasing and increasing, respectively.

Proposition 3. For any n ≥ 1 and k ≥ 1 we have

f_{n}(μ_{k}) > f_{n}(μ_{k+1}), f_{n}(m_{k}) < f_{n}(m_{k+1}).

Proof. The statement follows directly from the Lorch-Szegö-type Theorem 1, which will be proved below, and integrating the recurrence relation (5).

Proposition 4. For all n ≥ 2 and all k ≥ 1 we have |f′′_{n}(μ_{k})| > |f′′_{n}(m_{k})| > |f′′_{n}(μ_{k+1})| > 0. For all k ≥ 1 we have f′′_{1}(μ_{k}) < 0 and f′′_{1}(m_{k}) > 0.

Proof. Assume first that n ≥ 2. Since different f-functions are involved in the proof, we denote here μ_{k}^{n}, m_{k}^{n} and μ_{k+1}^{n} in order to emphasize their dependence on n. In view of the recursion formula (5), we know that

f_{n+2}(μ_{k}^{n}) = f_{n+2}(m_{k}^{n}) = f_{n+2}(μ_{k+1}^{n}) = 0 .  (10)

Since n + 2 ≥ 4, by Proposition 1 we then obtain

μ_{k}^{n}|f′_{n+2}(μ_{k}^{n})| > m_{k}^{n}|f′_{n+2}(m_{k}^{n})| > μ_{k+1}^{n}|f′_{n+2}(μ_{k+1}^{n})| .  (11)

By differentiating (5), we get f′′_{n}(η) = −f_{n+2}(η) − ηf′_{n+2}(η). This, combined with (10)-(11) gives

|f′′_{n}(μ_{k}^{n})| = μ_{k}^{n}|f′_{n+2}(μ_{k}^{n})| > m_{k}^{n}|f′_{n+2}(m_{k}^{n})| = |f′′_{n}(m_{k}^{n})| > μ_{k+1}^{n}|f′_{n+2}(μ_{k+1}^{n})| = |f′′_{n}(μ_{k+1}^{n})| .

The first two inequalities in the statement (for n ≥ 2) are so proved. The last one holds since if we would have equality we would violate |f′′_{n}(μ_{k+1})| > |f′′_{n}(m_{k+1})|. Assume now that n = 1. Then (6) tells us that f′′′_{1}(η) < 0 for η ∈ (z_{k}^{+}, z_{k}^{−}) so that the map η ↦ f′′_{1}(η) is strictly decreasing. Clearly, f′′_{1}(m_{k}) ≥ 0; if f′′_{1}(m_{k}) = 0, then the just mentioned monotonicity would imply f′′_{1}(η) < 0 for η ∈ (m_{k}, z_{k}^{−}], contradicting the fact that m_{k} is a relative minimum for f_{1}. Similarly, one may proceed to show that f′′_{1}(μ_{k}) < 0.

Proposition 5. Let n ≥ 1 and let 0 < α < β be two critical points for f_{n}. Then the two following identities hold:

\frac{1}{4} ∫_{α}^{β} ηf_{n}(η) dη = f′′_{n}(β) − f′′_{n}(α) ,

\frac{1}{4} ∫_{α}^{β} η^{n−1}f_{n}(η) dη = −2(n − 2) ∫_{α}^{β} η^{n−4}f′_{n}(η) dη + β^{n−2}f′′_{n}(β) − α^{n−2}f′′_{n}(α) ;

when n ≥ 3, the second identity also holds if α = 0.

Proof. An integration by parts yields

∫_{α}^{β} \frac{f′′_{n}(η)}{η} dη = ∫_{α}^{β} \frac{f′_{n}(η)}{η^{2}} dη .

The first identity then follows by integrating (6) over [α, β]. Next, notice that further integrations by parts yield

∫_{α}^{β} η^{n−2}f′′′_{n}(η) dη = (2 − n) ∫_{α}^{β} η^{n−3}f′′_{n}(η) dη + β^{n−2}f′′_{n}(β) − α^{n−2}f′′_{n}(α) ,
∫_{α}^{β} η^{n−3}f′′_{n}(η) dη = (3 − n) ∫_{α}^{β} η^{n−4}f′_{n}(η) dη .

The second identity then follows by multiplying (6) by η^{n−2} and integrating. By combining Propositions 4 and 5 we immediately obtain Corollary 1. Let n ≥ 2. Then, for any k ≥ 1, we have:

∫_{μ_{k}}^{μ_{k+1}} ηf_{n}(η) dη > 0

Another interesting property which holds on critical points is the following: Proposition 6. Let n ≥ 1 and let 0 < α < β be two critical points for f_{n}. Then

4f′′_{n}(β)^{2} + (2 − n)f_{n}(β)^{2} < 4f′′_{n}(α)^{2} + (2 − n)f_{n}(α)^{2} .

Proof. By using (7) and integrating by parts we find

[4f′′_{n}(β)^{2} + (2 − n)f_{n}(β)^{2}] − [4f′′_{n}(α)^{2} + (2 − n)f_{n}(α)^{2}]

= [4Δf_{n}(β)^{2} + (2 − n)f_{n}(β)^{2}] − [4Δf_{n}(α)^{2} + (2 − n)f_{n}(α)^{2}]
= ∫_{α}^{β} [4Δf_{n}(η)^{2} + (2 − n)f_{n}(η)^{2}]′ dη
= ∫_{α}^{β} [8Δf_{n}(η)(Δf_{n})′(η) + 2(2 − n)f_{n}(η)f′_{n}(η)] dη
= 2 ∫_{α}^{β} [ηf_{n}(η)Δf_{n}(η) + (2 − n)f_{n}(η)f′_{n}(η)] dη
= 2 ∫_{α}^{β} [ηf_{n}(η)f′′_{n}(η) + f_{n}(η)f′_{n}(η)] dη
= 2 ∫_{α}^{β}(ηf′_{n}(η))′f_{n}(η) dη
= −2 ∫_{α}^{β} ηf′_{n}(η)^{2} dη < 0 .

This proves the statement.

Proposition 7. Let n ≥ 1 and let k ≥ 1. Then,

∫_{m_{k}}^{μ_{k+1}} η^{n−1}f_{n}(η) dη < 0 , ∫_{μ_{k+1}}^{m_{k+1}} η^{n−1}f_{n}(η) dη > 0 .

If n ≥ 3, then the second inequality also holds when integrated over [μ_{1}, m_{1}] = [0, m_{1}].

Proof. Since the second inequality follows similarly, we only prove the first one. Consider first the case n = 1. We have

∫_{m_{k}}^{μ_{k+1}} f_{1}(η) dη = \frac{1}{z_{k}^{−}} ∫_{m_{k}}^{z_{k}^{−}} z_{k}^{−}f_{1}(η) dη + \frac{1}{z_{k}^{−}} ∫_{z_{k}^{−}}^{μ_{k+1}} z_{k}^{−}f_{1}(η) dη

< \frac{1}{z_{k}^{−}} ∫_{m_{k}}^{z_{k}^{−}} ηf_{1}(η) dη + \frac{1}{z_{k}^{−}} ∫_{z_{k}^{−}}^{μ_{k+1}} ηf_{1}(η) dη = \frac{1}{z_{k}^{−}} ∫_{m_{k}}^{μ_{k+1}} ηf_{1}(η) dη
= \frac{4}{z_{k}^{−}} (f′′_{1}(μ_{k+1}) − f′′_{1}(m_{k})) < 0 ,

where we used Proposition 5 in the last equality and Proposition 4 for the last inequality. Consider now the case n ≥ 2. Because of Proposition 2 and (5) we have f′_{n}(η) > 0 for all η ∈ (m_{k}, μ_{k+1}). Moreover, by Proposition 4 we know that μ_{k+1}^{n−2}f′′_{n}(μ_{k+1}) < 0 and m_{k}^{n−2}f′′_{n}(m_{k}) > 0. Therefore, by Proposition 5 we get

\frac{1}{4} ∫_{m_{k}}^{μ_{k+1}} η^{n−1}f_{n}(η) dη = −2(n − 2) ∫_{m_{k}}^{μ_{k+1}} η^{n−4}f′_{n}(η) dη + μ_{k+1}^{n−2}f′′_{n}(μ_{k+1}) − m_{k}^{n−2}f′′_{n}(m_{k}) < 0 .

The first inequality is so proved also for n ≥ 2.

3.3 Behaviour of the second derivative

Proposition 8. For any n ≥ 1 and any k ≥ 1, the following two facts hold:

f′′_{n}(η) < 0 and f′′′_{n}(η) > 0 for all η ∈ (z_{k}^{−}, μ_{k+1}] ,
f′′_{n}(η) > 0 and f′′′_{n}(η) < 0 for all η ∈ (z_{k}^{+}, m_{k}] .

Proof. Fix k ≥ 1 and take η ∈ (z_{k}^{−}, μ_{k+1}); then, f_{n}(η) > 0 and f′_{n}(η) > 0. Using (6) we then obtain

f′′′_{n}(η) + \frac{n − 1}{η}f′′_{n}(η) > 0  for all η ∈ (z_{k}^{−}, μ_{k+1}) .  (12)

Since μ_{k+1} is a maximum point for f_{n}, by Proposition 4 we infer that f′′_{n}(μ_{k+1}) < 0. By continuity, there exists \overline{η} < μ_{k+1} such that f′′_{n}(η) < 0 for all η ∈ (\overline{η}, μ_{k+1}]. The first statement follows if we show that \overline{η} ≤ z_{k}^{−}. For contradiction, assume that \overline{η} > z_{k}^{−}. Then, we would have f′′_{n}(\overline{η}) = 0 and f′′′_{n}(\overline{η}) ≤ 0 since f′′_{n} becomes negative for η > \overline{η}. This contradicts (12). Therefore, f′′(η) < 0 for all η ∈ (z_{k}^{−}, μ_{k+1}). By using again (6), we then also infer that f′′′(η) > 0 for all η ∈ (z_{k}^{−}, μ_{k+1}]. By reversing all the signs, we obtain similarly the statement in (z_{k}^{+}, m_{k}].

When n = 1 we have a more precise description of flex points:

Proposition 9. For any k ≥ 1 there exists a unique η_{k} ∈ [μ_{k}, m_{k}] and a unique η^{k} ∈ [m_{k}, μ_{k+1}] such that f′′_{1}(η_{k}) = f′′_{1}(η^{k}) = 0. Moreover, η_{k} ∈ (μ_{k}, z_{k}^{+}] and η^{k} ∈ (m_{k}, z_{k}^{−}].

Proof. Since the proof of the two statements is similar, we only prove the first one. By Proposition 4 we have f′′_{1}(μ_{k}) < 0 and f′′_{1}(m_{k}) > 0. Therefore, the equation f′′_{1}(η) = 0 has an odd number of solutions in (μ_{k}, m_{k}). By Proposition 8 the equation f′′_{1}(η) = 0 has no solutions in (z_{k}^{+}, m_{k}]. Assume for contradiction that f′′_{1}(η) = 0 has at least three solutions in [μ_{k}, z_{k}^{+}]. Then, in a descending flex η^{∗} ∈ (μ_{k}, z_{k}^{+}) we have f′′_{1}(η^{∗}) = 0 and f′′′_{1}(η^{∗}) ≤ 0. This contradicts (6) since f_{1}(η^{∗}) > 0.

3.4 Integral properties on intervals containing 0

Proposition 10. For all n ≥ 1 and all γ > −1 the following implication holds:

∫_{0}^{a} η^{γ+2}f_{n+2}(η) dη ≥ 0 for all a > 0 ⟹ ∫_{0}^{a} η^{γ}f_{n}(η) dη ≥ 0 for all a > 0 .

Proof. In order to prove the statement it suffices to show that if the first inequality is true, then

∫_{0}^{z_{k}^{−}} η^{γ}f_{n}(η) dη ≥ 0 for all k ≥ 1 .

By integrating by parts and (5) we have

∫_{0}^{z_{k}^{−}} η^{γ}f_{n}(η) dη = −\frac{1}{γ + 1} ∫_{0}^{z_{k}^{−}} η^{γ+1}f′_{n}(η) dη = \frac{1}{γ + 1} ∫_{0}^{z_{k}^{−}} η^{γ+2}f_{n+2}(η) dη ≥ 0

and the statement follows. Proposition 11. For all n ≥ 1, all k ≥ 1 and all γ > −1 the following identity holds:

\frac{1}{4} ∫_{0}^{z_{k}^{−}} η^{γ+4}f_{n}(η) dη = (γ+1)(γ+3)(n−γ−3) ∫_{0}^{z_{k}^{−}} η^{γ}f_{n}(η) dη+(z_{k}^{−})^{γ+3}Δf_{n}(z_{k}^{−})−(γ+3)(z_{k}^{−})^{γ+2}f′_{n}(z_{k}^{−}) .

Proof. Using (7) and several integrations by parts yield

\frac{1}{4} ∫_{0}^{z_{k}^{−}} η^{γ+4}f_{n}(η) dη = ∫_{0}^{z_{k}^{−}} η^{γ+3}(Δf_{n})′(η) dη

= −(γ + 3) ∫_{0}^{z_{k}^{−}}[η^{γ+2}f′′_{n}(η) + (n − 1)η^{γ+1}f′_{n}(η)] dη + (z_{k}^{−})^{γ+3}Δf_{n}(z_{k}^{−})
= (γ + 1)(γ + 3)(n − γ − 3) ∫_{0}^{z_{k}^{−}} η^{γ}f_{n}(η) dη + (z_{k}^{−})^{γ+3}Δf_{n}(z_{k}^{−}) − (γ + 3)(z_{k}^{−})^{γ+2}f′_{n}(z_{k}^{−}) .

This proves the statement.

4 Lorch-Szegö-type monotonicity results for the f-functions

Theorem 1. For any n ≥ 1 and k ≥ 1 we have that

∫_{z_{k}^{−}}^{z_{k+1}^{−}} ηf_{n}(η) dη > 0  and  ∫_{z_{k}^{+}}^{z_{k+1}^{+}} ηf_{n}(η) dη < 0.

If additionally n ≥ 2 then we also have that

∫_{0}^{z_{1}^{−}} ηf_{n}(η) dη > 0.

Proof. We first claim that for all k ≥ 1 we have

Δf_{n}(z_{k}^{−}) < 0,  Δf_{n}(z_{k}^{+}) > 0.  (13)

To this end, we remark that (7) shows that Δf_{n} is strictly increasing on ⋃_{k=1}^{∞} P_{k} and strictly decreasing on ⋃_{k=1}^{∞} N_{k}. In the local minima m_{k} ∈ N_{k} we have that Δf_{n}(m_{k}) = f′′_{n}(m_{k}) ≥ 0 and so for z_{k}^{+} = inf N_{k} we conclude that Δf_{n}(z_{k}^{+}) > 0. Analogously, μ_{k+1} ∈ P_{k+1}, Δf_{n}(μ_{k+1}) = f′′_{n}(μ_{k+1}) ≤ 0, z_{k}^{−} = inf P_{k+1}, Δf_{n}(z_{k}^{−}) < 0. Hence, (13) is proved. Next, we show that

Δf_{n}(z_{k}^{−}) < Δf_{n}(z_{k+1}^{−}) < 0.  (14)

In order to prove (14), we multiply (7) by Δf_{n}. Then, by integrating over (z_{k}^{−}, z_{k+1}^{−}) we obtain:

[(Δf_{n}(η))^{2}]_{z_{k}^{−}}^{z_{k+1}^{−}} = 2 ∫_{z_{k}^{−}}^{z_{k+1}^{−}} Δf_{n}(η)(Δf_{n})′(η) dη

= \frac{1}{2} ∫_{z_{k}^{−}}^{z_{k+1}^{−}} ηf′′_{n}(η)f_{n}(η) dη + \frac{n − 1}{2} ∫_{z_{k}^{−}}^{z_{k+1}^{−}} f′_{n}(η)f_{n}(η) dη

= −\frac{1}{2} ∫_{z_{k}^{−}}^{z_{k+1}^{−}} ηf′_{n}(η)^{2} dη < 0.

So, Δf_{n}(z_{k}^{−})^{2} > Δf_{n}(z_{k+1}^{−})^{2}. Together with (13), this proves (14). We may now obtain the first statement of Theorem 1 from the differential equation (7) by integration:

\frac{1}{4} ∫_{z_{k}^{−}}^{z_{k+1}^{−}} ηf_{n}(η) dη = ∫_{z_{k}^{−}}^{z_{k+1}^{−}}(Δf_{n})′(η) dη = (Δf_{n})(z_{k+1}^{−}) − (Δf_{n})(z_{k}^{−}) > 0.

In the last step we made use of (14). The inequality ∫_{z_{k}^{+}}^{z_{k+1}^{+}} ηf_{n}(η) dη < 0 may be proved similarly.

Assuming now n ≥ 2 and integrating over (0, z_{1}^{−}) we obtain:

[Δf_{n}(η)]_{0}^{z_{1}^{−}} = 2 ∫_{0}^{z_{1}^{−}} Δf_{n}(η)(Δf_{n})′(η) dη
= \frac{1}{2} ∫_{0}^{z_{1}^{−}} ηf′′_{n}(η)f_{n}(η) dη + \frac{n − 1}{2} ∫_{0}^{z_{1}^{−}} f′_{n}(η)f_{n}(η) dη
= −\frac{1}{2} ∫_{0}^{z_{1}^{−}} ηf′_{n}(η)^{2} dη + \frac{n − 2}{2} ∫_{0}^{z_{1}^{−}} f′_{n}(η)f_{n}(η) dη
= −\frac{1}{2} ∫_{0}^{z_{1}^{−}} ηf′_{n}(η)^{2} dη − \frac{n − 2}{4}f_{n}(0)^{2} < 0,

so that

Δf_{n}(0) < Δf_{n}(z_{1}^{−}) < 0.

The last statement of Theorem 1 follows now with the same argument as above.

When starting integration from η = 0, the following result holds:

Theorem 2. Let n ≥ 3 and assume that h ∈ C^{1}(0, +∞) satisfies h(η) > 0, h′(η) ≤ 0 for all η > 0 and

∫_{0} h(η)η^{(n−1)/2} dη < ∞.

Then, for any a > 0 we have

∫_{0}^{a} h(η) η^{(n−1)/2} f_{n}(η) dη > 0.

Proof. By making use of the definition of f_{n} we see that

I_{a} := ∫_{0}^{a} h(η) η^{(n−1)/2} f_{n}(η) dη = ∫_{0}^{a} h(η) η^{(1−n)/2} ∫_{0}^{∞} e^{−s^{4}}(ηs)^{n/2}J_{(n−2)/2}(ηs) ds dη .

Therefore, the change of variables z = ηs yields

I_{a} = ∫_{0}^{a} h(η) η^{−(n+1)/2} ∫_{0}^{∞} e^{−(z/η)^{4}}z^{n/2}J_{(n−2)/2}(z) dz dη .

In view of the integrability condition on η ↦ h(η)η^{(n−1)/2}, Fubini’s Theorem implies that

I_{a} = ∫_{0}^{∞} (\sqrt{z}J_{(n−2)/2}(z)) z^{(n−1)/2} ∫_{0}^{a} h(η) η^{−(n+1)/2} e^{−(z/η)^{4}} dη dz .

For all z > 0, let

g(z) := z^{(n−1)/2} ∫_{0}^{a} h(η) η^{−(n+1)/2} e^{−(z/η)^{4}} dη .

Then, by differentiating and integrating by parts

g′(z) = \frac{n − 1}{2}z^{(n−3)/2} ∫_{0}^{a} h(η) η^{−(n+1)/2} e^{−(z/η)^{4}} dη − z^{(n−1)/2} ∫_{0}^{a} h(η) η^{−(n+1)/2} \frac{4z^{3}}{η^{4}} e^{−(z/η)^{4}} dη
= z^{(n−3)/2} (∫_{0}^{a} h′(η) η^{(1−n)/2} e^{−(z/η)^{4}} dη − h(a) a^{(1−n)/2} e^{−(z/a)^{4}}) .

Since h′ ≤ 0 and h > 0, this shows that g is strictly decreasing. Since we assumed n ≥ 3, the Lorch-Szegö Theorem [10] (see also [1, Corollary 4.15.2]) applies and the proof is complete.

In the particular case of powers, we have the following statement

Theorem 3. Let n ≥ 3. Then for any γ ∈ (−1, \frac{n−1}{2}] and any a > 0 we have

∫_{0}^{a} η^{γ} f_{n}(η) dη > 0.

Let n ≥ 1. Then, for any k ≥ 1 we have

∫_{0}^{z_{k}^{−}} η^{n+1} f_{n}(η) dη < 0.

Proof. The first statement is a direct consequence of Theorem 2. By taking γ = n − 3 in Proposition 11, we obtain

\frac{1}{4} ∫_{0}^{z_{k}^{−}} η^{n+1}f_{n}(η) dη = (z_{k}^{−})^{n}Δf_{n}(z_{k}^{−}) − n(z_{k}^{−})^{n−1}f′_{n}(z_{k}^{−}) < 0 ,

where the inequality follows from (13) and from the fact that f′_{n}(z_{k}^{−}) > 0.

Remark 1. In view of the discussion in the introduction, see (3), it would be interesting to prove Theorem 1 not only for ηf_{n}(η) but also for η^{γ}f_{n}(η) with γ ≥ 1 as large as possible. This is related to finding the largest value γ_{n} of γ for which

∫_{0}^{a} η^{γ} f_{n}(η) dη ≥ 0 for all a > 0 .

Note that for any γ ∈ (−1, γ_{n}) the above integral is finite and strictly positive. By Theorem 3 we know that for any n ≥ 1 we have γ_{n} < n + 1. Moreover, if n ≥ 3 then γ_{n} ≥ \frac{n−1}{2}, see again Theorem 3. If n = 2, then γ_{2} ≥ 1, see Theorem 1. If n = 1, by repeating the arguments in the proof of [5, Lemma A.4], we have that

∫_{0}^{z_{1}^{−}} f_{1}(η) dη > 0

which, combined with Theorem 1, yields γ_{1} ≥ 0. By Proposition 10 we know that γ_{n+2} ≤ γ_{n} + 2. Although (8) holds for β = 0, numerical experiments suggest that γ_{n} < n−1, at least in high dimensions n. For instance, if n = 20, it seems that 16 < γ_{20} < 17.

5 Extensions to polyharmonic heat kernels

In this section we briefly sketch some properties of higher order polyharmonic heat kernels. The proofs can be obtained by slightly modifying the arguments in [5]. So, for m ≥ 2 consider the following Cauchy problem for the polyharmonic heat equation

\{ \begin{matrix}u_{t} + (−Δ)^{m}u = 0 & in ℝ_{+}^{n+1} \\ u (x, 0) = u_{0}(x) & in ℝ^{n} , \end{matrix}  (15)

where n ≥ 1 and u_{0} ∈ C^{0} ∩ L^{∞} (ℝ^{n}). Then, (1) admits a unique global in time solution explicitly given by

u(x, t) = αt^{−n/2m} ∫_{ℝ^{n}} u_{0}(x − y)f_{m,n} (\frac{|y|}{t^{1/2m}} ) dy (x, t) ∈ ℝ_{+}^{n+1} ,

where α = α_{m,n} > 0 is a suitable normalization constant and

f_{m,n}(η) = η^{1−n} ∫_{0}^{∞} e^{−s^{2m}}(ηs)^{n/2}J_{(n−2)/2}(ηs) ds .

By arguing as in [5] and using [1, Section 4.62], we find that (5) still holds, independently of m. Moreover, the following (2m − 1)-order differential equation generalizes (7):

[Δ^{m−1}f_{m,n}]′(η) = \frac{(−1)^{m}}{2m} η f_{m,n}(η)  for all n ≥ 1.  (16)

It is straightforward that (16) coincides with (7) if m = 2, whereas it reduces to f′(η) = −\frac{1}{2}ηf(η) whenever m = 1 (recall that in the latter case, the kernel f is independent of n). With these two identities, we can prove results similar to (8). We can show that

C_{m,n,β} := ∫_{0}^{∞} η^{n−1−β}f_{m,n}(η) dη > 0  for all integer n ≥ 1 and all β ∈ [0, n).  (17)

The proof of (17) follows the lines of [5], that is, we combine the use of Fubini’s Theorem, Lorch-Szegö Theorem, several integration by parts and two crucial recurrence formulas based on (5) and (16), namely

C_{m,n+2,β} = (n − β)C_{m,n,β} for all β ∈ (0, n)

C_{m,n,β} = 2m ∏_{j=1}^{m−1} [(2m + β − 2j)(n + 2j − 2m − β)]C_{m,n+2,β+2m} for all β ∈ (0, n + 2 − 2m).

Clearly, we expect that (17) may enable one to prove eventual local positivity results for (15) when u_{0} is as in (9).

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