On the Linearized System of Equations for the Condensate-Normal Fluid Interaction Near the Critical Temperature

The Cauchy problem for the linearization around one of its equilibria of a non linear system of equations, arising in the kinetic theory of a condensed gas of bosons near the critical temperature, is solved for radially symmetric initial data. As time tends to infinity, the solutions are proved to converge to an equilibrium of the same linear system, determined by the conservation of total mass and energy. The asymptotic limit of the condensate’s density is proved to be larger or smaller than its initial value under a simple and explicit criteria on the initial data. For a large set of initial data, and for values of the momentum variable near the origin, the linear approximation n(t) of the density of the normal fluid behaves instantaneously as the equilibria of the non linear system.


Introduction
Correlations between the superfluid component and the normal fluid in a uniform condensed Bose gas, at temperature below but close to the condensation temperature, and for a small number density of condensed atoms, may be described by the equation where n(t, p) represents the density of particles in the normal gas that at time t > 0 have momentum p and n c (t) is the density of the condensate at time t, that satisfies dn c dt (t) = −n c (t) R 3 I 3 (n(t))( p))d p t > 0. (1.4) The collision integral I 3 in (1.1) was first derived in [9] and [20] and their treatment was afterwards extended to a trapped Bose gas. By including Hartree-Fock corrections to the energy of the excitations the so called ZNG system was obtained (cf. [18]). On the interest of system (1.1), (1.4) for the description of condensed Bose gases see also [19,24,26]. Of course, even in presence of the superfluid component, interactions between particles in the normal gas continue to take place. In order to take them into account the Nordheim-Boltzmann collision operator should be added at the right hand side of the equation (1.1) as it is done in the references above. Such an operator is generally written as

7)
D (ω 1 ) = {(ω 3 , ω 4 ) : ω 3 > 0, ω 4 > 0, ω 3 + ω 4 ≥ ω 1 > 0} . (1.8) It may be of some interest to understand the respective effects of each of the two terms I 3 and I 4 at the different stages of the evolution of the condensate and its thermal cloud. It is known that the term I 4 provokes by itself the finite time blow up of some solutions of Nordheim equation (cf. [15]). The finite time blow up phenomena does not seem to be expected in presence of I 3 alone. On the contrary, as proved below, the Cauchy problem for the linearization of (1.1, 1.4) around an equilibria is globally well posed in a suitable functional space; its solutions conserve the natural physical quantities and converge to a suitable equilibrium. Some regularizing effects are also observed as shown in Theorem 1.9 and Theorem 3.2, below. Such effects are not observed for I 4 (cf. [12]). However, a detailed understanding of the respective mathematical properties of I 3 and I 4 is still missing, and that makes difficult to precisely foresee what may be expected in presence of both terms. Several of these questions are considered in [3] where in particular some regularizing property of the term I 3 is proved in a different functional setting.
Other theoretical models do exist to describe Bose gases in presence of a condensate (cf. [23]) but ZNG system, and (1.1), (1.4) in particular, are very appealing by their simplicity and are well suited for analytical PDE methods.
The two functions of time, n(t, p)| p| 2 d p, (1.9) give, respectively, the total number of particles and the total energy of the normal fluid part in the gas, with density function n(t, p). The total number of particles at time t in the system condensate-normal fluid is n c (t) + N (t) and its total energy is E(t). It formally follows from (1.1), (1.4) that these two quantities are constant in time: E(t) = E(0) and n c (t)+N (t) = n c (0)+N (0) for all t > 0. This corresponds to the conservation of the total mass and energy property that is satisfied by the particle system in the physical description (cf. for example [9]). It is also well known that equation (1.1) has a family of non trivial equilibria, where the mass of the particles is taken to be m = 1/2 and β is a positive constant related to the temperature of the gas whose particle's density is at the equilibrium n 0 . It is easily checked that R( p, p k , p ) ≡ 0 in (1.3) for n = n 0 . Our purpose is to prove the existence of classical solutions to the Cauchy problem for the "radially symmetric linearization" of (1.1)-(1.4) around an equilibrium n 0 , and describe some of their properties. Such linearization is deduced through the change of variables (cf. [10,13]) n(t, p) = n 0 ( p) + n 0 ( p)(1 + n 0 ( p)) (t, | p|) = n 0 ( p) + (t, | p|) and keeping only linear terms with respect to u in (1.1). Since equation (1.4) is linear with respect to n c its linearization (1.15) follows by just keeping the terms of I 3 (n 0 ( p) + n 0 ( p)(1 + n 0 ( p))| p| 2 u(t, | p|)) that are linear with respect to u. This finally reads, for dimensionless variables in units which minimize the number of prefactors, as (1.18) With some abuse of notation the function ν 0 (x) = n 0 ( p) is still denoted n 0 (x). For uniform condensed Bose gases at very low temperatures and large number density of condensed atoms, the limit of the ZNG system deduced in references [9] and [20] is slightly different. In that limit, the interactions involving only particles in the thermal cloud are neglected, ZNG system is reduced to (1.1), (1.4), up to lower order terms but with a different expression for R( p, p 1 , p 2 ) in (1.3) since the dispersion relation and scattering amplitude are different. Related works in the mathematical literature for the isotropic case may be found in [1,2,27]. The non isotropic linearized system around an equilibrium is treated in [11].

A Nonlinear Approximation
Another approximation of the system (1.1), (1.4) is possible where, in the equation (1.4), the function n is replaced by n 0 +n 0 (1+n 0 )x 2 u in the nonlinear collision term I 3 given by (1.2) to obtain the system x) x 2 dx, (1.27) instead of (1.13), (1.15). In that way the non linearity of I 3 in the equation forp c is kept, but the conservation in time ofp c (t)+ N (t) does not hold, and so an important global property of the original system (1.1)-(1.4) is lost. As a consequence the time existence of the solutions to system (1.26), (1.27) cannot be proved to be (0, ∞).
Then, system (1.26), (1.27) is not too satisfactory to describe global properties of the particle's system. But it may be a better approximation of the local properties of the solutions to the nonlinear system of equations (1.1), (1.4). In order to avoid any confusion, system (1.26), (1.27) is considered in the Appendix.

Further Motivation
It is known that for all non negative measure n in with a finite first moment, and for every constant ρ > 0, system (1.1)-(1.4) has a weak solution (n(t), n c (t)) with initial data (n in , ρ) that satisfies the conservation of mass and energy (cf. [7]). For all t > 0, n(t) is a non negative measure that does not charge the origin, with finite first moment, and n c (t) > 0. However, one basic aspect of the non equilibrium behavior of the system condensate-normal fluid is the growth of the condensate after its formation (cf. [4,18,23] and references therein). In the kinetic formulation (1.1)-(1.4), this behavior is driven by the integral of I 3 (n) in the right hand side of equation (1.4). A was shown in [25], the behaviour of that term crucially depends on the behavior of n(t, p) as | p| → 0 (this was discussed also in [7,21,26]). If, for example, the measure n(t) is a radially symmetric, bounded function near the origin then, from a simple use of Fubini's Theorem, for some constant C > 0 independent of n, and this would give a monotone decreasing behavior of n c (t). On the contrary, as it is shown in [25], if the measure n(t) is a function such that for some a(t) > 0, and satisfies some Hölder regularity property with respect to p in a neighborhood of the origin, then for some other constant C 1 > 0 independent of n, On the other hand, it was proved in [7] that if the measure | p| 2 n(t, p) has no atomic part and has an algebraic behavior as | p| → 0 then it satisfies (1.28). Both results in [25] and [7] assume some regularity of the solution n with respect to p, although no regular solutions to (1.1) are known yet. The existence of regular classical solutions to (1.1)-(1.4) satisfying (1.28) is one of the motivations of our present work. Since (1.28) is the behavior of the equilibrium n 0 (with a(t) ≡ β), it is natural to first consider the existence of such regular solutions for the linearization of (1.1) around n 0 . Because of the singular behavior (1.28) of n 0 near the origin, the linear operator L in (1.14) has regularizing effects. Similar regularizing effects may be expected also in the non linear equation (1.1).

Basic Arguments and Main Results
The function p c (t) in the right hand side of (1.13) may be absorbed by the change of variables, (1.30) to obtain This equation may be written as where, from (1.14), (1.33) and (1.35), the operator F is given by, (1.36) The equation (1.32) is then solved as a perturbation of with a forcing term in (1.35). Equation (1.37) is of interest by itself and has been considered in [10]. For example, one may consider what is seen somehow as the "classical approximation" of the nonlinear equation (1.1)-(1.3), where the factor of the Dirac's measure in (1.3) is replaced by n 1 n 2 − n(n 2 + n 1 ). The resulting equation also appears in the theory of wave turbulence for nonlinear optic waves (cf. for example [8,28]), related to the Schrödinger equation. The function | p| −2 is an equilibrium solution of that equation, and (1.37) is its isotropic linearization, around that equilibrium. It is proved in [10] that equation (1.37) has a fundamental solution ∈ C((0, ∞); L 1 (0, ∞)) that satisfies (1.37) in D ((0, ∞) × (0, ∞)) and almost every t > 0, x > 0. For all initial data f 0 ∈ L 1 , there exists a weak solution of (1.37), denoted S(t) f 0 , ) and (1.37) is satisfied pointwise, for t > 0 and x > 0. (cf. Appendix for more detailed statements).
Once the Cauchy problem for equation (1.32) is solved using the semigroup S(t), the change of time variable in (1.30) is inverted to obtain the function u(t), and deduce p c (t) using the conservation of mass of system and equation (1.15). Our first result, then, is as follows: for some θ ∈ (0, 1). Then, there exists a pair (u, p c ), such that, for each t > 0, u(t) is locally Lipschitz on (0, ∞) and, for all almost every t > 0 and x > 0, (1.44) and there exists a function H ∈ L ∞ ((δ, 0) × (0, δ)) for all δ > 0, defined in (5.12), such that x)dx belongs to W 1,1 loc (0, ∞) and for almost every t > 0, (1.48) Theorem 1.1, shows that the possible singular behavior as x −θ for some θ ∈ (0, 1) of the initial data u 0 at the origin is instantaneously regularized to u(t) ∈ L ∞ (0, ∞) for all t > 0. This is a direct consequence of the same property of the equation (1.37), where L may be seen as a pseudo differential operator with a symbol whose logarithmic growth at infinity induces a logarithmic regularizing effect (cf. Remark 6.6 below). The equation (1.37) enjoys a "local C 1 " regularizing property (cf. Lemma (6.4) below), that cannot be extended to the equation (1.42), due precisely to its local character. However, the equation (1.42) regularizes the initial data u 0 = δ 1 and gives a weak solution u ∈ C((0, ∞); L 1 (0, ∞)) (cf. Theorem 3.2). No regularizing effects have been observed for the isotropic linearization of the "classical approximation" of the Boltzmann Nordheim collision integral (1.5) around its stationary solutions ( [8,16]).
In view of (1.11), (1.12) and (1.30), if u is a solution of (1.32) given by Theorem 1.1, the pair of functions may be seen as an approximated solution of (1.1), (1.4), as far as n 0 ( p)(1 + n 0 ( p))| p| 2 u(t, | p|) remains small compared to n 0 . In view of (1.9) it is natural to look at the quantities These represent, respectively, the variation of the total number of particles and of energy caused by the initial perturbation n 0 ( p)(1 + n 0 ( p))| p| 2 u(0) of the equlibrium n 0 . Let us also define, The two following properties then hold true: It follows from Corollary 1.3 that the mass and the energy variations due to the perturbation n 0 (1 + n 0 )| p| 2 u(t) tend to the mass and energy of C * n 0 (1 + n 0 )| p| 2 , and this however small the perturbation is at infinity, even if, for example, u 0 is compactly supported. This kind of energy flux towards infinity could be expected, since it is well known to happen in the nonlinear homogeneous version of wave turbulence type of the system (1.1), (1.4) and is called direct energy cascade ( [8,28] and [16,27] (1.62)

Some Remarks
Several remarks follow from the previous results.

On the Formal Approximation
The approximation of (1.1), (1.4) by (1.13), (1.15) may be expected to be reasonable only as long as the perturbation remains small with respect to n 0 , (1.64) and the right hand side tends to zero as t → ∞. If, on the other hand, we had x 2 |u(t, x)| ≤ C, for some C > 0, R 0 > 0 and t 0 > 0 for all x > R 0 and all t > t 0 , and then, for R > C/|C * | and all t > t 0 , and this would contradict (1.64). System (1.13), (1.15) may then be considered "close to" (1.1), (1.4) only for small values of t. Of course, u could be such that, for some C(t) that tends to ∞ with t, x 2 |u(t, x)| ≤ C(t) for all x > 0.

The Function p c (t)
In view of Corollary 1.4, if the initial data u 0 is such that or equivalently, then lim t→∞ p c (t) > p c (0), and conversely. Condition (1.65) and its converse are both compatibles with n 0 (1 + n 0 )x 2 u 0 being a small perturbation of n 0 . For example and (1.67)

Very Low Temperature and Large n c
The linearization of system (1.1), (1.4) for large number density of condensed atoms and very low temperature may be performed following similar arguments as to those above (cf. [6] and [11]). No regularizing effects have been observed and the existence of a first positive eigenvalue and spectral gap for a suitable integrable operator ( [6] and [17]) provide a convergence rate to the equilibrium for a large set of initial data (cf. [11], Theorem 2.2) A necessary and sufficient condition on p c (0) to have a global solution.

The Operator F
Equation (1.32) may be treated as a perturbation of (1.37) only whenever the term F( f ) in (1.35) is bounded in spaces where the properties of the solutions of (1.37) may be used. The purpose of this section is to establish that this is the case.
Proposition 2.1 follows from estimates of the kernel T defined in (1.36), that we split as follows: The kernels T 1 and T 2 are estimated in the two next Propositions.
Using Taylor's expansion we have, for some ξ(x 2 , y 2 ) between x 2 and y 2 , and We deduce that, for all x ∈ (0, R) and y ∈ (0, R), Similarly, using the change x ↔ y, It follows that, for all x ∈ (0, R), y ∈ (0, R), On the other hand, since sinh is locally Lipschitz, and This proves (2.10).
3.-Proof of (2.12) We consider now the case where x + y > 1 and |x − y| < 1/8. Suppose again that 0 < y 2 < x. We may still use the Taylor's expansion around x 2 = y 2 , and write that If |x − y| < 1/8 and x + y > 1 then, for some and, On the other hand, if |x + y| > 1 and |x − y| < 1/8, for some positive C. It follows that and this shows (2.13). 5.-Proof of (2.14) and (2.15). Suppose now x + y > 1, |x − y| > x/2. As we have seen in the proof of Proposition 2.2, if y > x, and sinh x 2 sinh y 2 − Then, If, on the other hand, y < x/2, x 2 − y 2 > 3x 2 /4, we then have that We deduce that and this proves (2.14) and (2.15).
A simple inspection of the expression of T (x, y) given by (1.36 If x ≤ 2R, by (2.1) and (2.10),

Existence of Global Solution f
Using the properties of the operator L, Proposition 2.1 and a fixed point argument, classical solutions f ∈ C([0, ∞); L 1 (0, ∞)) of the Cauchy problem for However it is interesting to consider initial data slightly more general than in f 0 ∈ L 1 (0, ∞) ∩ L ∞ (0, ∞) but whose solutions are more regular than just integrable with respect to x in (0, ∞).
for some θ ∈ (0, 1). Then, there exists a function satisfying that 3) The function f also satisfies that Proof. Given f 0 fixed and satisfying the hypothesis, consider the operator On the other hand, By (iv) in Proposition (2.1) and Proposition 6.3 in the Appendix, for all t > 0, s ∈ (0, t) and x ∈ (0, 2), Since we also have, for x ∈ (0, 2), that and then Adding (3.8), (3.10) and (3.11), and we deduce If we denote that we have then proved that Let ρ > 0 and T > 0 be such that Then, for all f ∈ Z T such that || f || Z T ≤ ρ, and then On the other hand, and arguing as before, The map L is then a contraction form B Z T (0, ρ) into itself if T is small enough, and has a fixed point u ∈ B Z T (0, ρ) that satisfies (3.20) in Z T . Property (3.5) follows from and Gönwall's Lemma on (0, T ) and by (3.10) and (3.11): Then, there exists a constant C = C(T ) > 0 such that, (3.31) holds true.
On the other hand, since f 0 ∈ L 1 (0, ∞) and ||| f 0 ||| θ < ∞, by Proposition 6.3 in the Appendix, S(t) f 0 ∈ L ∞ (0, ∞). Moreover, by Proposition (6.3) and Proposition (2.1), for t ∈ (0, T ) and x ∈ (0, 2), It immediately follows that, for all t ∈ (0, T ) and x ∈ (0, 2), and then f (t) ∈ L ∞ (0, ∞) for all t ∈ (0, T ). We wish to extend now this function f for all t > 0. We notice, to this end that, for all x > 1, Since, by Proposition 2.1, It follows by Gönwall's Lemma, that, for some constant C depending on T and θ , On the other hand, for x ∈ (0, 2), using (3.23), and by (3.25), for all x ∈ (0, 2), By a classical argument it follows that the function f may be extended to a function, still denoted f , for all t > 0 such that f ∈ Z t for all t > 0 and satisfies (3.20) for all t > 0. The same arguments used to prove the estimates (3.5), (3.31) and (3.7) on the interval of time given by (3.15) may now be applied to obtain (3.5), (3.31), (3.7) on all finite interval (0, T ) for all T > 0.
Since 0 and s ∈ (0, t). This shows that The L 1 − L ∞ regularizing effect of the equation (1.30) observed in Theorem 3.1 follows from a similar property as to that of the equation (1.37) proved in [10]. Our next result shows how equation (1.30) regularizes the Dirac's delta given as initial data. This is a consequence of the same property of (1.37) proved Theorem 1.2 of [10] (cf. Section 6.1.1 in Appendix below). It also holds for example for the linearized coagulation equation around its equilibria ( [14]), but not for the linearization of the classical approximation (or wave turbulence "version") of the Nordheim equation [12].

Theorem 3.2.
There exists a function

29)
and such that 32) where the notation L( f (t)), ϕ is defined in Section 6.1.1.
Proof. By definition, S(t)δ 1 with t > 0 is nothing but the fundamental solution (t) of the equation (1.37). Consider the following operator T , defined on functions f ∈ C((0, T ); L 1 (0, ∞): By (6.5) in the Appendix below, there exists a constant C 0 > 0 such that, and then, by (6.13) in Section 6.1.2 and Proposition (2.1), for all f ∈ C((0, T ); Therefore, if C 0 + CT R < R and CT < 1, the operator T has a fixed point f ∈ C((0, T ); L 1 (0, ∞)) such that || f (t)|| 1 ≤ R for all t ∈ (0, T ). It easilly follows using Gönwal's Lemma that the solution f may be extended for all t > 0 and satisfies, for all t > 0, The proof of (3.32) follows now by classical arguments. Multiplication of (3.29) by ϕ ∈ D(0, ∞) yields where in the first term (cf. Section 6.1.1 in the Appendix below), The integral in the second term is denoted as and, in order to obtain the derivative of v, consider the quotient, On the other hand, since, for all g ∈ L 1 (0, ∞), A stronger regularizing effect of the equation (1.37) takes place for t > 0 and x ∈ (0, t), and is given below in Lemma 6.4 of the Appendix. However, because of its "local" feature, this property does not extends to the equation (1.30).
Using Proposition 6.8 in the Appendix it follows that We deduce that and for almost every t > 0 and x > 0, 0, ∞)) for all δ > 0. This ends the proof of (3.34) and (3.35), and proves (3.36).
On the other hand, if we multiply both sides of (3.20) by ϕ ∈ C 1 0 ([0, ∞)) and integrate, In order to derive this expression with respect to t, we use (3.38) and (3.43) again to obtain (3.46) and, for all t ∈ (0, T ), Identity (3.37) follows now, since

Further Properties of the Solution f
We describe in this section some further properties of the solutions f given by Theorem 3.1. We first consider what are the variations of mass and energy induced by the initial perturbation n 0 (1 + n 0 )x 2 f (τ ) of the equilibrium n 0 introduced in (1.11), (1.12). Then we prove that for all δ > 0, f ∈ L ∞ ((δ, ∞) × (0, ∞)) and that for every t > 0 the function f (t) has a limit as x → 0.

Mass and Energy
It will be sometimes denoted in what follows that With some abuse of notation the function ν 0 (x) = n 0 ( p) is still denoted n 0 (x). A first basic property is the following: Proof. Since f satisfies (3.34 ), (3.36), and x 6 n 0 (1 + n 0 )| f | p−2 f ∈ L 1 (0, ∞), multiplication of both sides of (3.36) and integration over (0, ∞) gives, using (1.31) and the symmetry of W (x, y) as follows: Since dμ is a non negative finite measure on (0, ∞),
The following property will show the boundedness of the variation of the mass: Proof. By Hölder's inequality, It follows by Proposition 4.1, for all t > 0, Since, it follows from Lemma 4.2 that and the Proposition follows.
From Proposition 4.4 we immediately have where C p is given in Proposition 4.4.
The following property also follows from similar arguments: Proof. Since L(C) ≡ 0, it follows that If we multiply the equation by n 0 (1 + n 0 )x 6 x) − C) + = 0 a.e. and the Proposition follows.
We have the following Corollary: Let let us also deduce the following:
Our next result concerns the long time behavior of the solution f . The L 2 (dμ) norm and D( f (τ )) play here the usual roles of entropy and entropy's dissipation through the identity (cf. (4.3)), (4.6) For every n ∈ N \ {0}, define the regularized kernel so that, dσ 1 (x, y) = W 1 (x, y)x 4 y 4 dydx is now a bounded measure on R 2 . Now let j be the function consider, for any pair of functions f, g defined on (0, ∞) the function U defined on (0, ∞) 2 as U (x, y) = ( f (x), g(y)) ∈ R 2 and denote that Lemma 4.9. The function J 1 is weakly l.s.c. on L 2 (dμ).
Proof. Let us show that J 1 is convex and continuous on L 2 (dμ) × L 2 (dμ). Since the Hessian of j is positive semi definite on R 2 , the function j is convex on R 2 and then so is J 1 . On the other hand, the estimate of W 1 (x, y)x 4 y 4 for x > 0 and y > 0 easily follows from this definition: , ∀y > 0, ∀x ∈ (0, y) , ∀y > 0, ∀x > y.
Then, for f ∈ L 2 (dμ), y)x 4 y 4 dxdy = I 1 + I 2 and the terms I i , i = 1, 2 are bounded as and Therefore, if { f n } n∈N and {g n } n∈N are two sequences in L 2 (dμ) converging, respectively, to f ∈ L 2 (dμ) and g ∈ L 2 (dμ), Using the previous estimate, we deduce in the second integral as In the first, Since the sequences { f n } n∈N and {g n } n∈N are bounded in L 2 (dμ) it follows that The function J 1 is then convex and continuous on L 2 (dμ) × L 2 (dμ) and the weakly l.s.c. on L 2 (dμ) × L 2 (dμ), (cf. for example [5], Corollary 3.9).
There exists then a sequence of {t j } k∈N and C(ϕ) such that Since The following auxiliary result is used in the proof of the next Proposition: Proof. There certainly exists C 1,ε > 0 such that Then, by continuity, and the result follows.
(4. 15) It was been proven in (4.8) that the first term in the right hand side of (4.15) tends to zero as k → ∞. In order to prove that the last term in the right hand side tends to zero the following Lemma is needed, whose proof is delayed after the end of the proof of Proposition 4.13.

Lemma 4.14.
For all ϕ ∈ C c (0, ∞) and all T > 0, The second term in the right hand side of (4.15) may be split as follows: The first term may be written as Since, by Lemma 4.14 with ϕ(x) = θ ε (y) sinh y 2 y 3 . The same argument shows that J 2 tends to zero too as k goes to ∞ and this shows that, for all ε > 0 fixed, We deduce from (4.14) and (4.16) that

Proposition 4.13 follows.
Proof of Lemma 4.14. For all ϕ ∈ C 0 (0, ∞), If ρ > 0 and R > 0 are such that suppϕ ⊂ (ρ, R), then |ϕ(y) − ϕ(x)| = 0 for x ∈ (0, ρ) and y ∈ (0, ρ). If, on the other hand, x ≥ ρ or y ≥ ρ, that If x > R and y > R, then |ϕ(y) − ϕ(x)| = 0 again, and therefore, The term I k,1 is easily estimated as Using Hölder's inequality and Proposition 4.1, The integral I k,2 may be split again as where, dy dx, and I k,2,1 is then estimated as I k,1 . The estimate of the term I k,2,2 uses that when y ≤ R < x/2 then sinh |x 2 − y 2 | ≥ sinh(3x 2 /4) as and using Hölder's inequality again, We have then, for all T > 0, that and by the compactness of the injection W 1, However, since, by Corollary 4.10, and by the fundamental Theorem of Calculus, It follows that h is the constant given by C * ∞ 0 ϕ(x)dμ(x) and this ends the proof of Lemma 4.14.
Proof. Arguing as in Proposition 4.4, for all ε > 0 and t > 1, On the other hand, , and the Corollary 4.15 follows from Corollary 4.13.

The Limit of f (t, x) as x → 0 for t > 0
We show now the existence of the limit for all f 0 and f as in Theorem 1.1, and describe its time evolution. We use the property, proven in Proposition 1.3 of [10], for some constants A 1 , A 2 and a function b 1 such that b 1 (t) = O(t −8 ) for t > 1. Slightly more precise information may be obtained and is shown here, since it is of further interest.

The Functions u(t) and p c (t)
We now return to the notation of the time variable as in sub Section 1.4. Then, given the function f (τ, x) obtained in Theorem 3.1, t = t (τ ) and p c (t) must be determined in order to define Proof. By Proposition 5.1, |M(τ )| < ∞ for all τ > 0 and then q c (τ ) ∈ (0, ∞) for all τ > 0 and the integral in the right hand side of (5.7) is well defined and convergent. Since q c (t) > 0 this integral is a monotone increasing function of τ . It only remains to check that its range is [0, ∞). By Corollary 4.15, for ε > 0 and τ ε large enough, Since, on the other hand, Therefore, the function e M(σ ) is not integrable at infinity, and and, for all t > 0, there exists a unique τ > 0 satisfying (5.7).
Proof of Theorem 1.1. For all t > 0, let τ > 0 be given by Proposition 5.2 and define that where f is obtained by Theorem 3.1 with initial data f 0 = u 0 . From the definition of p c , and (5.1) is satisfied. On the other hand, By (5.3) and (5.9), It follows from (5.6) that and then p c ∈ C 1 (0, ∞) and satisfies (1.15). On the other hand, by (5.1) and Theorem 3.3, where, a(t) = b t 0 p c (s)ds (5.14) satisfies, for some constant C > 0, Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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The Fundamental Solution
The equation (1.37) has a weak formulation, obtained as follows: if u is a regular function, Then, for u ∈ D (0, ∞), a possible definition of L(u) in the weak sense is Such a definition makes sense if the distribution u belongs to one of the spaces E p,q , with p < q, of Mellin transformable distributions whose Mellin transform is analytic on the strip of the complex plane defined by {s ∈ C; e(s) ∈ ( p, q)} (cf. [22]). The function , fundamental solution (1.37), is defined in Corollary 2.11 of [10] for all t > 0, as the inverse Mellin transform of a function U (t, ·). For each t > 0, the function U (t, ·) is holomorphic on C, analytic for e(s) ∈ (σ * 0 , 2) for some real negative constant σ * 0 ∈ (−2, −1) and for all T > 0 there exists a constant C > 0 for which 3) It follows that ∈ E 0,2 and satisfies the equation (1.37) in the weak sense described just above. For t > 1/2, and every α ∈ (0, 2), s)x −s ds. (6.4) As shown in Corollary 3.15 fof [10], ∈ C((0, ∞); L 1 (0, ∞)), and ∃C > 0; || (t)|| 1 ≤ C 1 + t 2 , ∀t > 0. (6.5) We recall in the next Proposition the results of Proposition 3.1, Proposition 3.2, Proposition 3.4 and Proposition 3.5 from [10] about the function , the fundamental solution of (1.37), that are frequently needed along the present work. Proposition 6.1. For all δ > 0 as small as desired there exists positive constants, denoted as C δ but whose numerical value may change from line to line, such that (i) For t > 1, (6.12) Properties (i) directly follow from Proposition 3.1, Proposition 3.2 and Proposition 3.3 of [10]. Properties (ii) come from Proposition 3.4 and those of (iii) from Proposition 3.5. (1.37). As proved in Theorem 1.4 of [10], for all f 0 ∈ L 1 (0, ∞), the function S(t) f 0 defined by (1.38) is such that S(t) f 0 ∈ C((0, ∞); L 1 (0, ∞)), there exists a constant C > 0 such that

The Cauchy Problem Associated to Equation
, and S(t) f 0 satisfies (1.37) for x > 0 and t > 0.

Some Further Properties of S(t)
We prove in this Appendix two properties of the solution S(t) f 0 of (1.37) with initial data f 0 , that are not given in [10]. The first is just an elementary continuity result. We look for the continuity of S(t) f 0 with respect to x in order to deduce the same property for the solution u of (1.13) and to be able later to speak of (u(t)). We first briefly recall, from [10], the function W : , s ∈ C, Re(s) ∈ (−2, 4). (6.14) where γ e is the Euler constant and ψ(z) = (z)/ (z) is the Digamma function. For any β ∈ (0, 2) fixed, the function (6.15) is analytic in the domain {s ∈ C; Re(s) ∈ (β, β + 1)} and satisfies Proof. By definition, and then We choose δ 1 small enough and R large enough, both depending on t, to get that using that f 0 ∈ L 1 and the asymptotics of for large and small arguments. Consider for example, the integral for y < δ, with δ < t. Then t/y > 1 and and, if we use that y > x/2, A similar argument yields the limit of the integral for y > R as R → ∞.
On the other hand, if we suppose y ∈ (δ 1 , R), |x − y| < δ, then Then, On the other hand, for |x − y| < δ, by (6.17), If t/R < 1/2, we must divide the domain D in two sub domains: With the previous argument, In the domain D − , with r = t/R > 0 and a = r/2, and then, A arguing as before, By (6.18), if |x − y| < δ and |1 − x/y| > ρ, then |1 − y/y| > ρ − δ δ 1 and ∈ C((0, ∞) ×D) It follows that, for all y ∈ (δ 1 , R) such that |1 − x/y| > ρ, the function y → t y , y y is continuous at x. Then, for all t > 0 fixed and y ∈ (δ 1 , R) such that |1 − x/y| > ρ, We deduce from Lebesgue's convergence Theorem, lim x→y J 1 = 0 and this gives the continuity of with respect to x. On the other hand, fix x > 0 and t > 0 and suppose that t n → t. Then In the first integral, for ρ ∈ (0, 1), we have In that range of values of y, we have the estimates since we may assume that t n ≥ t and 1 − ρ > 1/2. We deduce that This fixes ρ. We then have that and by Lebesque's convergence Theorem, J 2 → 0 as n → ∞.
The next result is useful to consider initial data f 0 that is unbounded near the origin. Proposition 6.3. Suppose that g ∈ L 1 (0, ∞) is such that, for some θ > 0, Then, S(t)g ∈ L ∞ (0, ∞), and, more precisely, Proof. By hypothesis, for all δ > 0, g ∈ L ∞ loc (δ, ∞) and for all x > 0, On the other hand, for x > 2, where x/y > 2 for y ∈ (0, 1). Several cases are now possible. If t > x, then t > y and it follows from by (6.6) in Proposition 6.1, The same argument yields, for t ∈ (2, x), that and, for 0 < y < t < 2, that By estimate (6.10) in Proposition 6.1, when t < 1, and y ∈ (t, 1), and this ends the proof of (6.20). When x ∈ (0, 2), we first write that When t > y and x < t, by (6.6) of Proposition 6.1, and for x > t, by (6.7) of Proposition 6.1, In both cases, Similar arguments show the same estimate for y ∈ (t, 1). Therefore, and (6.19) follows.
Proof. Suppose first that x ∈ (0, t), and write that In the term I 2 , x/y < t/y < 1 and x/y < x/t < 1. Then, by (6.12) in Proposition 6.1, there exists constant C such that Then, where the first term in the right hand side may estimated as follows. If t ∈ (0, 1), Since x ∈ (0, t) and σ * 0 < 0, it follows that and then
Since x/t < 1, t/y > 1 and |(x/y) − 1| > 1/(1 + δ) in I 1,1 and I 1,3 , by (6.8) of Proposition 6.1 may be applied to obtain that (y θ | f 0 (y)|), t < 1, A slightly different argument must be used to estimate I 1,2 . Since t/y > 1, by (6.3) and (6.4), for all α ∈ (0, 2), and then, from which, (6.26) and the last integral converges, since 2t (1+δ)x > 2, to This ends the proof of Lemma 6.4. Remark 6.5. When x > t the arguments of the proof of Lemma 6.4 give similar estimates for I 1,1 , I 1,3 and I 2 using the estimates in Proposition 6.1. However, this is not true for I 1,2 because when x > t the integral in the right hand side of (6.26) does not converge anymore. This is due of course to the fact that U (t, s) decreases slowly as |s| → ∞. All of that makes if impossible to extend the regularizing effect of Lemma 6.4 to the domain x > t nor to the solutions of equations (1.31) and (1.13). Remark 6.6. Under the change of variables x = e ξ/2 and xu(t, x) = w(t, ξ), Then, using it follows that L(u(t)) = P(w(t)) with and the operator P is then a pseudo differential operator with symbol p(ξ, k): Here (s) is the Digamma function and γ E denotes the Euler's constant. Use of Mathematica yields that , if 2x/3 < t < 2x.
On the other hand, we easily obtain, if t < 2x/3, that .
We now slightly rearrange the term A δ (h 0 , h) + A δ (h, h 0 ) as (6.36) The argument still follows as in [25], even if our function h satisfies slightly different conditions than (A.13) and (A.14). Indeed we claim that here also, the two functions under the integral signs in (6.35) and (6.36) are integrable on (0, ∞). The only delicate region is where bot ω and |ω − ω| are arbitrarily small.