Decay and eventual local positivity for biharmonic parabolic equations ^{∗}

Alberto FERRERO^{†}– Filippo GAZZOLA^{‡}– Hans-Christoph GRUNAU^{§}

Abstract We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator ∂_{t} + Δ^{2} is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost instantaneous change of sign. We determine conditions on the initial datum for which the corresponding solution exhibits some kind of positivity behaviour. We prove eventual local positivity properties both in the linear and semilinear case. At the same time, we show that negativity of the solution may occur also for arbitrarily large given time, provided the initial datum is suitably constructed.

1 Introduction and results

Contrary to the second order heat equation, no general positivity preserving property (ppp in the sequel) holds for the following Cauchy problem for fourth order parabolic equations

\{ \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}). By ppp, we mean here that positivity of the initial datum u_{0} implies positivity (in space and time) for the solution u = u(x, t) of (1). In fact, no global ppp for (1) may be expected at all, see [2, 4, 11]. This common feeling is based on observing the oscillatory behaviour of the biharmonic heat kernel. However, a more careful analysis shows that, under suitable assumptions, some restricted and somehow hidden versions of the ppp can be observed, see [10]. A better understanding of the behaviour of the kernels will certainly allow to reach stronger versions of the ppp. It is precisely the first goal of the present paper to analyze in some detail the biharmonic heat kernels: we determine explicitly their power series expansion, we obtain a third order differential equation satisfied by the kernels and we show that the kernels have infinitely many damped oscillations. With these tools we are then able to reach the second goal of the present paper, a rather complete description of versions of the ppp for (1). All these observations are then applied to the corresponding nonlinear problem

\{ \begin{matrix}u_{t} + Δ^{2}u = |u|^{p−1}u & in ℝ_{+}^{n+1} \\ u (x, 0) = u_{0}(x) & in ℝ^{n} \end{matrix}  (2)

where p > 1 + 4/n; the exponent 1 + 4/n is the analogue of the ”Fujita”-exponent (see [6, 15] and the references therein), arising in second order semilinear Cauchy problems. Our third purpose is to extend the global existence result and the decay estimates obtained in [9] to the case where the initial datum u_{0} has supercritical decay at infinity in order to show that in these cases, when u_{0} is positive and has a suitable asymptotic profile, global solutions of (2) are eventually locally positive.

∗Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.

†Dipartimento di Matematica dell’Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa (Italy)

‡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)

For the somehow related equation with positive nonlinearity u_{t} + Δ^{2}u = |u|^{p}, it is known [5] that solutions blow up in finite time in the sub-Fujita case 1 < p ≤ 1 + \frac{4}{n}, provided the initial datum has nonnegative integral over ℝ^{n}. For the very same equation in the super-Fujita case p > 1 + \frac{4}{n}, interesting and useful techniques were introduced by Galaktionov-Pohožaev [7] and Caristi-Mitidieri [3]. Although the nonlinearity in (2) may change sign, in some parts of our proofs we take advantage of these techniques. The corresponding second order semilinear parabolic problem has been widely studied; in this case and in contrast with our situation, powerful tools like strong maximum principles and construction of auxiliary functions satisfying suitable differential inequalities are available. For an overview of these results, we refer to the introduction in [9], to [15] and to references therein. Let us now explain in some more detail, what is known about (1) and (2) and what we are going to prove in the present work.

1.1 Positivity in the linear problem

For the biharmonic heat equation,

u_{t} + Δ^{2}u = 0  (t > 0),  (3)

either in ℝ^{n} or in a bounded smooth domain (then complemented with suitable boundary conditions) no positivity preserving of the solution with respect to the initial datum holds true. In general, one even has to expect instantaneous change of sign, see e.g. [2, 4, 11], which is a property of the differential equation and can be observed independently of a possible choice of boundary data and independently of whether it is considered in bounded or unbounded domains. On the other hand, there exist bounded domains with boundary conditions such that the corresponding elliptic first eigenvalue is simple and the first eigenfunction is of fixed sign and displays a nondegenerate behaviour at the boundary, see [12] and references therein. In these domains and for each positive initial datum, in dependence of this datum the solution of the initial boundary value problem is eventually positive. This positivity comes up almost immediately, since the higher modes in the expansion with respect to the eigenfunctions decay much faster than the fundamental mode. In a previous note [10], a related result - i.e. eventual local positivity - was proved for the Cauchy problem (1), assuming that the initial datum u_{0} is nonnegative and compactly supported. We consider here initial data u_{0} which are not compactly supported and which display a given decay behaviour as |x| → ∞. Not only we prove eventual local positivity for the solution to (1), but we also give quantitative lower bounds. We fix some arbitrary β ≥ 0 and consider the functional set

C_{β} := {g ∈ C^{0}(ℝ^{n}; ℝ_{+}) : g(0) > 0 , g(x) = o(|x|^{β}) as |x| → ∞} .

Our main positivity result for the linear Cauchy problem (1) is the following

Theorem 1. Let β ≥ 0 and let g ∈ C_{β}. Let

u_{0}(x) = \frac{1}{g(x) + |x|^{β}} .

Let u = u(x, t) be the corresponding solution of (1) and K ⊂ ℝ^{n} be a compact set.

(i) If β < n, then there exists C̃_{n,β} > 0 such that

lim_{t→+∞} t^{β/4}u(x, t) = C̃_{n,β} ,

uniformly with respect to x ∈ K. (ii) If β ≥ n and g(x) ≡ 1, then there exists D̃_{n,β} > 0 such that

lim_{t→+∞} t^{n/4}(log t)^{−1} u(x, t) = D̃_{n,n} if β = n
lim_{t→+∞} t^{n/4} u(x, t) = D̃_{n,β}  if β > n , (4)

uniformly with respect to x ∈ K.

The constants C̃_{n,β} and D̃_{n,β} in Theorem 1 do not depend on K. What does depend on K is the “speed of convergence”, namely how fast t^{β/4}u(x, t) − C̃_{n,β} converges to 0 (and similarly for D̃_{n,β}). Let us also mention that when β ≥ n, then for any g ∈ C_{β} (not necessarily constant) one still has that lim_{t→+∞} t^{β/4}u(x, t) = +∞ uniformly with respect to x ∈ K.

The quantitative positivity result of Theorem 1 provides strong enough information to be applied also to the semilinear problem (2). At a first glance, this appears somehow unexpected, since the techniques connected with the proof of Theorem 1 seem to be purely linear.

Theorem 1 does not clarify whether the eventual positivity for solutions of (1) is global or only local. It is shown in [10] that if the initial datum u_{0} is compactly supported, then the negativity for the solution of (1) always exists and shifts to infinity. This suggests that a similar phenomenon might occur for β sufficiently large. On the other hand, if u_{0} ≡ 1 then the solution of (1) is u (x, t) ≡ 1. This trivial example shows that if β = 0, presumably one has global eventual positivity for (1). We show that – at least in the case n = 1 – this is also true if β is positive but sufficiently small, see Proposition 4 in the appendix.

In any case, the following result shows that in general, we cannot expect neither global positivity nor uniform bounds for eventual positivity.

Theorem 2. Let β ∈ (0, n) . For any T > 1 there exists g ∈ C_{β} such that if

u_{0}(x) = \frac{1}{g(x) + |x|^{β}}

then, the corresponding solution u = u (x, t) of (1) satisfies u (x_{T} , T) < 0 for some x_{T} ∈ ℝ^{n}.

1.2 Global existence, decay and positivity in the nonlinear problem

We denote by b the biharmonic heat kernel, i.e. the fundamental solution of (3). Then, we introduce the definition of solution of (2):

Definition 1. We say that u is a solution of (2) over [0, T) if u ∈ C^{4,1} (ℝ^{n} × (0, T)) , u is bounded in ℝ^{n} × [0, t] for any t ∈ (0, T), u solves the equation in (2) in the classical sense in ℝ^{n} × (0, T) and ∥u (t) − b(t) ∗ u_{0}∥_{L^{∞}(ℝ^{n})} → 0 as t → 0. We denote by T^{∗} the supremum of all T ’s such that u is a solution of (2) in [0, T). Finally, we say that u is a global solution of (2) if T^{∗} = +∞.

For the nonlinear Cauchy problem (2) with initial data u_{0} as specified in Theorem 1, we shall construct global solutions, where on the long term, the contribution of the initial datum dominates the nonlinear term, provided β ∈ (4/(p − 1), n). This observation allows us to prove an eventual local positivity property even for solutions of the nonlinear problem (2):

Theorem 3. Let p > 1 + 4/n, 4/(p − 1) < β < n and g ∈ C_{β}. Then, for a > 0 small enough the initial datum

u_{0} (x) = \frac{a}{g(x) + |x|^{β}}

admits a global solution, which is eventually locally positive. More precisely, for any compact set K ⊂ ℝ^{n} there exists T_{K} > 0 such that u (x, t) > 0 for any x ∈ K and t ≥ T_{K}.

In order to show this eventual local positivity result, we prove first an existence and decay result for solutions of (2) covering the presumably full ranges for the exponent p and for the asymptotic decay of the initial datum, where global existence may be expected. The limiting case β = 4/(p − 1) was treated in [9]. The precise estimates in the range β ∈ (\frac{4}{p−1}, n), however, are crucial for proving positivity in Theorem 3.

Theorem 4. Let p > 1 + 4/n. There exists α > 0 such that if u_{0} satisfies

|u_{0} (x)| ≤ \frac{α}{1 + |x|^{β}}

for some β ≥ 4/(p − 1), then there exists a global solution of (2). Moreover, there exists a constant A = A (n, p, β) > 0 such that for any (x, t) ∈ ℝ_{+}^{n+1} we have

|u(x, t)| ≤ \{ \begin{matrix}\frac{A}{1+|x|^{β}+t^{β/4}} & if β ∈ [\frac{4}{p−1}, n) \\ \frac{A}{1+ \begin{matrix}|x| & ^{n} \\ log & (1+|x|) \end{matrix} + \frac{t^{n/4}}{log(1+t)}} & if β = n \\ \frac{A}{1+|x|^{n}+t^{n/4}} & if β > n. \end{matrix}  (5)

For any ε > 0 the constant A > 0 may be chosen independently of β ∈ [4/(p − 1), n − ε] ∪ [n + ε, +∞).

Remark 1. The shape of u_{0} in the statements of Theorems 1 and 3 can be slightly modified. By using the results in [10] one can see that the proof still works if we add to any such u_{0} some nonnegative continuous compactly supported function.

2 Basic properties of the heat kernels

The kernel of the linear operator v ↦ v_{t} + Δ^{2}v in ℝ^{n} is given by

\begin{matrix}b(x, t) = α_{n} \frac{f_{n}(η)}{t^{n/4}} ,  η = \frac{|x|}{t^{1/4}} , \\ f_{n}(η) = η^{1−n} ∫_{0}^{∞} e^{−s^{4}}(ηs)^{n/2}J_{(n−2)/2}(ηs) ds , \end{matrix}  (6)

where J_{ν} denotes the ν-th Bessel function and α_{n} > 0 is a normalization constant. More precisely, if ω_{n} denotes the surface measure of the n-dimensional unit ball (so that ω_{1} = 2), then α_{n} is given by

α_{n}^{−1} = ω_{n} ∫_{0}^{∞} r^{n−1}f_{n}(r) dr.

Thanks to Galaktionov-Pohožaev [7], we know that these 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.  (7)

Let us now recall the definition of the Gamma function and the power series expansion of the Bessel function J_{ν}:

Γ(y) = ∫_{0}^{∞} e^{−s} s^{y−1} ds (y > 0) ,  J_{ν}(y) = ∑_{k=0}^{∞} \frac{(−1)^{k}(y/2)^{2k+ν}}{k! Γ(k + ν + 1)} (ν > −1).  (8)

For these definitions and for further properties of Γ and J_{ν} we refer to [1]. Here, we just recall the last formula on p.13 in [1]:

Γ(x + ℓ) = (x + ℓ − 1)(x + ℓ − 2) ⋯ xΓ(x)  for all x > 0 , ℓ ∈ ℕ .  (9)

Moreover, we will need the following property, obtained through the change of variable z = s^{4}:

∫_{0}^{∞} e^{−s^{4}} s^{α} ds = \frac{1}{4} ∫_{0}^{∞} e^{−z} z^{−1+(α+1)/4} dz = \frac{1}{4} Γ (\frac{α + 1}{4})  (α > −1).  (10)

The f-functions obey the following recurrence formula:

f′_{n}(η) = −η f_{n+2}(η) ,  for all n ≥ 1.  (11)

This follows by direct computation:

\frac{d}{dη} f_{n}(η) = \frac{d}{dη} [∫_{0}^{∞} e^{−s^{4}}s^{n−1}(ηs)^{(2−n)/2}J_{(n−2)/2}(ηs) ds]
{by [1, (4.6.2)]} = − ∫_{0}^{∞} e^{−s^{4}}s^{n}(ηs)^{(2−n)/2}J_{n/2}(ηs) ds = −ηf_{n+2}(η).

As a consequence of (11) we have the following recursion formula, which relates integrals of f-functions in annuli of the corresponding space:

∫_{r_{1}}^{r_{2}} r^{n−1}f_{n}(r) dr = [\frac{r^{n}}{n} f_{n}(r)]_{r_{1}}^{r_{2}} + \frac{1}{n} ∫_{r_{1}}^{r_{2}} r^{n+1}f_{n+2}(r) dr.

Next, we show that a representation of f_{n} through power series is available:

Theorem 5. For any integer m ≥ 1, we have

f_{2m}(η) = ∑_{k=0}^{∞} (−1)^{k} \frac{Γ (\frac{k+m}{2})}{2^{2k+m+1} k! (k + m − 1)!} η^{2k} .  (12)

For any nonnegative integer m, we have

f_{2m+1}(η) = \frac{2^{m}}{\sqrt{8π}} ∑_{k=0}^{∞} (−1)^{k} \frac{(k + m)! Γ (\frac{2k+2m+1}{4})}{k! (2k + 2m)!} η^{2k} .  (13)

In particular, f_{n}(0) > 0 for all n and

f_{1}(η) = \frac{1}{\sqrt{8π}} ∑_{k=0}^{∞} (−1)^{k} \frac{Γ (\frac{2k+1}{4})}{(2k)!} η^{2k} , f_{2}(η) = \frac{1}{4} ∑_{k=0}^{∞} (−1)^{k} \frac{Γ (\frac{k+1}{2})}{[2^{k} k!]^{2}} η^{2k} .

Proof. Throughout this proof we use the convention that for any real numbers a_{k} and any integer i: ∏_{k=i}^{i−1} a_{k} = 1. By exploiting the power series expansion (8), for any integer n we find

f_{n}(η) = ∑_{k=0}^{∞} \frac{(−1)^{k}}{k! Γ (k + \frac{n}{2}) 2^{2k−1+n/2}} (∫_{0}^{∞} e^{−s^{4}}s^{2k+n−1} ds) η^{2k} .

By (9) we obtain

Γ (k + \frac{n}{2}) = 2^{−k} Γ (\frac{n}{2}) ∏_{h=0}^{k−1}(n + 2h) for all k ∈ ℕ

which, combined with (10), yields

f_{n}(η) = ∑_{k=0}^{∞} (−1)^{k} \frac{Γ (\frac{2k+n}{4})}{k! Γ (\frac{n}{2}) 2^{k+1+n/2} ∏_{h=0}^{k−1}(n + 2h)} η^{2k} .  (14)

Assume now that n = 2m + 1 for some integer m ≥ 0. Then, recalling that Γ(1/2) = \sqrt{π} and using again (9), we obtain

Γ (\frac{n}{2}) = Γ (m + \frac{1}{2}) = 2^{−m} \sqrt{π} ∏_{h=0}^{m−1}(2h + 1).

Moreover, as may be proved by induction over (m + k), one has

2^{k} ∏_{h=0}^{m+k−1}(2h + 1) = \frac{(2k + 2m)!}{2^{m} (k + m)!} for any integer m ≥ 0 .

Using these two identities, the denominator of the coefficient of η^{2k} in (14) becomes

k! Γ (\frac{n}{2}) 2^{k+1+n/2} ∏_{h=0}^{k−1}(n + 2h) = \sqrt{8π} k! 2^{k} ∏_{h=0}^{m+k−1}(2h + 1) = \sqrt{8π} \frac{k! (2k + 2m)!}{2^{m} (k + m)!} .

This proves (13). On the other hand, if n = 2m for some integer m ≥ 1, then

Γ (\frac{n}{2}) = Γ(m) = (m − 1)!  and  ∏_{h=0}^{k−1}(n + 2h) = 2^{k} ∏_{h=0}^{k−1}(m + h)

so that

Γ (\frac{n}{2}) ∏_{h=0}^{k−1}(n + 2h) = 2^{k} (k + m − 1)!

and (12) follows by replacing into (14).

Basing upon Bessel’s equation, we deduce a third order ordinary differential equation for the function f_{n}, which will be essential for proving their positivity/oscillatory properties.

Theorem 6. For any integer n ≥ 1, the function f_{n} solves the equation

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.  (15)

Proof. In the proof of this lemma we use the following notation: keeping n ∈ ℕ as space dimension fixed, for any function g ∈ C^{2} ((0, ∞)) we write

Δg (ρ) = g′′ (ρ) + \frac{n − 1}{ρ}g′ (ρ)  for all ρ > 0.  (16)

One may check by direct computation that the Bessel function J_{(n−2)/2} satisfies the identity

Δ (ρ^{1−n/2}J_{(n−2)/2}) = −ρ^{1−n/2}J_{(n−2)/2}.  (17)

If we put

F (ρ) = ρ^{1−n/2}J_{(n−2)/2} (ρ) for all ρ > 0

then, by (6) and (17) we find

Δf_{n} (η) = Δ (∫_{0}^{∞} s^{n−1}e^{−s^{4}}F (ηs) ds) = ∫_{0}^{∞} s^{n−1}e^{−s^{4}}s^{2} (ΔF) (ηs) ds
= − ∫_{0}^{∞} s^{n−1}e^{−s^{4}}s^{2}F (ηs) ds.

Using the divergence formula and (17) we obtain

(Δf_{n} (η) )′ = − ∫_{0}^{∞} s^{n−1}e^{−s^{4}}s^{3}F′ (ηs) ds = \frac{1}{4ω_{n}} ∫_{ℝ^{n}} ∇ (e^{−|y|^{4}}) ∇F (η |y|) dy
= −\frac{1}{4ω_{n}} ∫_{ℝ^{n}} e^{−|y|^{4}}ηΔF (η |y|) dy = \frac{1}{4ω_{n}} ∫_{ℝ^{n}} e^{−|y|^{4}}ηF (η |y|) dy
= \frac{η}{4} ∫_{0}^{∞} s^{n−1}e^{−s^{4}}F (ηs) ds = \frac{η}{4}f_{n} (η) .

By (16) we immediately obtain (15).

Figure 1: The function f_{1}

Finally, we show that the f_{n} and so, the biharmonic heat kernel, has infinitely many oscillations:

Theorem 7. As η → ∞, the function η ↦ f_{n}(η) changes sign infinitely many times.

Proof. Assume first that n = 1 so that (15) reads

f′′′_{1} (η) = \frac{η}{4}f_{1} (η) .  (18)

For contradiction assume that f_{1}(η) ≥ 0 for η sufficiently large, say η ≥ \overline{η}. Then, (18) states that also f′′′_{1}(η) ≥ 0 for η ≥ \overline{η}. This implies that f′′_{1}(η) admits a limit as η → ∞. In view of (7) this limit is necessarily zero. And since f′′_{1} is non-decreasing, this means that f′′_{1}(η) ≤ 0 for η ≥ \overline{η}. Therefore, f_{1} is concave, nonnegative and vanishes at infinity in view of (7): this is impossible. Similarly, we reach a contradiction if we assume that f_{1}(η) ≤ 0 for η sufficiently large. Assume now that n = 2. By putting g(t) = f_{2}(e^{t}), (15) becomes

\frac{d}{dt}(e^{−2t}g′′(t)) = \frac{e^{2t}}{4} g(t) .  (19)

For contradiction, assume that g(t) ≥ 0 for t sufficiently large, say t ≥ T. Then, with the same argument used above we exclude that g′′(t) ≤ 0 for all t ≥ T. Take τ > T such that g′′(τ) > 0 and integrate (19) over [τ, t] for any t ≥ τ:

e^{−2t}g′′(t) ≥ e^{−2τ} g′′(τ) ⟹ g′′(t) ≥ e^{−2τ} g′′(τ)e^{2t}

showing that g′′(t) → +∞ as t → ∞. But then also g(t) → +∞ as t → ∞, contradicting (7). Similarly one reaches a contradiction by assuming that g(t) ≤ 0 for t sufficiently large. The statement is so proved for n = 1 and n = 2. This means that f_{1} and f_{2} have an increasing sequence of local maxima and of local minima. Hence, for larger values of n it suffices to recall (11) and to argue by induction.

Remark 2. In order to gain a more precise impression of the asymptotic behaviour of the f-function for η → ∞, we define

g(y) := f_{n} (y^{3/4}) ,  f_{n}(η) = g (η^{4/3}) ,  η = y^{3/4} ≥ 0.

So, we have

f′_{n}(η) = \frac{4}{3}η^{1/3}g′ (η^{4/3})
f′′_{n}(η) = \frac{16}{9}η^{2/3}g′′ (η^{4/3}) + \frac{4}{9}η^{−2/3}g′ (η^{4/3})
f′′′_{n}(η) = \frac{64}{27}ηg′′′ (η^{4/3}) + \frac{16}{9}η^{−1/3}g′′ (η^{4/3}) − \frac{8}{27}η^{−5/3}g′ (η^{4/3}) .

In terms of g, the differential equation (15) for f_{n} reads:

g(y) = \frac{256}{27}g′′′(y) + \frac{64}{9}n\frac{g′′(y)}{y} − \frac{32}{27}(3n − 2)\frac{g′(y)}{y^{2}}.  (20)

According to [13, Chapter X], the asymptotic behaviour of solutions to the “almost autonomous” equation (20) is to a “certain extent” determined by the corresponding autonomous equation

h(y) = \frac{256}{27}h′′′(y).  (21)

Bounded solutions to (21) are given by

h(y) = γ exp (−\frac{3}{16} \sqrt[3]{2}y) cos (\frac{3}{16}\sqrt{3}\sqrt[3]{2}y + δ)

where γ > 0 and δ ∈ ℝ. So, we expect the following asymptotic behaviour of f_{n}(η):

(γ + o(1)) exp (−\frac{3}{16} \sqrt[3]{2}η^{4/3}) cos (\frac{3}{16}\sqrt{3}\sqrt[3]{2}η^{4/3} + δ + o(1))

for suitable constants γ and δ. Numerical experiments strongly support validity of such an asymptotic expansion while on the other hand, [13, Chapter X.13–X.17] does not directly yield such a statement.

3 Linear decay and proof of Theorem 1

The solution u of the linear Cauchy problem (1) is 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}.  (22)

With the change of variables y = t^{1/4}z, we obtain

u(x, t) = α_{n} ∫_{ℝ^{n}} u_{0}(x − t^{1/4}z)f_{n}(|z|) dz = α_{n}t^{−β/4} ∫_{ℝ^{n}} \frac{f_{n}(|z|)}{t^{−β/4}g(x − t^{1/4}z) + |z − t^{−1/4}x|^{β}} dz . (23)

We distinguish two cases.

3.1 The case β < n

We first prove Lemma 1. Let K ⊂ ℝ^{n} be a compact subset and let 0 ≤ β < n. Then, there exists C_{n,β} ∈ ℝ such that

lim_{t→+∞} t^{β/4}u(x, t) = α_{n}ω_{n}C_{n,β} ,

the limit being uniform with respect to x ∈ K. In particular, C_{n,0} = (ω_{n}α_{n})^{−1}. Proof. By (7) and since β < n and g ∈ C_{β}, we may apply Lebesgue’s Theorem and exploit local uniform convergence in ℝ^{n} \ {0} to obtain

lim_{t→+∞} (∫_{ℝ^{n}} \frac{f_{n}(|z|)}{t^{−β/4}g(x − t^{1/4}z) + |z − t^{−1/4}x|^{β}} dz) = ∫_{ℝ^{n}} \frac{f_{n}(|z|)}{|z|^{β}} dz = ω_{n} ∫_{0}^{∞} η^{n−1−β}f_{n}(η) dη.

Hence, if we put

C_{n,β} := ∫_{0}^{∞} η^{n−1−β}f_{n}(η) dη ,  (24)

the statement follows by replacing into (23). By Lemma 1, the proof of Theorem 1 (in the case β < n) will be complete once we show: Proposition 1. For all integer n ≥ 1 and all β ∈ [0, n) we have C_{n,β} > 0. Since the proof of Proposition 1 is quite involved we postpone it to the next section.

3.2 The case β ≥ n

Here we cannot apply Lebesgue’s Theorem as in Lemma 1 because we must isolate the singularity. More precisely, if γ > 0 denotes the least zero of f_{n}, we split the integral in (23) as follows:

\frac{t^{β/4}u(x, t)}{α_{n}} = ∫_{ℝ^{n}} \frac{f_{n}(|z|)}{t^{−β/4} + |z − t^{−1/4}x|^{β}} dz = ∫_{|z−t^{−1/4}x|>γ} + ∫_{|z−t^{−1/4}x|<γ} =: I_{1}(t) + I_{2}(t)

and we estimate the two integrals. In view of (7), to I_{1} we may apply Lebesgue’s Theorem and obtain

lim_{t→+∞} I_{1}(t) = ∫_{|z|>γ} \frac{f_{n}(|z|)}{|z|^{β}} dz = δ_{n,β} ∈ ℝ .  (25)

The estimate of I_{2} appears slightly more delicate. With the change of variables y = t^{−1/4}x − z, we obtain

I_{2}(t) = ∫_{|y|<γ} \frac{f_{n}(|t^{−1/4}x − y|)}{t^{−β/4} + |y|^{β}} dy = ∫_{|y|<γ} \frac{f_{n}(|y|) + o(1)}{t^{−β/4} + |y|^{β}} dy = ω_{n} ∫_{0}^{γ} r^{n−1} \frac{f_{n}(r)}{t^{−β/4} + r^{β}} dr + o(I_{2}(t))

so that, with the change of variables r = t^{−1/4}s, we obtain

(1 + o(1))I_{2}(t) = ω_{n} t^{(β−n)/4} ∫_{0}^{γt^{1/4}} s^{n−1} \frac{f_{n}(t^{−1/4}s)}{1 + s^{β}} ds .

If β = n the last equality implies I_{2}(t) ∼ D_{n,n} log t as t → ∞, whereas if β > n it implies I_{2}(t) ∼ D_{n,β}t^{(β−n)/4} as t → ∞, for suitable positive constants D_{n,β}. By combining these estimates with (25), we readily obtain (4).

4 Proof of Proposition 1

The proof of Proposition 1 is somehow inductive. It needs a statement for large β in order to be started and two recursion formulae in order to be continued.

Lemma 2. For all integer n ≥ 4 and all β ∈ (\frac{n+1}{2}, n) we have C_{n,β} > 0.

Proof. According to (6), we may rewrite C_{n,β} in (24) as

C_{n,β} = ∫_{0}^{∞} η^{−β} ∫_{0}^{∞} e^{−s^{4}}(ηs)^{n/2}J_{(n−2)/2}(ηs) ds dη .  (26)

Absolute integrability with respect to η near ∞ is ensured by the condition β > (n+1)/2. Observe that \sqrt{ρ}J_{(n−2)/2}(ρ) is bounded at ∞ and that ρ^{1−n/2}J_{(n−2)/2}(ρ) is bounded near ρ = 0. For every η > 0, with the change of variable s = z/η we obtain

∫_{0}^{∞} e^{−s^{4}}(ηs)^{n/2}J_{(n−2)/2}(ηs) ds = \frac{1}{η} ∫_{0}^{∞} e^{−z^{4}/η^{4}}z^{n/2}J_{(n−2)/2}(z) dz

which, inserted into (26), yields

C_{n,β} = ∫_{0}^{∞} η^{−β−1} ∫_{0}^{∞} e^{−z^{4}/η^{4}}z^{n/2}J_{(n−2)/2}(z) dz dη .

By Fubini’s Theorem we then get

C_{n,β} = ∫_{0}^{∞} z^{n/2}J_{(n−2)/2}(z) ∫_{0}^{∞} \frac{e^{−z^{4}/η^{4}}}{η^{β+1}} dη dz .  (27)

For every z > 0, with the change of variable η = z/s we obtain

∫_{0}^{∞} \frac{e^{−z^{4}/η^{4}}}{η^{β+1}} dη = \frac{1}{z^{β}} ∫_{0}^{∞} e^{−s^{4}}s^{β−1} ds .

Inserting this into (27) shows that

C_{n,β} = γ_{n,β} ∫_{0}^{∞} z^{\frac{n−1}{2} −β} \sqrt{z} J_{(n−2)/2}(z) dz  (28)

with

γ_{n,β} = ∫_{0}^{∞} e^{−s^{4}}s^{β−1} ds = \frac{1}{4}Γ (\frac{β}{4} ) > 0

where the second equality follows from (10). Then, we argue as in [10, Lemma 1]: since the map z ↦ z^{\frac{n−1}{2} −β} is decreasing, we may apply the Lorch-Szegö Theorem [1, Corollary 4.15.2] to (28) in order to show that C_{n,β} > 0. Here we use n ≥ 4 so that for the index of the Bessel function, we have that \frac{n−2}{2} > \frac{1}{2}.

Lemma 3. For any n ≥ 1 and β ∈ [0, n) we have

C_{n+2,β} = (n − β)C_{n,β}.  (29)

For any n ≥ 3 and β ∈ [0, n − 2) we have

C_{n,β} = 4(β + 2)(n − 2 − β)C_{n+2,β+4}.  (30)

Proof. By (11) and integration by parts (recall (7)) we have

C_{n+2,β} = ∫_{0}^{∞} η^{n+1−β}f_{n+2} (η) dη = − ∫_{0}^{∞} η^{n−β}f′_{n} (η) dη
= (n − β) ∫_{0}^{∞} η^{n−β−1}f_{n} (η) dη = (n − β)C_{n,β}.

This proves (29). By exploiting the differential equation for f_{n}, see Theorem 6, we have

C_{n,β} = ∫_{0}^{∞} η^{n−1−β}f_{n} (η) dη
= 4 ∫_{0}^{∞} η^{n−2−β}f′′′_{n} (η) dη  (31)
+4(n − 1) ∫_{0}^{∞} η^{n−3−β}f′′_{n} (η) dη − 4(n − 1) ∫_{0}^{∞} η^{n−4−β}f′_{n} (η) dη.

Note that the last integral in (31) is well defined since β < n − 2 and by Theorem 5 we have f′_{n} (η) ∼ cη as η → 0.

By (11) we have

∫_{0}^{∞} η^{n−4−β}f′_{n} (η) dη = − ∫_{0}^{∞} η^{n−3−β}f_{n+2} (η) dη = −C_{n+2,β+4}.  (32)

Hence, after an integration by parts and using again (7), (11) and the fact that f′_{n} (η) ∼ cη as η → 0, we get

∫_{0}^{∞} η^{n−3−β}f′′_{n} (η) dη = [η^{n−3−β}f′_{n} (η)]_{0}^{∞} − (n − 3 − β) ∫_{0}^{∞} η^{n−4−β}f′_{n} (η) dη
= (n − 3 − β) C_{n+2,β+4}.  (33)

With the aid of (33) and integration by parts we obtain

∫_{0}^{∞} η^{n−2−β}f′′′_{n} (η) dη = [η^{n−2−β}f′′_{n} (η)]_{0}^{∞} − (n − 2 − β) ∫_{0}^{∞} η^{n−3−β}f′′_{n} (η) dη
= −(n − 2 − β) (n − 3 − β) C_{n+2,β+4}.  (34)

Combining (31)-(34) the proof of (30) follows.

We are now ready to prove Proposition 1. We prove it separately in the cases where n is odd or even.

Proof of Proposition 1 for odd integers. By Lemma 2 we have

C_{9,β} > 0  for all β ∈ (5, 9) .  (35)

In turn, by (35) and (30) we have C_{7,β} > 0 for any β ∈ (1, 5) while by Lemma 2 it follows that C_{7,β} > 0 for any β ∈ (4, 7). This yields

C_{7,β} > 0  for all β ∈ (1, 7) .  (36)

Then, according to (36) and (30), we obtain C_{5,β} > 0 for any β ∈ [0, 3) and according to (36) and (29) we obtain C_{5,β} > 0 for any β ∈ (1, 5). This yields

C_{5,β} > 0  for all β ∈ [0, 5) .  (37)

By (37) and (29) we also have for n = 1, 3

C_{n,β} > 0 for all β ∈ [0, n) .

It remains to consider odd dimensions n ≥ 5. It is already proved for n = 5 in view of (37). Suppose by induction that the conjecture is true for some odd integer n ≥ 5. By (29) we have C_{n+2,β} > 0 for any β ∈ [0, n) and by Lemma 2 we deduce that C_{n+2,β} > 0 for any β ∈ (\frac{n+3}{2}, n + 2). Since n > \frac{n+3}{2} for any n ≥ 4, this proves that

C_{n+2,β} > 0 for all β ∈ [0, n + 2) .

The proof of Proposition 1 for odd integers is now complete. □

Proof of Proposition 1 for even integers. By Lemma 2 we have

C_{8,β} > 0  for all β ∈ (9/2, 8) .  (38)

By (38) and (30) we have C_{6,β} > 0 for any β ∈ (1/2, 4) while by Lemma 2 it follows that C_{6,β} > 0 for any β ∈ (7/2, 6). This yields

C_{6,β} > 0  for all β ∈ (1/2, 6) .  (39)

According to (39) and (30) we obtain C_{4,β} > 0 for any β ∈ [0, 2) while according to (39) and (29) we obtain C_{4,β} > 0 for any β ∈ (1/2, 4). This yields

C_{4,β} > 0  for all β ∈ [0, 4) .  (40)

By (40) and (29) we also have

C_{2,β} > 0 for all β ∈ [0, 2) .

It remains to consider even dimensions n ≥ 4. This proof by induction is performed exactly as in the case of odd n. The proof of Proposition 1 is so complete also for even integers. □

5 Proof of Theorem 2

Let β ∈ (0, n) and T > 1 be fixed. According to Theorem 7 we know that f_{n} is changes sign infinitely many times. Let γ_{1} < γ_{2} be the first two zeroes of f_{n} so that we have

f_{n} (η) > 0 for all η ∈ (0, γ_{1}) and f_{n} (η) < 0 for all η ∈ (γ_{1}, γ_{2}) .

Let d > 0 and 0 < δ < (γ_{2} − γ_{1}) /2 be such that

f_{n} (η) < −d  for all η ∈ (γ_{1} + δ, γ_{2} − δ) .  (41)

Put b = (γ_{2} − γ_{1}) /2 and a = b − δ and define for a fixed M > 1

g (x) = \{ \begin{matrix}1 & if |x| ≤ a \\ 1 + \frac{M−1}{b−a} (|x| − a) & if a < |x| < b \\ M & if |x| ≥ b. \end{matrix}

If

u_{0}(x) = \frac{1}{g(x) + |x|^{β}}

then by (22) the corresponding solution u = u(x, t) of (1) reads

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

By (7) and (41) we have

\frac{u (x, t)}{α_{n}} < ∫_{\{γ_{1}+δ<|z|<γ_{2}−δ\}} \frac{−d}{g (|x − t^{1/4}z|) + |x − t^{1/4}z|^{β}} dz  (42)

+K ∫_{\{|z|<γ_{1}\}∪\{|z|>γ_{2}\}} \frac{e^{−μ|z|^{4/3}}}{g (|x − t^{1/4}z|) + |x − t^{1/4}z|^{β}} dz.

Take x_{0} ∈ ℝ^{n} such that |x_{0}| = (γ_{1} + γ_{2}) /2 and put R = T^{−1/4}a and x_{T} = T^{1/4}x_{0}. Then by (42) we have

\frac{u (x_{T} , T)}{α_{n}} < ∫_{B_{R}(x_{0})} \frac{−d}{g (T^{1/4} |x_{0} − z|) + T^{β/4} |x_{0} − z|^{β}} dz

+K ∫_{\{|z|<γ_{1}\}} \frac{1}{g (|x − t^{1/4}z|)} dz + K ∫_{\{|z|>γ_{2}\}} \frac{e^{−μ|z|^{4/3}}}{g (|x − t^{1/4}z|)} dz.  (43)

Note that if z ∈ B_{R} (x_{0}), then

T^{1/4} |x_{0} − z| < RT^{1/4} = a and g (T^{1/4} |x_{0} − z|) = 1.  (44)

On the other hand if |z| < γ_{1}, then

|x_{0} − z| > |x_{0}| − |z| > \frac{γ_{1} + γ_{2}}{2} − γ_{1} = b.

Since T > 1 we also have T^{1/4} |x_{0} − z| > b and

g (T^{1/4} |x_{0} − z|) = M.  (45)

Similarly for |z| > γ_{2} we have

|x_{0} − z| > |z| − |x_{0}| > γ_{2} − \frac{γ_{1} + γ_{2}}{2} = b

and

g (T^{1/4} |x_{0} − z|) = M.  (46)

Inserting (44)-(46) into (43) we obtain

\frac{u (x_{T} , T)}{α_{n}} < −\frac{ω_{n}R^{n}d}{n (1 + a^{β})} + \frac{Kω_{n}γ_{1}^{n}}{nM} + \frac{K}{M} ∫_{\{|z|>γ_{2}\}} e^{−μ|z|^{4/3}} dz.

It is clear that if we choose M sufficiently large then u (x_{T} , T) < 0.

6 Proof of Theorems 3 and 4

Let u_{0} be a function such that

|u_{0} (x)| ≤ \frac{α}{1 + |x|^{β}}  (47)

for some α > 0. Let u be a solution of the integral equation

u(t) = b (t) ∗ u_{0} + ∫_{0}^{t} b (t − s) ∗ |u|^{p−1} u (s) ds.  (48)

By a contraction mapping argument (see [14] Section 3.3), we know that (48) admits a unique solution u(t) defined in the maximal interval of existence [0, T^{∗}) with 0 < T^{∗} ≤ +∞ and such that u ∈ C^{0} ∩ L^{∞} (ℝ^{n} × [0, T]) for any T ∈ (0, T^{∗}). Moreover, it is well known (see Proposition A4 in [6]) that any bounded solution of the integral equation (48) is a solution of (2). In order to overcome the difficulties which arise from the oscillatory behaviour of the kernel b (see also Theorem 7 above), Galaktionov and Pohožaev [7] introduced the following majorizing kernel b̃(x, t):

b̃(x, t) = θ_{n}t^{−n/4} exp (−μη^{4/3})

with μ as in (7) and

θ_{n}^{−1} = ω_{n} ∫_{0}^{∞} r^{n−1} exp (−μr^{4/3}) dr.

From this definition we have

|b (x, t)| ≤ D_{n}b̃(x, t)  (49)

for a suitable constant D_{n} > 0. Let v_{0} be defined by

v_{0} (x) = D_{n} |u_{0} (x)| .  (50)

Besides (48), we also consider the integral equation

v(t) = b̃(t) ∗ v_{0} + D_{n} ∫_{0}^{t} b̃(t − s) ∗ v^{p} (s) ds.  (51)

If v is a solution of (51), by (49) and (50) we infer that

|u(x, t)| ≤ v (x, t)  (52)

as long as v(t) exists. In particular, the maximal interval of existence for (48) contains the corresponding interval for (51). We define the operator

B̃v(x, t) = b̃(t) ∗ v_{0} + D_{n} ∫_{0}^{t} b̃(t − s) ∗ |v (s) |^{p−1}v (s) ds  (53)

over the space C^{0} ∩ L^{∞} (ℝ^{n} × [0, T]) =: C_{b}^{0} (ℝ^{n} × [0, T]) for any T > 0. Since the initial datum v_{0} = v_{0}(x) is nonnegative, any fixed point v = v(x, t) of the operator B̃ is a nonnegative function so that v also satisfies (51). As for the “linear” contribution of the initial datum to B̃ in (53), we have

0 ≤ b̃(t) ∗ v_{0} = θ_{n} ∫_{ℝ^{n}} t^{−n/4}e^{−μ( \frac{|x−y|^{4}}{t} )^{1/3}}D_{n} |u_{0}(y)| dy

≤ D_{n}θ_{n}α ∫_{ℝ^{n}} t^{−n/4}e^{−μ( \frac{|y|^{4}}{t} )^{1/3}} \frac{1}{1 + |x − y|^{β}} dy =: D_{n}θ_{n}αG_{1} (x, t) .  (54)

Here, any β ≥ 0 makes sense. Next, we estimate the second term in the right hand side of (53). Let M > 0 and let v ∈ C^{0} (ℝ_{+}^{n+1}) be such that

0 ≤ v (x, t) ≤ \frac{M}{1 + |x|^{β} + t^{β/4}} for all (x, t) ∈ ℝ_{+}^{n+1}.

Then, we have

0 ≤ ∫_{0}^{t} b̃(t − s) ∗ v^{p} (s) ds = θ_{n} ∫_{0}^{t} ∫_{ℝ^{n}} s^{−n/4}e^{−μ( \frac{|y|^{4}}{s} )^{1/3}}v^{p}(x − y, t − s) dyds

≤ θ_{n}M^{p} ∫_{0}^{t} ∫_{ℝ^{n}} s^{−n/4}e^{−μ( \frac{|y|^{4}}{s} )^{1/3}} \frac{1}{[1 + (t − s)^{β/4} + |x − y|^{β}]^{p}} dyds =: θ_{n}M^{p}G_{2} (x, t) . (55)

In the following two propositions we formulate precise decay estimates for G_{1} (x, t) and G_{2} (x, t).

Proposition 2. Let β ≥ 0. Let G_{1} be the function introduced in (54) . There exists a constant C_{1} = C_{1} (n, β) > 0 such that for all (x, t) ∈ ℝ_{+}^{n} we have

G_{1} (x, t) ≤ \{ \begin{matrix}\frac{C_{1}}{1+|x|^{β}+t^{β/4}} & if β ∈ [0, n) \\ \frac{C_{1}}{1+ \begin{matrix}|x| & ^{n} \\ log & (1+|x|) \end{matrix} + \frac{t^{n/4}}{log(1+t)}} & if β = n \\ \frac{C_{1}}{1+|x|^{n}+t^{n/4}} & if β > n. \end{matrix}  (56)

Moreover, for any ε > 0 the constant C_{1} may be chosen independently of β ∈ [0, n − ε] ∪ [n + ε, +∞).

Proposition 3. Let p > 1 + 4/n and β ≥ \frac{4}{p−1}. Let G_{2} be the function introduced in (55) . There exists a constant C_{2} = C_{2} (n, p, β) > 0 such that for all (x, t) ∈ ℝ_{+}^{n} we have

G_{2} (x, t) ≤ \{ \begin{matrix}\frac{C_{2}}{1+|x|^{βp−4}+t^{(βp/4)−1}} & if β ∈ [\frac{4}{p−1}, \frac{n+4}{p}) \\ \frac{C_{2}}{1+ \frac{|x|^{n}}{log(1+|x|)} + \frac{t^{n/4}}{log(1+t)}} & if β = \frac{n+4}{p} \\ \frac{C_{2}}{1+|x|^{n}+t^{n/4}} & if β > \frac{n+4}{p}. \end{matrix}  (57)

Moreover, for any ε > 0 the constant C_{2} may be chosen independently of β ∈ [\frac{4}{p−1}, \frac{n+4}{p} − ε] ∪

[\frac{n+4}{p} + ε, +∞) .

The proofs of Propositions 2 and 3 are postponed to Sections 8 and 9, respectively.

6.1 Proof of Theorem 4 in the case β ∈ [\frac{4}{p−1}, n)

We fix β_{0} ∈ [4/(p − 1), n) arbitrary and remark that the following arguments are uniform in β ∈ [4/(p − 1), β_{0}]. We proceed as in [9]. Let M > 0 to be fixed below. For any T > 0 we introduce the set

S_{T} = \{v ∈ C^{0} (ℝ^{n} × [0, T]) : 0 ≤ v (x, t) ≤ \frac{M}{1 + |x|^{β} + t^{β/4}} \} .

It is clear that the set S_{T} is a nonempty, closed and convex subset of the Banach space C_{b}^{0} (ℝ^{n} × [0, T]) of continuous and bounded functions in ℝ^{n} × [0, T]. We show that, for any T > 0, S_{T} is an invariant set for B̃, i.e.

B̃S_{T} ⊂ S_{T}

for a suitable choice of M > 0 and α > 0 in (47). Note that for any β ∈ (\frac{4}{p−1}, n) the exponents of t which appear in (57) are strictly larger than β/4 which appears in (56), while they are equal for β = 4/(p − 1). Hence, we have

G_{2} (x, t) ≤ \frac{C̃_{2}}{1 + |x|^{β} + t^{β/4}}  (58)

with a suitable constant C̃_{2} = C̃_{2} (n, p, β) > 0. To see that C̃_{2} may be chosen independently of β ∈ [\frac{4}{p−1}, β_{0}] , we observe that in (55), β may be replaced by β̃ := \frac{β+4}{p} ≤ \frac{β_{0}+4}{p} < \frac{n+4}{p}. Applying

Proposition 3 to this β̃ yields (58) with C̃_{2} to be chosen uniformly for β̃ ∈ [\frac{4}{p−1}, \frac{β_{0}+4}{p}], i.e. independently

of β ∈ [\frac{4}{p−1}, β_{0}]. Therefore, inserting (56) and (58) into (54) and (55), we obtain

B̃v (x, t) ≤ \frac{D_{n}θ_{n}αC_{1} + θ_{n}D_{n}M^{p}C̃_{2}}{1 + |x|^{β} + t^{β/4}} = \frac{M}{1 + |x|^{β} + t^{β/4}}

for

M = (C̃_{2}θ_{n}D_{n}p)^{−1/(p−1)}

and

α = \frac{p − 1}{D_{n}^{p/(p−1)}θ_{n}^{p/(p−1)}C_{1}C̃_{2}^{1/(p−1)}p^{p/(p−1)}} .

The rest of the proof is based on an application of the Schauder fixed point Theorem as in the proof of Theorem 1 in [9]. With this procedure we find a function v ∈ C^{0} (ℝ_{+}^{n+1}) such that B̃v = v and

0 ≤ v (x, t) ≤ \frac{M}{1 + |x|^{β} + t^{β/4}} for all (x, t) ∈ ℝ_{+}^{n+1}.

By (52) we conclude that (48) admits a solution u ∈ C^{0}(ℝ_{+}^{n+1}) such that

0 ≤ |u(x, t)| ≤ v (x, t) for all (x, t) ∈ ℝ_{+}^{n+1}.

Finally by Propositions 2 and (58) we infer that the constants C_{1} and C̃_{2} may be chosen independently of β ∈ [\frac{4}{p−1}, β_{0}] so that also α and M may be chosen independently of β ∈ [\frac{4}{p−1}, β_{0}] . So, Theorem 4 is proved for β ∈ [4/(p − 1), n).

6.2 Proof of Theorem 4 in the case β ≥ n

If β > n, for suitable M we work within the set

S_{T} = \{v ∈ C^{0} (ℝ^{n} × [0, T]) : 0 ≤ v (x, t) ≤ \frac{M}{1 + |x|^{n} + t^{n/4}} \} ,

while for β = n we put

S_{T} = \{v ∈ C^{0} (ℝ^{n} × [0, T]) : 0 ≤ v (x, t) ≤ \frac{M}{1 + \begin{matrix}|x| & ^{n} \\ log & (1+|x|) \end{matrix} + \frac{t^{n/4}}{log(1+t)}} \} .

This yields the proof of Theorem 4 in the case β ≥ n exactly as in the case \frac{4}{p−1} ≤ β < n. □

6.3 Proof of Theorem 3

Let u (x, t) be the global solution of (2) with the initial condition u_{0} as in the statement of Theorem 3. Provided a is sufficiently small, this global solution exists in view of Theorem 4. Moreover, by (5),

(55) and Proposition 3 we infer that for any ε > 0 there exists t_{1} > 0 (independent of x) such that

|∫_{0}^{t} b (t − s) ∗ |u|^{p−1} u (s) ds| ≤ D_{n} ∫_{0}^{t} b̃(t − s) ∗ |u(s)|^{p} ds
≤ D_{n}θ_{n}A^{p}G_{2} (x, t) ≤ εt^{−β/4}  for all t > t_{1} and x ∈ ℝ^{n}. (59)

This fact occurs since for any 4/(p − 1) < β < n we have β/4 < min {(βp/4) − 1, n/4} . On the other hand, by Theorem 1 we deduce that for any compact set K ⊂ ℝ^{n} and for any ε > 0 there exists t_{2} > 0 (independent of x ∈ K) such that

b (t) ∗ u_{0} ≥ (aC̃_{n,β} − ε) t^{−β/4} for all t > t_{2} and x ∈ K.  (60)

Combining (59)-(60) we infer that

u (x, t) = b (t) ∗ u_{0} + ∫_{0}^{t} b (t − s) ∗ |u|^{p−1} u (s) ds ≥ (aC̃_{n,β} − 2ε) t^{−β/4} for all t > t_{3} and x ∈ K

with t_{3} = max {t_{1}, t_{2}} . Since C̃_{n,β} > 0 we may choose ε > 0 small enough such that aC̃_{n,β} − 2ε > 0. This proves the eventual local positivity of u. □

7 Some technical lemmas

We recall here the trivial inequalities

min \{\frac{1}{a}, \frac{1}{b}\} ≤ \frac{2}{a + b}  for all a, b > 0.  (61)

For any m ∈ ℕ and q > 0 there exist γ_{1}, γ_{2} > 0 such that

γ_{1} (∑_{i=1}^{m} α_{i})^{q} ≤ ∑_{i=1}^{m} α_{i}^{q} ≤ γ_{2} (∑_{i=1}^{m} α_{i})^{q}  for all α_{i} ≥ 0, i = 1, ..., m.  (62)

We provide some technical estimates which will be fundamental for our results.

Lemma 4. Let p > 1 + 4/n and β ∈ [\frac{4}{p−1}, \frac{n+4}{p}) . There exists a constant C = C (n, p, β) such that for n ≥ 2 we have

Γ_{1,n} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{0}^{3/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{σ^{3}}{[\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} ]^{\frac{βp}{4}}} dσ dρ dw ≤ C.

(63)

For n = 1 and any T > 0 we have

Γ_{1,1} := ∫_{0}^{3/T} z^{−4}e^{−μz^{4/3}} ∫_{0}^{T z} \frac{σ^{3}}{[\frac{T^{3}}{z} (T z − σ) + (σ − 1)^{4}]^{\frac{βp}{4}}} dσ dz ≤ C.  (64)

Moreover, for any β_{1} ∈ (\frac{4}{p−1}, \frac{n+4}{p}) the constant C may be chosen independently of β ∈ [\frac{4}{p−1}, β_{1}] .

Proof. The estimates (63) and (64) may be obtained with the same procedure introduced in the proof of Lemmas 1 to 4 in [9]. There, the integral was split as follows:

∫_{0}^{∞} ∫_{0}^{1/(2T)} ∫_{0}^{T ρ} + ∫_{0}^{∞} ∫_{1/(2T)}^{3/T} ∫_{0}^{T ρ/2} + ∫_{0}^{∞} ∫_{1/(2T)}^{1/T} ∫_{T ρ/2}^{T ρ} + ∫_{0}^{∞} ∫_{1/T}^{3/T} ∫_{T ρ/2}^{T ρ} .

Most delicate are the integrals

∫_{0}^{∞} ∫_{1/(2T)}^{1/T} ∫_{T ρ/2}^{T ρ} and ∫_{0}^{∞} ∫_{1/T}^{3/T} ∫_{T ρ/2}^{T ρ},

where the σ-integral is divided further by a ρ dependent polynomial of order 8. This enables us to find suitable estimates for terms which become strongly singular for certain values of ρ. The condition β < \frac{n+4}{p} is used in order to ensure finiteness of certain integrals which arise in the course of our estimates.

Lemma 5. Let p > 1 + 4/n and β ∈ [\frac{4}{p−1}, n) . There exists a constant C = C (n, p, β) such that:

– for n ≥ 2 and for any T > 0 we have

Γ_{2,n} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{3/T}^{∞} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{σ^{3}}{[\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} ]^{\frac{βp}{4}}} dσ dρ dw ≤ C;

(65)

– for n = 1 and for any T > 0 we have

Γ_{2,1} := ∫_{3/T}^{∞} z^{−4}e^{−μz^{4/3}} ∫_{0}^{T z} \frac{σ^{3}}{[\frac{T^{3}}{z} (T z − σ) + (σ − 1)^{4}]^{\frac{βp}{4}}} dσ dz ≤ C.  (66)

Moreover, for any β_{1} ∈ (\frac{4}{p−1}, n) the constant C may be chosen independently of β ∈ [\frac{4}{p−1}, β_{1}] .

Proof. Proceeding as in the proof of Lemma 5 in [9] we arrive at

Γ_{2,n} ≤ C + \frac{C}{T^{n−4+βp}} ∫_{3}^{∞} τ^{n−5+ \frac{βp}{4}} e^{−μ(\frac{τ}{T})^{4/3}}dτ =: C + Cg(T)  (67)

for any n ≥ 1. We need to prove that the function g = g(T) defined above is bounded for T ∈ (0, ∞) . Since g is continuous in (0, ∞) we have only to prove that g remains bounded as T → 0 and T → ∞. If T < 1 we have

g(T) = \frac{1}{T^{n−4+βp}} ∫_{3}^{∞} τ^{n−5+ \frac{βp}{4}} e^{− \frac{μ}{2}(\frac{τ}{T})^{4/3}}e^{− \frac{μ}{2}(\frac{τ}{T})^{4/3}} dτ

≤ \frac{e^{− \frac{μ}{2}(\frac{3}{T})^{4/3}}}{T^{n−4+βp}} ∫_{3}^{∞} τ^{n−5+ \frac{βp}{4}} e^{− \frac{μ}{2} τ^{4/3}}dτ = C \frac{e^{− \frac{μ}{2}(\frac{3}{T})^{4/3}}}{T^{n−4+βp}} .

The last term tends to 0 as T → 0. If T > 1 we split the integral in (67) as follows

g(T) = \frac{1}{T^{n−4+βp}} [∫_{3}^{3T^{α}} τ^{n−5+ \frac{βp}{4}} e^{−μ(\frac{τ}{T})^{4/3}}dτ + ∫_{3T^{α}}^{∞} τ^{n−5+ \frac{βp}{4}} e^{−μ(\frac{τ}{T})^{4/3}}dτ]

for some α > 1 which will be fixed below. Since

τ ≤ 3T^{α} ⟹ \frac{1}{T} ≤ \frac{C}{τ^{1/α}}

and

τ ≥ 3T^{α} ⟹ \frac{τ}{T} ≥ 3^{1/α}τ^{\frac{α−1}{α}} ,

we have

g(T) ≤ C ∫_{3}^{3T^{α}} \frac{τ^{n−5+ \frac{βp}{4}}}{τ^{\frac{n−4+βp}{α}}} dτ + \frac{C}{T^{n−4+βp}} ∫_{3}^{∞} τ^{n−5+ \frac{βp}{4}} e^{−μCτ ^{\frac{4(α−1)}{3α}}} dτ.

If we choose α > 1 sufficiently close to 1 then τ^{n−5+ \frac{βp}{4} − \frac{n−4+βp}{α}} ∈ L^{1} (3, ∞) so that g(T) remains bounded when T → ∞. This completes the proof of the lemma.

8 Proof of Proposition 2

We proceed along the lines of the proof of [9, Proposition 2]. We make the change of variables y = t^{1/4}z to obtain

G_{1} (x, t) = ∫_{ℝ^{n}} \frac{e^{−μ|z|^{4/3}}}{1 + |x − t^{1/4}z|^{β}} dz.

We start with the following

Lemma 6. There exists a constant K_{1} = K_{1}(n, β) > 0 such that for all |x| ≤ 1 and t ≥ 0 we have

G_{1} (x, t) ≤ \{ \begin{matrix}\frac{K_{1}}{1+|x|^{β}+t^{β/4}} & if β ∈ [0, n) \\ \frac{K_{1}}{1+|x|^{n}+ \frac{t^{n/4}}{log(1+t)}} & if β = n \\ \frac{K_{1}}{1+|x|^{n}+t^{n/4}} & if β > n. \end{matrix}  (68)

Moreover, for any ε > 0 the constant K_{1} may be chosen independently of β ∈ [0, n − ε] ∪ [n + ε, +∞) .

Proof. For |x| ≤ 1 and t ≥ 0 we immediately obtain

G_{1} (x, t) ≤ ∫_{ℝ^{n}} e^{−μ|z|^{4/3}}dz ≤ \frac{A_{1}}{1 + |x|^{β}} .  (69)

with a suitable constant A_{1} = A_{1}(n, β) > 0. If β ∈ [0, n) then following closely the proof of Lemma 6 in [9] we obtain

G_{1} (x, t) ≤ A_{2}t^{−β/4}  for all (x, t) ∈ ℝ_{+}^{n},  (70)

where A_{2} = A_{2}(n, β) > 0. If β = n then for any (x, t) ∈ ℝ_{+}^{n} we have

G_{1} (x, t) ≤ t^{−n/4} [∫_{|z−t^{−1/4}x|≤1} \frac{1}{t^{−n/4} + |z − t^{−1/4}x|^{n}} dz + ∫_{|z−t^{−1/4}x|>1} e^{−μ|z|^{4/3}}dz]

≤ t^{−n/4} [∫_{|w|≤1} \frac{1}{t^{−n/4} + |w|^{n}} dw + ∫_{ℝ^{n}} e^{−μ|z|^{4/3}}dz] ≤ A_{3}t^{−n/4} [1 + log (1 + t)] , (71)

where A_{3} = A_{3}(n) > 0. If β > n then for x ∈ ℝ^{n} and t > 1 , we have

G_{1} (x, t) ≤ t^{−β/4} [∫_{|z−t^{−1/4}x|≤1} \frac{1}{t^{−β/4} + |z − t^{−1/4}x|^{β}} dz + ∫_{|z−t^{−1/4}x|>1} e^{−μ|z|^{4/3}}dz]

≤ t^{−β/4} [∫_{|w|≤1} \frac{1}{t^{−β/4} + |w|^{β}} dw + ∫_{ℝ^{n}} e^{−μ|z|^{4/3}}dz] .

After the change of variables w = t^{−1/4}η we obtain

G_{1} (x, t) ≤ t^{−β/4} [t^{β/4−n/4} ∫_{ℝ^{n}} \frac{1}{1 + |η|^{β}} dη + ∫_{ℝ^{n}} e^{−μ|z|^{4/3}}dz] ≤ A_{4}t^{−n/4}

with A_{4} = A_{4} (n, β) > 0. On the other hand, for |x| ≤ 1 and t ∈ [0, 1] the first inequality in (69) holds true so that for any x ∈ ℝ^{n} and t > 0 we have

G_{1} (x, t) ≤ A_{5}t^{−n/4}.  (72)

for A_{5} = A_{5} (n, β) > 0. Combining (69) with (70) if β ∈ [0, n), with (71) if β = n and with (72) if β > n, by (61) we immediately obtain (68).

Assuming now |x| > 1 we prove:

Lemma 7. There exists a constant K_{2} = K_{2}(n, β) > 0 such that for all |x| > 1 and t ≥ 0 we have

G_{1} (x, t) ≤ \{ \begin{matrix}\frac{K_{2}}{1+|x|^{β}+t^{β/4}} & if β ∈ [0, n) \\ \frac{K_{2}}{1+ \begin{matrix}|x| & ^{n} \\ log & (1+|x|) \end{matrix} + \frac{t^{n/4}}{log(1+t)}} & if β = n \\ \frac{K_{2}}{1+|x|^{n}+t^{n/4}} & if β > n. \end{matrix}  (73)

Moreover, for any ε > 0 the constant K_{2} may be chosen independently of β ∈ [0, n − ε] ∪ [n + ε, +∞) .

Proof. For β ∈ [0, n) the proof of (73) is exactly the same as in the proof of Lemma 7 in [9]. In the rest of this proof we assume β ≥ n. We proceed as in the proof of [3, Lemma 2.2]. Given R > 1/2, let |x| > R such that

G_{1}(x, t) = ∫_{ℝ^{n}} t^{−n/4} \frac{e^{−μ( \frac{|x−y|^{4}}{t} )^{1/3}}}{1 + |y|^{β}} dy

= ∫_{|y|>R} t^{−n/4} \frac{e^{−μ( \frac{|x−y|^{4}}{t} )^{1/3}}}{1 + |y|^{β}} dy + ∫_{|y|≤R} t^{−n/4} \frac{e^{−μ( \frac{|x−y|^{4}}{t} )^{1/3}}}{1 + |y|^{β}} dy

≤ \frac{C(n)}{1 + R^{β}} + ∫_{|y|≤R} \frac{1}{|x − y|^{n}} (\frac{|x − y|^{4}}{t} )^{n/4} \frac{e^{−μ( \frac{|x−y|^{4}}{t} )^{1/3}}}{1 + |y|^{β}} dy.  (74)

Since we have

|x − y| ≥ |x| − |y| ≥ |x| − R > 0 for |y| ≤ R, |x| > R

and

z^{n/4}e^{−μz^{1/3}} ≤ C(n)e^{−(μ/2)z^{1/3}} for all z ≥ 0,

by (74) we obtain

G_{1}(x, t) ≤ \frac{C (n)}{1 + R^{n}} + \frac{C (n)}{(|x| − R)^{n}} ∫_{|y|≤R} \frac{e^{−(μ/2)( \frac{|x−y|^{4}}{t} )^{1/3}}}{1 + |y|^{β}} dy
≤ \frac{C (n)}{1 + R^{n}} + \frac{C (n)}{(|x| − R)^{n}} ∫_{|y|≤R} \frac{1}{1 + |y|^{β}} dy.

If β = n, we infer

G_{1} (x, t) ≤ \frac{C (n)}{1 + R^{n}} + \frac{C (n)}{(|x| − R)^{n}} log (1 + R) .

If β > n, we find that

G_{1} (x, t) ≤ \frac{C (n)}{1 + R^{n}} + \frac{C (n, β)}{(|x| − R)^{n}} .

Choosing R = |x| /2, then for any |x| > 1 and t ≥ 0 we obtain with a suitable constant C (n, β)

G_{1} (x, t) ≤ \{ \begin{matrix}\frac{C(n,β)}{1+|x|^{n}} log (1 + |x|) & if β = n \\ \frac{C(n,β)}{1+|x|^{n}} & if β > n. \end{matrix}

Combining this estimate with (71), (72) and (61), we obtain (73) also for β ≥ n. The proof of Proposition 2 follows combining the estimates of Lemmas 6 and 7.

9 Proof of Proposition 3

With the change of variables y = s^{1/4}z we obtain

G_{2} (x, t) = ∫_{0}^{t} ∫_{ℝ^{n}} \frac{e^{−μ|z|^{4/3}}}{[1 + (t − s)^{β/4} + |x − s^{1/4}z|^{β}]^{p}} dzds.  (75)

We start with the following time decay estimate. Lemma 8. Let p > 1+4/n. Then, there exists a constant B_{1} = B_{1} (n, p, β) > 0 such that for all x ∈ ℝ^{n} and t > 0 we have

G_{2} (x, t) ≤ \{ \begin{matrix}B_{1}t^{− \frac{βp}{4} +1} & if β ∈ [\frac{4}{p−1}, \frac{n+4}{p}) \\ B_{1}t^{−n/4} [1 + log (1 + t)] & if β = \frac{n+4}{p} \\ B_{1}t^{−n/4} & if β > \frac{n+4}{p}. \end{matrix}  (76)

Moreover, for any ε > 0 the constant B_{1} may be chosen independently of β ∈ [\frac{4}{p−1}, \frac{n+4}{p} − ε] ∪ [\frac{n+4}{p} + ε, +∞) . Proof. We split the integral in the right hand side of (75) in the following way

G_{2}(x, t) ≤ ∫_{0}^{t} ∫_{|z−s^{−1/4}x|≤1/2} + ∫_{0}^{t} ∫_{|z−s^{−1/4}x|>1/2} = I_{1} + I_{2}.

In the rest of this proof, we denote C = C(n, p, β). If we put w = z − s^{−1/4}x then, making use of (62), we find for the integral defined by I_{1}:

I_{1} = ∫_{0}^{t} ∫_{|z−s^{−1/4}x|≤1/2} \frac{e^{−μ|z|^{4/3}}}{[1 + (t − s)^{β/4} + s^{β/4} |z − s^{−1/4}x|^{β}]^{p}} dzds

≤ ∫_{0}^{t} ∫_{|w|≤1/2} \frac{1}{[1 + (t − s)^{β/4} + s^{β/4} |w|^{β}]^{p}} dw ds

≤ C ∫_{0}^{t} ∫_{|w|≤1/2} \frac{1}{(1 + t − s + s |w|^{4})^{\frac{βp}{4}}} dwds = C ∫_{|w|≤1/2} ∫_{0}^{t} \frac{1}{(1 + t − s + s |w|^{4})^{\frac{βp}{4}}} dsdw

≤ C ∫_{|w|≤1/2} \frac{1}{(1 − |w|^{4}) (1 + t |w|^{4})^{\frac{pβ}{4} −1}} dw.  (77)

We distinguish the three cases \frac{4}{p−1} ≤ β < \frac{n+4}{p}, β = \frac{n+4}{p}, β > \frac{n+4}{p}.

If \frac{4}{p−1} ≤ β < \frac{n+4}{p}, then |w|^{−pβ+4} / (1 − |w|^{4}) ∈ L^{1} (B_{1/2} (0)) so that by (77) we obtain

I_{1} ≤ \frac{C}{t^{\frac{βp}{4} −1}} ∫_{|w|≤1/2} \frac{|w|^{−βp+4}}{(1 − |w|^{4})} dw ≤ \frac{C}{t^{\frac{βp}{4} −1}} .  (78)

If β = \frac{n+4}{p}, by (77) we have

I_{1} ≤ C ∫_{|w|≤1/2} \frac{1}{(1 + t |w|^{4})^{n/4}} dw ≤ \frac{C}{t^{n/4}} ∫_{|w|≤1/2} \frac{1}{(\frac{1}{t} + |w|^{4})^{n/4}} dw ≤ \frac{C [1 + log (1 + t)]}{t^{n/4}}. (79)

Finally, if β > \frac{n+4}{p} by (77) and the change of variables η = t^{1/4}w we infer that

I_{1} ≤ \frac{C}{t^{n/4}} ∫_{|η|≤t^{1/4}/2} \frac{1}{(1 + |η|^{4})^{\frac{βp}{4} −1}} dη ≤ \frac{C}{t^{n/4}} ∫_{ℝ^{n}} \frac{1}{(1 + |η|^{4})^{\frac{βp}{4} −1}} dη.  (80)

Next, we consider the integral I_{2}. By (62) we have

I_{2} ≤ ∫_{0}^{t} ∫_{|z−s^{−1/4}x|>1/2} \frac{e^{−μ|z|^{4/3}}}{[(t − s)^{β/4} + s^{β/4} |z − s^{−1/4}x|^{β}]^{p}} dzds

≤ ∫_{0}^{t} ∫_{|z−s^{−1/4}x|>1/2} \frac{e^{−μ|z|^{4/3}}}{[(t − s)^{β/4} + (\frac{s}{16})^{β/4}]^{p}} dzds

≤ C ∫_{ℝ^{n}} e^{−μ|z|^{4/3}}dz · ∫_{0}^{t} \frac{1}{(t − s + \frac{s}{16})^{\frac{βp}{4}}} ds ≤ \frac{C}{t^{\frac{βp}{4} −1}} .

Combining this estimate with (78), (79), (80) we obtain (76).

We now study the decay of G_{2} (x, t) with respect to x.

Lemma 9. Let p > 1 + 4/n and β ≥ 4/(p − 1). Then, there exists a constant B_{2} = B_{2} (n, p, β) > 0 such that

G_{2} (x, t) ≤ \frac{B_{2}}{1 + |x|^{βp−4}} for all |x| ≤ 1, for all t ≥ 0.

Moreover, the constant B_{2} may be chosen independently of β.

Proof. See the proof of Lemma 9 in [9].

Lemma 10. Let p > 1 + 4/n and β ∈ [\frac{4}{p−1}, \frac{n+4}{p}). Then, there exists a constant B_{3} = B_{3} (n, p, β) > 0 such that

G_{2} (x, t) ≤ \frac{B_{3}}{1 + |x|^{βp−4}}  for all |x| > 1, for all t ≥ 0.  (81)

Moreover, for any β_{1} ∈ (\frac{4}{p−1}, \frac{n+4}{p}) the constant B_{3} may be chosen independently of β ∈ [\frac{4}{p−1}, β_{1}] .

Proof. Define F_{2} (x, t) = (1 + |x|^{βp−4}) G_{2} (x, t) . We will show that F_{2} is bounded in ℝ_{+}^{n+1}. We distinguish the cases n ≥ 2 and n = 1. The case n ≥ 2. The argument is almost the same as in the proof of Lemma 10 in [9]. We obtain for any |x| > 1 and t ≥ 0

F_{2} (x, t) ≤ C ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{0}^{∞} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{σ^{3}}{[\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} ]^{\frac{βp}{4}}} dσ dρ dw.

(82)

We split the integral in the right hand side of (82) in the following way

∫_{0}^{∞} ∫_{0}^{3/T} ∫_{0}^{T ρ} + ∫_{0}^{∞} ∫_{3/T}^{∞} ∫_{0}^{T ρ} .

By Lemmas 4 and 5 we infer that F_{2} (x, t) is bounded for |x| > 1 and t ≥ 0. This implies (81). The case n = 1. We argue again as in Lemma 10 in [9] to obtain for any |x| > 1 and t ≥ 0

F_{2} (x, t) ≤ C ∫_{0}^{∞} z^{−4}e^{−μz^{4/3}} ∫_{0}^{T z} \frac{σ^{3}}{[\frac{T^{3}}{z} (T z − σ) + (σ − 1)^{4}]^{\frac{βp}{4}}} dσ dz.  (83)

As above, we split the integral in the right hand side of (83) into

∫_{0}^{3/T} ∫_{0}^{T z} + ∫_{3/T}^{∞} ∫_{0}^{T z} .

Again, by Lemmas 4 and 5 we infer that F_{2} (x, t) is bounded for |x| > 1 and t ≥ 0. This implies (81) also for n = 1.

Lemma 11. Let p > 1 + 4/n and β ≥ \frac{n+4}{p}. Then, there exists a constant B_{4} = B_{4} (n, p, β) > 0 such that for all |x| > 1 and t ≥ 0, we have:

G_{2} (x, t) ≤ \{ \begin{matrix}\frac{B_{4}}{1 + |x|^{n}} & if β > \frac{n+4}{p} \\ B_{4} \frac{log (1 + |x|)}{1 + |x|^{n}} & if β = \frac{n+4}{p}. \end{matrix}

Moreover, for any β_{1} > \frac{n+4}{p} the constant B_{4} may be chosen independently of β ∈ [β_{1}, ∞).

Proof. Define F_{2} (x, t) = (1 + |x|^{n}) G_{2} (x, t). We split the corresponding integral expression as follows:

F_{2} (x, t) = ∫_{0}^{t} ∫_{|x−s^{1/4}z|≥ \frac{|x|}{2}} \frac{(1 + |x|^{n}) exp (−μ|z|^{4/3})}{(1 + (t − s)^{β/4} + |x − s^{1/4}z|^{β})^{p}} dz ds

+ ∫_{0}^{t} ∫_{|x−s^{1/4}z|< \frac{|x|}{2}} \frac{(1 + |x|^{n}) exp (−μ|z|^{4/3})}{(1 + (t − s)^{β/4} + |x − s^{1/4}z|^{β})^{p}} dz ds =: F_{2,1} (x, t) + F_{2,2} (x, t) .

In what follows, we repeatedly use (62). As for the the first term, we have

F_{2,1} (x, t) ≤ C ∫_{0}^{t} ∫_{|x−s^{1/4}z|≥ \frac{|x|}{2}} \frac{(1 + |x|^{n}) exp (−μ|z|^{4/3})}{(1 + t − s + |x|^{4})^{βp/4}} dz ds
≤ C(1 + |x|^{n}) [1 + t − s + |x|^{4})^{1−βp/4}]_{0}^{t}
≤ C(1 + |x|)^{n+4−βp} ≤ C,

where we used the fact that n + 4 − βp ≤ 0. As for the second integral, we observe that

|x − s^{1/4}z| < \frac{|x|}{2} ⇒ |z| > \frac{|x|}{2s^{1/4}} .

Taking into account that for any α ≥ 0 there exists C_{α} > 0 such that η^{α} exp(−η) ≤ C_{α} for all η ≥ 0, we conclude that

F_{2,2} (x, t) ≤ C ∫_{0}^{t} (\frac{|x|^{4}}{s} )^{n/4} ∫_{|w|<|x|/2} exp (−\frac{μ}{2^{4/3}} (\frac{|x|^{4}}{s} )^{1/3}) \frac{1}{(1 + t − s + |w|^{4})^{βp/4}} dw ds

≤ C ∫_{0}^{|x|/2} ∫_{0}^{t} \frac{ρ^{n−1}}{(1 + t − s + ρ^{4})^{βp/4}} ds dρ
= C ∫_{0}^{|x|/2} ρ^{n−1} [(1 + t − s + ρ^{4})^{1−βp/4}]_{0}^{t} dρ
≤ C ∫_{0}^{|x|/2} ρ^{n−1}(1 + ρ^{n})^{(4−βp)/n} dρ

= \{ \begin{matrix}C [log(1 + ρ^{n})]_{0}^{|x|/2} ≤ C log(1 + |x|) & if β = \frac{n + 4}{p} \\ C [−(1 + ρ^{n})^{(n+4−βp)/n}]_{0}^{|x|/2} ≤ C < ∞ & if β > \frac{n + 4}{p}, \end{matrix}

thereby proving the claim.

The proof of Proposition 3 follows combining the estimates of Lemmas 8-11.

A Appendix: polynomial approximations of the heat kernels

By Theorem 5, we may rewrite the f-function as

f_{n}(η) = c_{n} ∑_{k=0}^{∞}(−1)^{k}a_{k}η^{2k},

where

a_{k} = \{ \begin{matrix}\frac{(k+m)!Γ(\frac{2k+2m+1}{4})}{k!(2k+2m)!} & if n = 2m + 1 is odd, \\ \frac{Γ(\frac{k+m}{2})}{2^{2k+m+1}k!(k+m−1)!} & if n = 2m is even; \end{matrix}

and

c_{n} = \{ \begin{matrix}\frac{2^{m}}{\sqrt{8π}} & if n = 2m + 1 is odd, \\ 1 & if n = 2m is even. \end{matrix}

Moreover, for our purposes, the quotient of two consecutive coefficients

b_{k} := \frac{a_{k+1}}{a_{k}}

will be important. We first prove

Lemma 12. For any integer n ≥ 1, the sequence (b_{k})_{k∈ℕ} is strictly decreasing.

Proof. Consider first the case where the space dimension is odd, namely n = 2m + 1 for some integer m ≥ 0. Then, we have

\frac{b_{k+1}}{b_{k}} = \frac{a_{k+2}a_{k}}{a_{k+1}^{2}} = \frac{(k + 1)(2k + 2m + 1)Γ (\frac{2k+2m+5}{4}) Γ (\frac{2k+2m+1}{4})}{(k + 2)(2k + 2m + 3)Γ (\frac{2k+2m+3}{4})^{2}}

= \frac{(k + 1)Γ (\frac{2k+2m+5}{4})^{2}}{(k + 2)Γ (\frac{2k+2m+3}{4}) Γ (\frac{2k+2m+7}{4})} < \frac{k + 1}{k + 2} < 1

due to the log–convexity of the Gamma–function. Similarly, when the space dimension is even, namely n = 2m for some integer m ≥ 1, we have

\frac{b_{k+1}}{b_{k}} = \frac{(k + 1)Γ (\frac{k+m+2}{2})^{2}}{(k + 2)Γ (\frac{k+m+1}{2}) Γ (\frac{k+m+3}{2})} < \frac{k + 1}{k + 2} < 1

where we used again the log–convexity of the Gamma–function.

For K ∈ ℕ_{0}, we define the following approximations for the f-functions:

F_{K}(η) = c_{n} ∑_{k=0}^{K}(−1)^{k}a_{k}η^{2k} .

We do not indicate the dependence of F_{K} on n by an additional index. We are now ready to prove polynomial approximations of the f-functions:

Lemma 13. Let n ≥ 1 and assume that K ∈ ℕ is even. Then, for 0 ≤ η ≤ \frac{1}{\sqrt{b_{K}}} one has

F_{K−1}(η) ≤ f_{n}(η) ≤ F_{K}(η).

Moreover, we have the following error estimates:

for all |η| ≤ \frac{1}{\sqrt{b_{K}}} : max\{|f_{n}(η) − F_{K}(η)|, |f_{n}(η) − F_{K−1}(η)|\} ≤ |F_{K−1}(η) − F_{K}(η)| = c_{n}a_{K}η^{2K}.

Proof. One certainly has f_{n}(η) ≤ F_{K}(η), if for all even k ≥ K one has that

a_{k+2}η^{2k+4} − a_{k+1}η^{2k+2} ≤ 0 ⇔ η^{2} ≤ \frac{1}{b_{k+1}}

which, thanks to Lemma 12, is equivalent to

η^{2} ≤ \frac{1}{b_{K+1}} .

Similarly, one has F_{K−1}(η) ≤ f_{n}(η), provided for all odd k ≥ K − 1

a_{k+1}η^{2k+2} − a_{k+2}η^{2k+4} ≥ 0 ⇔ η^{2} ≤ \frac{1}{b_{k+1}} ⇔ η^{2} ≤ \frac{1}{b_{K}} .

This completes the proof. We now show how this polynomial approximation may be used to determine some quantitative properties of the f-function in space dimension n = 1, where it takes a particularly simple form:

f_{1}(η) = \sqrt{\frac{2}{π}} ∫_{0}^{∞} e^{−s^{4}} cos(ηs) ds.

We first refine (7) by estimating its modulus in a quantitative way: Lemma 14. For any η ≥ 0, the following holds

\frac{1}{c_{1}} |f_{1}(η)| = 4 |∫_{0}^{∞} e^{−s^{4}} cos(ηs) ds| ≤ 4.4 exp (−0.15η^{4/3}) .

Proof. We have

\frac{1}{c_{1}} f_{1}(η) = 2 ∫_{ℝ} e^{−s^{4}} cos(ηs) ds = ∫_{ℝ} e^{−s^{4}}e^{iηs} ds + ∫_{ℝ} e^{−s^{4}}e^{−iηs} ds = 2 ∫_{ℝ} e^{−s^{4}}e^{iηs} ds,

where we used the change of variables s ↦ −s. By Cauchy’s Theorem we have for any real ε > 0: ∫_{ℝ} e^{−s^{4}}e^{iηs} ds = ∫_{ℝ} e^{−(s+iε)^{4}}e^{iη(s+iε)} ds ,

|∫_{ℝ} e^{−s^{4}}e^{iηs} ds| ≤ e^{−ηε} ∫_{ℝ} e^{−s^{4}}e^{6ε^{2}s^{2}}e^{−ε^{4}} ≤ e^{−ηε}e^{17ε^{4}} ∫_{ℝ} e^{−s^{4}/2} ds.

Choosing ε = (η/34)^{1/3} so that 17ε^{3} = η/2 yields

|∫_{ℝ} e^{−s^{4}}e^{iηs} ds| ≤ (∫_{ℝ} e^{−s^{4}/2} ds) exp (−η^{4/3}/ (2\sqrt[3]{34})) ≤ 2.2 exp (−0.15η^{4/3}) .

The last inequality follows from the fact that

∫_{ℝ} e^{−s^{4}/2} ds = 2 ∫_{0}^{∞} e^{−s^{4}/2} ds = \frac{\sqrt[4]{2}}{2} Γ (\frac{1}{4}) = 2.1558...

which, in turn, follows by arguing as for (10). We prove a crucial positivity property for f_{1}:

Lemma 15. For any positive nonincreasing function h : (0, ∞) → [0, ∞) satisfying

∫_{0} h(η)dη < +∞

we have

∫_{0}^{∞} h(η)f_{1}(η) dη > 0.

Proof. Note first that by Lemma 14 and under the assumptions on h, we have hf_{1} ∈ L^{1}(0, ∞). Then, we apply Lemma 13 with K = 6 so that, for |η| ≤ 10:

F_{5}(η) ≤ f_{1}(η) ≤ F_{6}(η).

F_{5} has its first zero beyond 3.43, F_{6} before 3.46, so that f_{1} has its first zero in the interval [3.43, 3.46]. We conclude by means of explicit calculations and Lemma 14

\frac{1}{c_{1}} ∫_{0}^{∞} h(η)f_{1}(η) dη = \frac{1}{c_{1}} ∫_{0}^{3.43} h(η)f_{1}(η) dη + \frac{1}{c_{1}} ∫_{3.43}^{∞} h(η)f_{1}(η) dη
≥ h(3.43) (\frac{1}{c_{1}} ∫_{0}^{3.43} F_{5}(η) dη − \frac{1}{c_{1}} ∫_{3.43}^{∞} |f_{1}(η)| dη)
≥ h(3.43) (6.93 − 4.4 ∫_{3.43}^{∞} exp (−0.15η^{4/3}) dη)
≥ h(3.43) (6.93 − \frac{4.4}{\sqrt[3]{3.43}} ∫_{3.43}^{∞} η^{1/3} exp (−0.15η^{4/3}) dη)
≥ h(3.43) (6.93 − 6.72) > 0.

For suitable profiles h, the monotonicity assumption in the previous lemma may be relaxed.

Lemma 16. There exists β_{0} > 0 such that if 0 < β < β_{0}, then

∫_{ℝ} f_{1}(|η|)|ξ − η|^{−β} dη > 0  for all ξ ∈ ℝ.  (84)

Proof. Let γ_{1} be the smallest positive zero of f_{1} and take β ≤ 1/2. Let g(η) = 4.4c_{1} · exp (−0.15η^{4/3}) so that by Lemma 14, we have |f_{1}(|η|)| ≤ g(η). We choose ξ_{0} > 0 large enough so that

ξ_{0} ≥ 2γ_{1} and for all ξ ≥ ξ_{0} : 4ξg(ξ/2) ≤ 0.1c_{1}.

First, we consider ξ ≥ ξ_{0} and split the integral over ℝ in (84) into

∫_{−∞}^{−γ_{1}} + ∫_{−γ_{1}}^{γ_{1}} + ∫_{γ_{1}}^{ξ/2} + ∫_{ξ/2}^{2ξ} + ∫_{2ξ}^{∞} .

We obtain by making use of the approximations for ∫ f_{1} and ∫ g from Lemma 15:

∫_{ℝ} f_{1}(|η|)|ξ − η|^{−β} dη ≥ −|ξ|^{−β} ∫_{−∞}^{−γ_{1}} g(η) dη + 2(3/2)^{−β}|ξ|^{−β} ∫_{0}^{γ_{1}} f_{1}(η) dη
−(|ξ|/2)^{−β} ∫_{γ_{1}}^{∞} g(η) dη − 2g(ξ/2) ∫_{0}^{ξ} η^{−β} dη
≥ (2(3/2)^{−β} ∫_{0}^{γ_{1}} f_{1}(η) dη − (1 + 2^{β}) ∫_{γ_{1}}^{∞} g(η) dη − 4|ξ|g(ξ/2)) |ξ|^{−β}
≥ c_{1} (2(3/2)^{−β}6.93 − (1 + 2^{β})6.72 − 0.1) |ξ|^{−β} > 0

uniformly in ξ ≥ ξ_{0} for β close enough to 0. The same estimate holds for ξ ≤ −ξ_{0}. Finally, for ξ ∈ [−ξ_{0}, ξ_{0}], we have uniform convergence of

∫_{ℝ} f_{1}(|η|)|ξ − η|^{−β} dη → ∫_{ℝ} f_{1}(|η|)dη = α_{1}^{−1} > 0 as β → 0.

Therefore, the existence of β_{0} > 0 as in the statement follows.

As an immediate consequence we have even global positivity for a special class of initial data:

Proposition 4. We assume that n = 1 and u_{0}(x) = |x|^{−β}. For β > 0 sufficiently small, the corresponding solution of (1) given by (23) is positive in ℝ_{+}^{2}.

Acknowledgement. We are grateful to a referee of [9], who suggested to study not only the limiting case β = 4/(p − 1) but the full range β ≥ 4/(p − 1) and the dependence of the decay properties on β.

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