Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity γ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma < 1$$\end{document}, such that K(x, y) is approximately xγ+λy-λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{\gamma + \lambda } y^{-\lambda }$$\end{document}, when x is larger than y. We restrict the analysis to the case γ+2λ≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma + 2 \lambda \ge 1 $$\end{document}. In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case considered in Ferreira et al. (Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 2023), where γ+2λ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma + 2 \lambda <1$$\end{document}. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case γ+2λ≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma +2\lambda \ge 1$$\end{document}, the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.


Aim of the paper
In this paper we study the long-time behavior of the coagulation equation with injection: where and where η ≥ 0. We will assume in all the following that η(x) ≡ 0 and that it is either compactly supported or it decays fast enough with the cluster size.
The study of problems with this form arises naturally in problems of aerosols and atmospheric science ( [12,18,21]).In this context the function f (t, x) denotes the density of clusters with size x at time t.
The collision operator K[f ](t, x) in (1.2) was introduced by Smoluchowski (see [19]).The kernel K(x, y) encodes the information about the mechanism driving the coagulation of clusters.In this paper we are interested in kernels arising in atmospheric science applications.A common feature of these kernels is the homogeneity property.Indeed, in many cases the rate of aggregation scales like a power law with the cluster size.This means that the coagulation kernel K(x, y) satisfies K(ax, ay) = a γ K(x, y) for a > 0 and (x, y) ∈ (0, ∞) 2 , ( for some γ ∈ R. On the other hand, since the coagulation process does not depend on the order in which the clusters of size x and y are chosen we have the symmetry property K(x, y) = K(y, x) for (x, y) ∈ (0, ∞) 2 . (1.4) We will assume that the coagulation kernel K satisfies c 2 x γ+λ y λ + y γ+λ x λ ≤ K (x, y) ≤ c 1 x γ+λ y λ + y γ+λ x λ and γ + 2λ ≥ 0, ( with 0 < c 2 ≤ c 1 < ∞, where γ is the homogeneity parameter introduced in (1.3) and λ ∈ R.
The polynomial bounds (1.5) are satisfied by many of the most relevant collision kernels arising in aerosol and atmospheric science, such as the diffusive coagulation kernel and the free molecular kernel (see for instance [12]).Notice that the condition γ + 2λ ≥ 0 in (1.5) does not imply any loss of generality.This can be seen from the fact that the function x α y β + x β y α can be written as x γ+λ y λ + y γ+λ x λ with γ + λ = max{α, β} and −λ = min{α, β}.Since we will consider solutions of (1.1), (1.2) in which f (t, •) is a Radon measure, it is convenient to impose the following condition on the kernel K: K ∈ C((0, ∞) 2 ). (1.6) It is well known that part or all the mass of the solutions of (1.1), (1.2) can escape towards x = ∞ in finite, or even zero time.This phenomenon is known as gelation.For a more detailed analysis on this matter, see for example [3,4].In order to guarantee that gelation does not take place, we will assume in all the following that γ < 1 and γ + λ < 1. (1.7) See [5] for a more detailed discussion on the gelation regimes.We expect that gelation does not take place under the weaker assumptions γ ≤ 1, γ+λ ≤ 1 (see [3]).However, in the critical cases γ = 1 or γ + λ = 1, the self-similar solutions are not defined using power laws to scale the particle sizes but most likely using exponential functions.Given that the analysis of these solutions would require arguments different from the ones in this paper, we will not consider this case here.
The existence of solutions for equation (1.1), (1.2) has been considered in [6] and in [7].In this paper we study the long time behavior of the solutions of equation (1.1).Due to the presence of the source we can expect the solutions to (1.1), (1.2) to converge to a stationary non-equilibrium solution f = f (x) as t → ∞, i.e. to a solution of (1.8) However, it turns out that if η ≡ 0 and γ + 2λ ≥ 1, a solution for (1.8) does not exist.In [13], the discrete stationary coagulation model has been studied for the explicit coagulation kernel when |γ + λ| < 1, |λ| < 1, |γ| < 1. Formal asymptotics for the large size behavior of the solutions f k of (1.9) has been obtained in [13].The results in that paper indicate that a solution of (1.9) exists if and only if γ + 2λ < 1.
In the case of general kernels satisfying the assumptions (1.4), (1.5), (1.6), and source terms η decreasing fast enough, it has been proved in [10] that the solutions of (1.8) (as well as its discrete counterpart) exist, if and only if γ + 2λ < 1.
It is worth to remark that it has been proved in [10] that the solutions of (1.8) can be estimated, up to a multiplicative constant, from above and below by the power law x − 3+γ 2 for large values of x.
Since in the case γ +2λ ≥ 1 a stationary solution of (1.8) does not exist, we cannot expect the solutions to (1.1), (1.2) to behave as the stationary solution f of (1.8) as t → ∞ for x of order one.It is then natural to ask what is the long time asymptotics of the solutions to (1.1), (1.2) for large values of t and x.The scaling hypothesis that has been extensively used in the study of coagulation equations suggests that the mass of the particle distributions f (t, x) is concentrated in cluster sizes x of order t p for a suitable exponent p that would be determined from dimensional considerations, which take into account the way in which the mass rescales in time.In the case of kernels K (x, y) = x γ+λ y −λ + y −λ x γ+λ , −1 < λ < 0 with 0 ≤ γ + λ < 1 and γ < 1, it was suggested in [5], using a combination of matched asymptotics and numerical simulations, that the long time behavior of the solutions of (1.1), (1.2) is given by self-similar solutions with the form (1.11) The approximation (1.11) can be expected to be valid for large cluster sizes, i.e. x 1.We will use from now the notation with x 1 to indicate large cluster sizes x.In the case considered in [5] we have that γ + 2λ < 1 and therefore stationary solutions f s solving (1.8) exist.In this case, (1.8) and (1.11) suggest that Φ (ξ) behaves for small values of ξ as Kξ − 3+γ 2 , for a suitable constant K > 0.More precisely, plugging (1.11) in (1.1), (1.2), it follows that Φ solves where Φ satisfies the following boundary condition at ξ → 0 that guarantees that there is a constant flux of particles from the origin: with J = ∞ 0 xη (x) dx.The existence of solutions of equation (1.12) satisfying the constant flux solution condition at ξ = 0, (1.13), has been rigorously proved in [9] for kernels K satisfying (1.4), (1.5), (1.6), (1.7) with γ + 2λ < 1.
The picture described above, which combines the stationary behavior f , (cf.(1.8)) for cluster sizes x of order one, and the self-similar behavior (1.18) for large cluster sizes, provides a rather complete description of the long time behavior of the solutions to (1.1), (1.2) in the case γ + 2λ < 1.However, the same scenario cannot yield a description of the long time asymptotics of the solutions to (1.1), (1.2) if γ + 2λ ≥ 1, because, as explained above, in this case a solution of (1.8) does not exist.
Notice that the existence/non-existence of solutions to (1.8) is related to the existence of stationary solutions of (1.8) yielding a constant flux of particles with the form of a power law, i.e. f (x) = cx − γ+3 2 .These solutions exist for γ + 2λ < 1 and they do not exist for γ + 2λ ≥ 1.In the framework of the wave turbulence it would be stated that the kernels K with γ + 2λ < 1 satisfy the locality property, while the kernels K with γ + 2λ ≥ 1 do not have the locality property, see [22].
(1. 15) In order to prove that in the case γ + 2λ ≥ 1 there are no solutions of (1.8), the main idea used in [10] is based on the fact that, for this range of exponents, the transfer of clusters of size x of order one towards very large cluster sizes is so fast that the concentration of clusters with size of order one would become zero.
In the case of time dependent solutions having initially finite mass, this increases linearly due to the fact that For large times, due to the increase of the average cluster size, we might expect that, if γ + 2λ ≥ 1, there should be a fast transport of the newly injected clusters of order one towards much larger cluster sizes.This almost instantaneous transport results in small concentrations of clusters of order one for large times.
We now remark that the part of the coagulation operator which describes the coagulation between particles of different sizes can be approximated by means of a transport operator in the space of cluster sizes.These arguments, that will be described in detail using formal asymptotics in Section 2, show that the solutions of (1.1), (1.2) for large cluster sizes can be approximated by means of the following equation if γ + 2λ ≥ 1 We emphasize that the non-local transport term )dz is a consequence of the presence of the source η (x) in (1.1).The fact that the contributions to the coagulation operator K [f ] which are due to the aggregation of particles with very different sizes can be approximated by a differential operator has been extensively used in the literature of coagulation equations (cf.[12]).The resulting first order terms are often referred to represent heterogeneous condensation (cf.[12]).On the other side, the term K [f ] in (1.16) describes the aggregation of clusters of comparable sizes.
The solutions of (1.16) are expected to describe the asymptotic behavior of the solution of equation (1.1) both when γ + 2λ > 1 and when γ + 2λ = 1, even if in these two cases we have two slightly different scenarios.Namely, when γ + 2λ > 1, we will have that if x is of order 1, then f (t, x) → 0 as t → ∞, while when γ + 2λ = 1 we will have that if x is of order 1, then f (t, x) → f s (x) as t → ∞, where f s is a solution of (compare with (1.8)).Due to the presence of the term x −λ f (x), equation (1.17) might have solutions when γ + 2λ = 1.
The scaling properties of (1.16) suggest that this equation is compatible with the existence of self-similar solutions for equation (1.16) with the form (1.18) Here the self-similar profile Φ satisfies the following equation, obtained by substituting equality (1.18) in equation (1.16) and using the self-similar change of variables The main result of this paper is to determine the range of exponents γ and λ satisfying γ + 2λ ≥ 1 and the non-gelation conditions (1.7) for which self-similar solutions of (1.16) with the form (1.18) exist (see Figure 1 for a classification of these exponents).Specifically, we will prove the following.Suppose that γ + 2λ ≥ 1 and that (1.7) holds.Then • If γ > −1, there exists at least one self-similar solution of (1.16) with the form (1.18).
• If γ ≤ −1 and γ + 2λ = 1, we prove that there are no self-similar solutions f s of the form (1.18) such that The meaning of the condition 1 0 x −λ Φ(x)dx < ∞ is that the number of clusters removed by the coagulation process in any bounded time interval is finite.We therefore do not exclude the existence of a self-similar solution of equation (1.16) with A remarkable property of the self-similar solutions of (1.16), that we constructed in this paper, is that they vanish identically for 0 < ξ < ρ(M γ+λ ) := The fact solutions Φ of (1.19) vanish in an interval (0, ρ(M γ+λ )) means that, for large times t, the injected particles are transferred almost instantaneously to clusters with sizes x ≥ ρ(M γ+λ )t 2 1−γ .Moreover, the fraction of clusters with sizes x < ρ(M γ+λ )t 2 1−γ becomes negligible for very long times.The existence of this "minimal" cluster size for large times is a remarkable feature that, to our knowledge, has not been observed in the literature on self-similar solutions for the coagulation equation (see for instance [7], [8], [11], [17], [20]).In particular, it is worth to notice that this behavior of the self-similar solutions is very different from the one exhibited by the self-similar solutions obtained in the case γ + 2λ < 1 in [9].
The results that we obtain in this paper in the critical case γ + 2λ = 1 are more fragmentary than those obtained for γ + 2λ > 1.In the case γ + 2λ = 1, we only prove the existence of a solution Φ of (1.19) for γ > −1 with Φ = 0 on (0, ρ(M γ+λ )), but we do not prove that any solution vanishes in the interval (0, ρ(M γ+λ )).
On the other hand, we will prove in this paper that the function Φ which describes the self-similar profile in (1.18) decreases exponentially as ξ → ∞ in the same manner as the self-similar solutions constructed in [9] and as it usually happens for the self-similar solutions of coagulation equations in problems without injection, see [11] and [7].
As indicated above, if γ ≤ −1 and γ + 2λ > 1, there are no self-similar solutions of (1.16) with the form (1.18) and, if γ ≤ −1 and γ + 2λ = 1, there are no self-similar solutions of (1.16) with the form (1.18) satisfying It is natural to ask what is the long time asymptotics of the solutions of (1.1), (1.2) in this case.This question will be the subject of study of a future work.

Figure 1
Coefficients for existence and non-existence of self-similar profiles in the case γ + 2λ > 1 In both cases the coagulation term K [f ] in (1.16) which describes the aggregation of particles with comparable sizes is negligible for large times, and the long time behavior of the distribution of clusters is determined by the coagulation of particles of size x of order one with large particles.The main difference between the cases γ + λ ≥ 0 and γ + λ < 0 arises from the fact that in the second case the transfer of clusters (not monomers) from the region where x is of order one to x 1, heterogeneous condensation, is relevant.There are several results in the physical literature which are related, and are consistent with the ones in this paper, see for instance [2], [14], [15], [16].In [2] a coagulation model with kernels satisfying (1.5) with parameters γ, λ such that γ + 2λ ≥ 1 and including also a source term and a removal of particles term has been studied.In that problem, the cluster concentrations as t → ∞ are determined by the coalescence of particles of very different sizes.It is then possible to approximate the coagulation-removal model for large clusters by means of the equation where K is the maximum particle size in the system.The numerical simulations in [2] show that the concentration of cluster sizes of order one approach to a stationary solution that converges to zero if K is sent to infinity.
In [14] a coagulation model with injection and with kernels satisfying (1.5) with γ = 0 and λ ∈ (1/2, 1] has been considered.Numerical simulations and formal computations in [14] suggest that the cluster concentration for clusters of order one tends to zero as t → ∞.
In addition, it is suggested that the concentration for large clusters are described by a selfsimilar solution.
In [15], [16] coagulation equations with injection are considered with kernels satisfying (1.5) with −λ = γ ≤ −1.It is seen there that for γ + 2λ ≥ 1, the solutions of the corresponding coagulation equation behave in a non self-similar manner and decay logarithmically.This is in agreement with the non-existence of self-similar solutions that we obtained in this paper for γ ≤ −1.

Notation and plan of the paper
We use the notation R * := (0, ∞) and R + := [0, ∞).Given an interval I ⊂ R + we denote with C c (I) the Banach space of the functions on I that are continuous and compactly supported.We endow the space C c (R * ) with the supremum norm denoted with • ∞ .We denote with M + (I) the space of the non-negative Radon measures on I. Given a measure µ ∈ M + (I) we denote with µ T V the total variation norm of µ.For a compact interval I ⊆ R * and two bounded measures µ, ν, we denote the Wasserstein metric by W 1 , namely where the supremum is taken over the Lipschitz functions and where To keep the notation lighter, we denote with C and c constants that might change from line to line in the computations.In addition, we use the notation f g, for two functions f, g, to mean that there exists a constant C > 0 such that f ≤ Cg.
To simplify the notation, in some cases, we write M α instead M α (µ) if the choice of the measure µ is clear in the argument.We use the notation f ∼ g as x → x 0 to indicate that lim x→x0 f (x) g(x) = 1, while we use the notation f ≈ g to say that there exists a constant M > 0 such that As previously mentioned, we use the notation x 1 for large cluster sizes x.Additionally, for cluster sizes x, y, we denote x y or y x to mean that x is much larger than y.For two terms, A and B, we use the notation A B to mean informally that A can be approximated in terms of B in the region under consideration in the respective formula.
The paper is organized as follows.In Section 2 we present a heuristic motivation to study the existence of the self-similar solutions considered in this paper.In Section 3 we present the main results of the paper regarding the existence and non-existence of a self-similar profile and its properties.In Section 4 we explain the main ideas behind the proofs of existence and non-existence, skipping the technical difficulties of the proofs.Section 5 deals with the proof of the existence of a self-similar solution for equation (1.19) when γ > −1.We also prove in this section that the solution decays exponentially for large values.Section 6 deals with the non-existence of self-similar solutions for equation (1.19) when γ ≤ −1.
2 Asymptotic description of the long time behavior 2.1 The case γ + 2λ > 1 In this section we describe the long time asymptotics of the solutions of (1.1), (1.2) if γ + 2λ > 1 using formal asymptotic arguments.
As indicated in the Introduction, we expect f (t, x) to converge to zero as t → ∞ for x of order one.Therefore, the contribution due to the term 1 2 x 0 K (x − y, y) f (t, x − y) f (t, y) dy can be expected to be negligible in this region since this term is quadratic in f and we can expect the linear term ∞ 0 K (x, y) f (t, y) dyf (t, x) to give a larger contribution.We will check that these assumptions are self-consistent, in the sense that they will predict an asymptotic behavior for f for which the assumptions made hold.
We examine the asymptotic behavior of the linear term ∞ 0 K (x, y) f (t, y) dyf (t, x) when x is of order one.Due to the effect of the coagulation, we expect the distribution f (t, y) to be concentrated for long times in the larger cluster sizes y as t → ∞.Using the assumptions (1.14) and (1.15) we obtain the following asymptotic behavior of K (x, y) for x y K (x, y) y γ+λ x −λ . (2.1) We then expect to have the following asymptotics as t → ∞, due to the concentration of f in the large cluster sizes for x of order one.Then, considering the dominant terms in (1.1), (1.2) for x of order one, we obtain the following equation As explained in the introduction, if γ + 2λ > 1, since steady states describing the cluster concentrations with x of order one do not exist, we expect to have f (t, x) → 0 as t → ∞.This suggest that we should have M γ+λ as t → ∞ (something that would happen if M γ+λ behaves like a power law, as we will see to be the case).Then (2.2) implies the following asymptotic behavior for for x of order one.We notice that the higher order contributions to (2.3) due to the contributions of the term 1 2 x 0 K (x − y, y) f (t, x − y) f (t, y) dy are of order 1 (M γ+λ ) 2 or smaller.Therefore, these contributions will be neglected in the following.
The equation (2.3) yields an approximate formula for the concentration of clusters with x of order one.We now approximate the part of the coagulation operator which is due to the aggregation of particles with size one with very large particles.To this end we introduce a characteristic length L = L (t) 1 such that we can approximate f (t, y) by means of (2.3) for y ≤ L. We attempt to approximate the evolution of the distribution f (t, x) for x L.Then, the coagulation operator in (1.2) can be approximated, using the symmetry properties of the first term on the right of (1.2), as follows We can rewrite this formula as ) where χ [L,∞) denotes the characteristic function of the interval [L, ∞) .We now approximate K [f ] (t, x) for large values of x and t → ∞ and more precisely for x L. To this end we use (2.3) and we assume also that K and f are sufficiently regular for large values of x.Then, using the fact that y x we obtain the following approximation Moreover, (2.1) yields an approximation for K (x, y) if x y.Exchanging the roles of x and y we obtain K (x, y) ∼ x γ+λ y −λ as x y → ∞ and plugging this formula into (2.5)we obtain Using this approximation in (2.4) we then obtain for x 1 and t → ∞.We can now use (2.3) to derive a formula for L 0 y 1−λ f (t, y) dy.We then obtain the approximation Notice that we use that L 0 yη (y) dy ∞ 0 yη (y) dy since by assumption η (y) decreases sufficiently fast for large values of y.
We will denote as f out the distribution of particles in the region where x 1.More precisely we write f out = f χ [L,∞) .Combining (1.1), (1.2) with (2.6) we obtain the following evolution equation for f out Notice that we use the approximation In the rest of the paper we will study the properties of the self-similar solutions associated to the equation (2.7).It is worth to remark that the transport term on the left of (2.7) is the way in which the injection of particles with size x of order one affects the outer distribution of clusters f out .Indeed, multiplying (2.7) by x and integrating we obtain The mass conservation property associated to the coagulation kernel yields On the other hand, integrating by parts in the second term on the left of (2.8) we obtain Combining these results we obtain (2.9) The identity (2.9) states that the total mass of the clusters in the outer region is equal to the injection rate.This result, that holds for long times, could be expected because in the regime described in this section, the injected particles are transferred instantaneously to large cluster sizes.This is also consistent with f (t, x) → 0 as t → ∞ when x ≈ 1.

The case γ + 2λ = 1
In the case γ + 2λ = 1, the approximation of the concentrations of clusters f (t, •) in the region where x is of order one must be obtained in a different manner.The reason is that in this case we cannot expect the moment M γ+λ = ∞ 0 x γ+λ f (t, x)dx to converge to infinity as t → ∞.Indeed, suppose that most of the mass of the monomers is distributed in a characteristic length L(t) that increases as t → ∞.Then, if we denote as M 0 and M 1 the moments of f of order zero and one, respectively, (i.e. Assuming that ∞ 0 xη (x) dx = 1 and hence that M 1 t as t → ∞, we deduce that M 0 L(t) = t.The rescaling properties of (1.1), (1.2) suggest that M0 t (M 0 ) 2 L(t) γ .Therefore, plugging the identity M 0 = t L(t) in this formula we obtain (2.10) It then follows that the self-similar rescaling ansatz implies that M γ+λ remains of order one for large times.A consequence of this is that we cannot approximate (1.1), (1.2) for clusters of order one by means of the equation (2.2).Instead of this we will use a different approximation by splitting f in an outer part which describes the cluster distribution for x of order L and an inner part that describes the cluster distribution for x of order one.More precisely, we write where Using (1.1), (1.2) we would then obtain the following evolution equation for f inner where we have used the decomposition (2.11) in the lost term of the coagulation operator K [f ].We use also the fact that in order to compute the gain term for x of order one we need to use only f inner .
By assumption, the main contribution of f outer (t, y) is due to clusters with size L 1. On the other hand, for x of order one, we have the approximation K (x, y) f outer (t, y) y γ+λ x −λ f outer (t, y) and (2.12) becomes The scaling argument above, (2.10), suggests that ∞ 0 y γ+λ f outer (t, y) dy approaches to a positive constant as t → ∞ if f outer behaves in a self-similar manner and γ + 2λ = 1.We will write On the other hand, we will assume that the function f inner , which is described by means of (2.13), approaches to a stationary solution for large times.We then obtain the following equation which would be expected to describe the behavior of the solutions of (2.13) for long times We can now derive an equation describing the evolution of f outer .To this end we argue as in the case of γ + 2λ > 1 in order to approximate the effect in f outer due to the collisions of clusters with size L with clusters with size of order one.Using approximations analogous to the ones used in the derivation of (2.7) (cf.(2.4), (2.5)) we obtain the following approximation for K for x of order L.
On the other hand, multiplying (2.15) by x and integrating in (0, ∞) we obtain, since Using this formula to eliminate ∞ 0 y 1−λ f inner (y) dy in (2.16) we obtain Therefore, we obtain that f outer satisfies the equation (2.7).
The whole asymptotic behavior derived here relies on the existence of solutions of the equation (2.15).In this equation, the value of M γ+λ is chosen as one associated to a selfsimilar solution of (2.7).The equation (2.15) can then be interpreted as a stationary solution for a coagulation equation with source η and a removal term −M γ+λ f inner (x) x −λ .The results in [10] imply that no solutions of (2.15) exist if M γ+λ = 0 and γ + 2λ = 1, but when M γ+λ > 0 the existence/non-existence of stationary solutions for equation (2.15) is still an open problem.

Setting and main results
The main results of this paper concern the existence and the non-existence of solutions to the following equation We write now precisely what we mean by a solution of equation (3.1) and then we state the main theorems on the existence of a self-similar profile under certain assumptions on the parameters γ and λ (Theorem 5.1), on its properties (Theorem 6.1) and on the non-existence of the self-similar solutions under different assumptions on the parameters γ and λ (Theorem 3.5).
Definition 3.1.Let K be a homogeneous symmetric coagulation kernel satisfying (1.5), (1.6), with homogeneity γ < 1 and with γ + λ < 1 and γ + 2λ ≥ 1.A self-similar profile of equation (2.7) with respect to the kernel K, is a measure x γ+λ Φ(dx) < ∞ and (0,1) and such that it satisfies the following equation 3) are finite.The integrals in the left-hand side of equation (3.3) are bounded due to the fact that ϕ is compactly supported and that Φ is a Radon measure.We analyse now the right hand side.Since γ Using (3.2), the fact that −λ ≤ γ + λ, as well as the fact that ϕ is compactly supported the desired conclusion follows.
Theorem 3.3 (Existence of the self-similar profiles).Let K be a homogeneous symmetric coagulation kernel, of homogeneity γ, satisfying (1.5), (1.6), with γ, λ such that (1.7) holds and such that Then there exists a self-similar profile Φ as in Definition 3.1.Moreover, Φ is such that for some L > 0 and it is absolutely continuous with respect to the Lebesgue measure.Then Φ(dx) = φ(x)dx and the density φ is such that In this paper we do not prove the uniqueness of the self-similar profiles.Therefore, it makes sense to understand if the proven properties for the self-similar profile constructed in Theorem 5.1 hold for each self-similar profile as in Definition 3.1.When γ + 2λ > 1 we prove that each self-similar profile as in Definition 3.1 is zero in the set (0, ρ(M γ+λ )), where ρ(M γ+λ ) is given by (3.4), see Theorem 6.1 for more details.In contrast, when γ + 2λ = 1, we only prove that the self-similar profile constructed in the proof of Theorem 3.3 is such that Φ((0, ρ(M γ+λ ))) = 0.However, we do not know if this property holds for every self-similar profile as in Definition 3.1.
Remark 3.6.Notice that if γ + 2λ > 1 we prove that self-similar solutions as in Definition 3.1 do not exist when γ ≤ −1.Instead, when γ + 2λ = 1 and γ ≤ −1, we do not exclude the existence of a self-similar solution Φ as in Definition 3.1 with

Main ideas of the proofs
In this section we explain the main ideas for the proofs of existence/non-existence of self-similar solutions.Both in the case γ + 2λ > 1 and γ + 2λ = 1, to prove that a self-similar solution exists, we find an invariant region for the evolution equation corresponding to (3.1), namely the following equation By Tychonoff fixed point theorem the existence of an invariant region implies that there exists a solution of equation (3.1).We prove that the set given by To prove that the set P is invariant we proceed as follows.
1. We prove that , for every t > 0. This is done multiplying by x equation (4.1), integrating from zero to infinity and then studying the ODE for the zeroth moment obtained in this manner.
2. As a second step we prove that there exists an upper bound for M γ+λ .To this end we multiply both sides of equation (3.1) by x γ+λ and we integrate over x in (0, ∞) to obtain an ODE for the γ +λ moment.Using Grönwall's lemma, the fact that γ +2λ ≥ 1 and that γ > −1, the desired conclusion follows.
4. Using the fact that γ + λ < 1, γ > −1, Φ(t, (0, ρ(C 1 ))) = 0, as well as the fact that Finally, from the upper bound for M 2−γ−λ , we derive a lower bound for M γ+λ .Indeed, Cauchy-Schwarz inequality implies that Hence 1 C2 ≤ M γ+λ .In order to prove non-existence we proceed by contradiction.Due to the contribution of the coagulation operator in equation (1.16), we expect the zeroth moment of f to decay in time.However, assuming the self-similar change of variable (1.18), we have that dx is constant or increasing in time and this gives a contradiction.Hence we cannot expect self-similar solutions to exist.To make the argument rigorous we will have to prove that 0 < ∞ 0 Φ(dx) < ∞.When γ + 2λ > 1 we do this by proving that for each self-similar profile there exists a δ > 0 such that Φ((0, δ)) = 0 and that Φ tends to zero sufficiently fast as x → ∞.Instead, when γ + 2λ = 1, the methods used in this paper do not allow to prove that Φ is equal to zero near the origin, hence we use (3.5) as well as (3.2) to prove that 0

Existence of a self-similar profile
We aim to prove the existence of a solution of equation (4.1).Namely, we will prove the following theorem, which is just a reformulation of Theorem 3.3.
To this end we prove the existence of a (time-dependent) solution for a suitably truncated and regularized version of equation (4.1): The operator K R is the truncated operator of parameter R > 0 defined as where the kernel K R is a truncated kernel, i.e. it is a continuously differentiable, bounded and symmetric function K R : R 2 * → R + , such that for every (x, y) ∈ R 2 * we have that K R (x, y) ≤ K(x, y) and Notice that the kernel K R can be obtained starting from the kernel K by means of standard truncations and mollifying arguments.
Using Tychonoff fixed point theorem we prove the existence of a stationary solution Φ R for equation (5.1) and we will prove that there exists the limit Φ of {Φ R } R as R tends to infinity.To conclude we will prove that the measure Φ satisfies equation (1.19).

Existence of a time dependent solution for the truncated equation
Since in this section we work only with the truncated equation, we omit the label R in Φ R .We will reintroduce the label R in Section 5.3.We start this section by introducing a definition of solutions for equation (5.1).Definition 5.2.Let K R be the truncated kernel defined as in (5.3) as a function of the homogeneous symmetric coagulation kernel K satisfying (1.5), (1.6), with homogeneity γ < 1 and with γ + λ < 1 and Given the positive numbers k 1 , k 2 , ρ * , we define the subset (5.5) Theorem 5.3.Assume K R to be a truncated kernel defined as in (5.3) as a function of a homogeneous symmetric kernel K satisfying (1.5), (1.6), with parameters γ, λ ∈ R such that (1.7) holds and γ + 2λ ≥ 1 and γ > −1.Then there exist two constants k 1 , k 2 > 0 and a constant given by (4.2), then there exists a unique solution Φ ∈ C 1 ([0, T ]; M + (R * )) of equation (5.1) in the sense of Definition 5.2, for T > 0 sufficiently small.For every t ∈ [0, T ], we have that for some positive constants k 1 and k 2 .Consider the ODE given by (4.2), then (5.7) has a unique solution X(t, x 0 , α) ≥ ρ(k 2 ).Let X(t, x 0,1 , α 1 ) and X(t, x 0,2 , α 2 ) be the solutions of equation (5.7) with respect to the functions α 1 and α 2 satisfying (5.6) and ) where and where we denote by Proof.Since V (x, α) > 0, for every x ≤ ρ(k 2 ), we deduce that the set {x ≥ ρ(k 2 )} is an invariant region of (5.7).This also implies that a unique solution exists.Equation (5.7) is a Bernoulli equation that can be reduced to the linear ODE via the change of variable y = x 1−(λ+γ) .We deduce that Since α 1 , α 2 satisfy (5.6) and since x 0,1 , x 0,2 ∈ [ρ(k 2 ), 4R], the above formula implies that Since the set {x ≥ ρ(k 2 )} is invariant for (5.7), the above inequality, together with the definition of y as a function of x, implies (5.8).
We will solve equation (5.1) using Lagrangian coordinates.To this end we introduce the notation , where α is defined for x > 0, y > 0 and t ≥ 0 as X(t, α (x, y, t), α) = X(t, x, α) + X(t, y, α). (5.9) The function α is well defined because for every fixed time t ≥ 0 the function x → X(t, x, α) is increasing.
Lemma 5.5.Let K R be a truncated kernel defined as in (5.3) as a function of a homogeneous symmetric kernel K satisfying (1.5), (1.6), with γ, λ ∈ R satisfying (1.7).Let k 1 , k 2 , R be three positive constants.Assume that the initial condition Φ 0 is such that given by (4.2).For a sufficiently small time T > 0, depending on R, there exists a function The function F is such that k 1 ≤ α(t) ≤ k 2 and such that (5.12) Proof.We define the set and the set We endow the set M + (R * ) with the Wasserstein metric W 1 .The reason for this choice will become clear in the proof of the contractivity of the evolution operator corresponding to (5.10), (5.11).We endow the set C([0, T ], M + (R * )) with the metric induced by the distance Similarly, we endow C([0, T ]) with the norm f [0,T ] := sup t∈[0,T ] |f (t)|.
For every (F, α) ∈ X T , we define the operator Moreover, given (F, α) ∈ X T , we define the operator We can now rewrite (5.10), (5.11) in a fixed point form, namely as (F, α) = F[F, α] where We prove now that F : X T → X T .Since for every (F, α) ∈ X T , the operator T [F, α](t) is linear and continuous, we deduce that it can be identified with an element of M +,b (R * ).
For x, y such that X(s, x, α) ≥ R or X(s, y, α) ≥ R, we have that K R (X(s, x, α), X(s, y, α)) = 0. Thus, the second and the third terms above are equal to zero.
We now prove that To this end notice that Notice that Finally, we have that the term can be arbitrarily small by taking T small.This is due to the definition of the map T [F, α] and the upper bound for F in the definition of the space X T .We deduce that, for small time T , we have that where the constant C(T ) can be made arbitrarily small.We now check that the map F is a contraction.To this end, we use the fact that |e where X(s, y, β))]F (s, dy) dsΦ 0 (dx) and Then, the differentiability of the kernel K R , together with inequality (5.8), implies that sup Using again the differentiability of the kernel K R , inequality (5.8) and the definition of α , we obtain that sup where T can be selected so that C(T, R)(1 We remark that, in order to obtain this bound, we used the properties of the Wasserstain distance.Similar estimates cannot be obtained by endowing M + (R * ) with the total variation norm.
To understand this better, let α, Using the fact that ||ϕ|| Lip ≤ 1 and that we can assume we work on a sufficiently nice compact set due to the support of the measures, we can derive the following estimate Using (5.8) and (5.9), we have and thus (5.13) More precisely, inequality (5.13) was needed to prove the contractivity of the map T 2 .
In order to prove the upper bound for the map T [F, α], we first observe that and then that for T sufficiently small.Thus, Finally, where the constant T C(T, R)(1+||Φ 0 || T V ) 2 can be made smaller than 1 4 by selecting T small.The operator F is therefore a contraction and, by Banach fixed point theorem, we conclude that it has a unique fixed point (F, α) ∈ X T .
Hence, using again the equality (5.14), we have that We conclude that Φ satisfies (5.4).

Existence of a stationary solution for the truncated equation
Theorem 5.6.Assume K R to be a truncated kernel as in (5.3) defined as a function of the homogeneous symmetric kernel K that satisfies (1.5), (1.6), for parameters γ, λ ∈ R such that (1.7) holds and such that γ + 2λ ≥ 1, γ > −1.There exists R > 0 such that, for every truncation parameter R > R, there exists a Φ ∈ M + (R * ) satisfying the equation for some constants C 1 , C 2 > 0 that do not depend on the truncation R. Additionally, Φ is such that Φ((0, ρ(C 1 ))) = 0 where ρ(C 1 ) is given by (4.2).
To prove the theorem we introduce the semigroup {S(t)} {t≥0} defined as S(0)Φ 0 = Φ 0 and S(t)Φ 0 = Φ(t, •), where Φ is the solution of equation (5.4) constructed in the previous section.We want to apply Tychonoff fixed point theorem to prove the existence of a stationary solution for equation (5.1).To this end we need to find an invariant region for S(t) (Proposition 5.7) and prove the continuity in the weak- * topology of the map Φ → S(t)Φ for every time t (Proposition 5.12).In this way we prove that for every time t there exists a fixed point Φt such that S(t) Φt = Φt .We will then prove that S is continuous (Proposition 5.8) in time and conclude that the limit as t tends to zero of Φt is a solution of equation (5.16).This is a standard method used in the study of coagulation equations to prove existence of self-similar profiles.For more details, see [8, Theorem 1.2].Proposition 5.7 (Invariant region).Assume K R to be a truncated kernel as in (5.3) defined as a function of the homogeneous symmetric kernel K that satisfies (1.5), (1.6), for parameters γ, λ ∈ R such that (1.7) holds and such that γ + 2λ ≥ 1, γ > −1.Then there exist some positive constants C 1 , C 2 that do not depend on the truncation R such that the set is invariant under the evolution operator S(t), where ρ(C 1 ) was defined in (4.2).
Proof.Let Φ 0 and K R be as in Lemma 5.5.Let T be as in Lemma 5.5.We prove that there exist two constants C 1 , C 2 > 0 that do not depend on the truncation parameter R such that the solution Φ obtained in Theorem 5.3 is such that for every time t ∈ [0, T ] Additionally, we prove that we can conclude from (2) that Φ(t, (0, ρ(C 1 )) = 0, for every t ∈ [0, T ].
We first notice that, since Φ(t, •) has compact support, we can consider in equation (5.4) test functions ϕ such that ϕ(x) = x k , for k ∈ R.
Let us prove (1).We consider in equation (5.4) a test function ϕ such that ϕ(x) = x.We deduce that d dt We prove (2).We start by proving this for γ + 2λ > 1.Consider a test function ϕ such that ϕ(x) = x γ+λ in equation (5.4).Then, we can see that the moment M γ+λ satisfies the following since, due to the definition of K R in (5.3), we have that on the set [ρ(2C 1 ), R 4 ] 2 , there exists a constant C > 0 such that K R ≥ K C .We denote by z := y x .Notice that since γ + λ > 0 when y ≤ x we have that As a consequence, by symmetry, we deduce that (5.20) Notice we can take R sufficiently large so that − 2λ+γ−1 1−γ + ε < 0, for every R ≥ R.
Proposition 5.8 (Time-continuity of the semigroup).Assume K R to be a truncated kernel as in (5.3) defined as a function of the homogeneous symmetric kernel K that satisfies (1.5), (1.6), for parameters γ, λ ∈ R such that (1.7) holds and such that γ > −1 and γ + 2λ ≥ 1.
Proof.Let T > 0. We want to estimate the value of Assume without loss of generality that s ≤ t.By the definition of the operator S we know that We have that there exists a constant C 2 > 0 such that M γ+λ (Φ(r)) ≥ 1 C2 , for every r ∈ [0, T ].As Φ has compact support, we can conclude that the exists a constant C > 0, which may depend on ρ(C 1 ), R, T, γ, λ, but independent on s, t, such that thus giving us the desired continuity of the semigroup.Lemma 5.9 (Dual equation).Assume K R to be a truncated kernel as in (5.3) defined as a function of a homogeneous symmetric kernel K that satisfies (1.5), (1.6), for parameters γ, λ ∈ R satisfying (1.7) and with γ > −1 and γ + 2λ ≥ 1.Let Φ 1 , Φ 2 ∈ P be two solutions of (5.4) with initial conditions Φ in,1 , Φ in,2 ∈ P , respectively, and T > 0 be as in Theorem 5.3.Then there exists a unique solution ϕ ∈ , which solves the following equation: where Remark 5.10.We prove the statement of the lemma for a modified operator L(ϕ) which preserves the continuity in the variable ξ, is equal to zero when ξ ≥ 8R and L(ϕ Due to the support of Φ 1 and Φ 2 , when proving the continuity of the map S(t) in the weak- * topology, it suffices to analyse the operator L(ϕ).Thus, it is enough to prove the statement of the lemma only for the operator L(ϕ).Notice that, for example, the operator L(ϕ) does not preserve compactness because of the presence of the term We keep the notation L(ϕ) for simplicity.We omit further details as the proof consists of standard methods used in the study of coagulation equations, see, for example, [10].
Proof of Lemma 5.9.First, we use the method of characteristics.We define X(t, ξ) to be the solution of the ODE x γ+λ with initial condition x(0) = ξ.
In this way equation ( 5.27) can be rewritten in the following fixed point form: where Our strategy for proving the statement of the lemma is to apply Banach fixed point theorem.The operator To this end we notice that We notice that From this we deduce that and hence L is a contraction for sufficiently small times T. We can extend the solution to all possible times noting that the contraction constant c(ρ(C 1 ), Φ 1 , Φ 2 , R) does not depend on the final condition ψ.We thus deduce that there exists a solution ϕ of the fixed point We now prove that the found solution is Lipschitz continuous.
Proposition 5.11.Assume K R to be a truncated kernel as in (5.3) defined as a function of a homogeneous symmetric kernel K such that it satisfies (1.5), (1.6), for parameters γ, λ ∈ R satisfying (1.7) and such that γ > −1 and γ + 2λ ≥ 1.Let T > 0 be as in Theorem 5.
In order to bound the term we need to estimate and For I 1 , by the definition of K R in (5.3), we have K R is C 1 and we can assume it has compact support as we are only interested in the region [ρ(C 1 ), 8R] 2 .Thus, we have that the first derivative of K R is bounded from above.Moreover, there exist constants If we prove that there exists a constant C > 0, which can depend on ρ(C 1 ), R, and the norms of Φ 1 , and Φ 2 , but does not vary depending on the choice of Φ 1 , Φ 2 such that sup then there exists a constant C > 0, which can depend on ρ(C 1 ), R and the norms of Φ 1 and Φ 2 such that Let us now prove (5.29).From equation (5.28) and the fact that K R is in C 1 with compact support, we obtain an upper bound for |∂ 2 ϕ(t, X(t, ξ))|.
Proof.Let δ > 0. In order to prove continuity in the weak- * topology of the semigroup and because of the support of Φ 1 and Φ 2 , it is enough to prove that, if for every then we have that for every where Φ 1 and Φ 2 are the solutions of equation (5.4) with initial conditions Φ in,1 and Φ in,2 , respectively.To this end we notice that since Φ 1 and Φ 2 satisfy equation (5.4) then To simplify the computations, we adopt the notation Φ := Φ 1 − Φ 2 and we notice that Φ satisfies the following equation for every First we make the following notation: Notice that C norm can be bounded from above by Let now ϕ be the solution found in Lemma 5.9 with coagulation kernel K R .By Proposition 5.11, we know that there exists a constant C(t, R, Φ 1 , Φ 2 ), that depends only on the norm of Φ 1 and Φ 2 , but is otherwise independent of the choice of Φ 1 and Φ 2 , such that |ϕ(s, We thus look at the set: The set K ρ(C1),R is totally bounded, thus there exist N ∈ N and χ 1 , . . ., χ Cnorm ).Then we obtain that: ( We can bound T 1 by where in the last step we used (5.31).
Proof of Theorem 5.6.By Proposition 5.12 we know that the operator Φ → S(t)Φ is continuous in the weak- * topology.Additionally, thanks to Proposition 5.7, we know that P is an invariant region for S(t).Since P is also convex and compact and since we have proven that the map t → S(t) is continuous, we apply Theorem 1.2 in [8] to deduce that there exists a Φ such that S(t)Φ = Φ.

Existence of the self-similar profile
To keep the notation lighter, in the previous sections we denoted with Φ the solution of the truncated problem.In this section, since we pass to the limit as R tends to infinity, we add the label R to Φ and we will denote with Φ the self-similar profile.For simplicity, we denote by ρ := ρ(C 1 ), where ρ(C 1 ) was defined in (4.2).
Proof of Theorem 5.1.For every R > 1, sufficiently large, there exist T > 0 and Φ where K R is the coagulation kernel defined in (5.3).Notice that the bounds in Proposition 5.7 are independent of R > 1.
We now prove that the measure Φ in (5.34) satisfies equation (3.3) by taking the limit as R → ∞.Fix ϕ ∈ C c (R * ).We start with passing to the limit in the coagulation term.
Let ε ∈ (0, 1).We first show that for sufficiently large R. We then prove that as R → ∞.For (5.35), we have that: For the first term, we have where we used the definition of K R in (5.3) and the fact that the total mass of the measures is equal to 1.
which gives a small contribution due to the uniform estimates for Φ R if we make R sufficiently large.We now analyse (5.36).We consider a continuous function g : R + → [0, 1] such that g(x) = 1, when x ≤ 1, and g(x) = 0, when x ≥ 2. We define the function p M as as R → ∞.We have that We can analyse the two terms separately.For the first term, we know by (5.25) that the M γ+λ (Φ R ) moment is uniformly bounded from below and thus and then we use that x γ+λ Φ(dx) → 0 by (5.34).
For the second term, we split the integral into a region with compact support and the remainder regions.More precisely, x γ+λ Φ(dx) + 2C [M,∞) x γ+λ Φ R (dx) .
We (5.34).For the regions close to infinity, we use as the total mass is uniformly bounded.Since γ + λ < 1, the remaining term goes to zero as M → ∞.

Properties of the constructed self-similar profile
In this section we aim at proving Theorem 3.3.In particular we prove that the self-similar profile, whose existence has been proven in Theorem 5.1, satisfies the properties stated in Theorem 3.3.In other words we have to prove the following theorem.
Theorem 5.13.Assume K to be a homogeneous symmetric coagulation kernel, of homogeneity γ, satisfying (1.5), (1.6), with γ, λ such that (1.7) holds and such that Let Φ be the self-similar profile constructed in Theorem 5.1.Then R * e Lx Φ(dx) < ∞ for some L > 0 and Φ is absolutely continuous with respect to the Lebesgue measure.Then Φ(dx) = φ(x)dx and φ is such that To prove this theorem we start by proving that every self-similar profile Φ in the sense of Definition 3.1 and such that Φ((0, δ)) = 0, for a positive δ, is absolutely continuous with respect to the Lebesgue measure and satisfies some moment bounds.Proposition 5.14 (Regularity).Assume K to be a homogeneous symmetric coagulation kernel, of homogeneity γ, satisfying (1.5), (1.6), with γ, λ such that (1.7) holds and such that γ + 2λ ≥ 1.Let Φ ∈ M + (R * ) be a self-similar profile as in Definition 3.1.Assume additionally that Φ((0, δ)) = 0 for δ > 0. Then Φ is absolutely continuous with respect to the Lebesgue measure.Its density φ ∈ L 1 ([δ, ∞)) satisfies the following equation (5.44) Proof.We just sketch the proof as similar arguments have been repeatedly used in the analysis of coagulation equations, see for instance the proof of Lemma 4.9 in [9].An analogous proof will be presented in the proof of Proposition 6.2.Let Φ be a solution of (3.1), then Using the fact that dη is a bounded measure in every compact set, we deduce that the measure is absolutely continuous with respect to the Lebesgue measure on the set (δ, ∞) if δ > ρ(M γ+λ ), while it is absolutely continuous with respect to the Lebesgue measure in the set (δ, ρ(M γ+λ )) ∪ (ρ(M γ+λ ), ∞) if δ ≤ ρ(M γ+λ ).In the latter case, we therefore have that Φ(dx) = φ(x)dx + c 0 δ ρ(M γ+λ ) (x) with density φ ∈ L 1 (R * ) and with c 0 ∈ R. To exclude that the self-similar solution Φ has a Dirac in ρ(M γ+λ ), we notice that is non-zero only if ξ = 2ρ(M γ+λ ).Hence the coagulation operator applied to measure c 0 δ ρ(M γ+λ ) produces a Dirac in 2ρ(M γ+λ ), contradicting the fact that Φ |(ρ(M γ+λ ),∞) is absolutely continuous with respect to the Lebesgue measure.As a consequence, we deduce that c 0 = 0.
Lemma 5.16 (Exponential bound).Assume K to be a homogeneous symmetric coagulation kernel, of homogeneity γ, satisfying (1.5), (1.6), with γ, λ such that (1.7) holds and such that γ + 2λ ≥ 1.Let Φ ∈ M + (R) be as in Proposition 5.14 and let φ be its density.Then there exist two positive constants L and M such that ∞ M φ(z)e Lz dz < ∞. (5.49) Proof.Adapting the approach used in [11] we define the function Ψ a as where M > δ.Hence, by its definition We consider a function ψ in (5.48) with ψ (x) := e L min{x,a} to deduce that (5.50) Thus, there exists a c ≥ 0 such that x+y y e L min{w,a} dwφ(x)φ(y)dxdy.
As in [9] and [11] we can deduce, using Jensen's inequality together with the fact that γ + λ < 1 and that −λ < 1, that This, together with the fact that we can take M arbitrary large, implies that for a positive constant c and for c(γ, λ) If we prove that lim sup a→∞ Ψ a (L) < ∞ for some L, then we can conclude.Indeed we would have Let us prove that lim sup a→∞ Ψ a (L) < ∞ for some L. First of all notice that this is true if Ψ a (L) ≤ 1 as in this case Ψ a (L) ≤ L + lim a→∞ Ψ a (0) and the desired conclusion follows.If instead Ψ a > 1, then Without loss of generality we can assume α > 0 such that lim a→∞ Ψ a (0) = α.This implies that a solution of the ODE Ψ exists for small intervals around 0 and is such that lim a→∞ Ψ a (L) < ∞ for L in that interval.
Then there exists a positive constant M such that lim sup z→∞ φ(z)e M z < ∞. (5.52) Proof.From equation (5.43) we deduce that We now show that J φ decays exponentially and that the term First of all, we prove that there exists a constant M 1 > 0 such that J φ (z) ≤ e −M1z for large z.To this end we notice that the bound (5.49) implies that On the other side, for large values of z we have that where M 2 > 0.
We deduce that for large values of z The conclusion follows since, for every z > 6 Non-existence results and properties of the self-similar profiles To study the non-existence of the self-similar solutions we proceed by contradiction and start by assuming that a self-similar solution exists.To find a contradiction we need to analyse the properties of each self-similar profile.In the case γ + 2λ > 1 the fundamental properties that we prove to be true are that Φ((0, δ)) = 0 for some positive δ > 0 and that Φ decays sufficiently fast for large sizes.When γ + 2λ = 1, it is possible to prove that Φ decays fast for large sizes, but we have not been able to prove that Φ((0, δ)) = 0 for some δ > 0. This is the reason why we require the additional condition (3.5) in Theorem 3.5.

Properties of the self-similar profiles
In this section we study the properties of each self-similar solution that do not rely on the existence of the self-similar solution and hence do not rely on the assumption γ > −1.Theorem 6.1 (Properties of the self-similar profiles).Let K be a homogeneous symmetric coagulation kernel, of homogeneity γ, satisfying (1.5), (1.6), with γ, λ such that (1.7) holds.
First of all we prove, see Proposition 6.2, that when γ + 2λ > 1, each solution of equation (1.19) in the sense of Definition 3.1 is zero near the origin.The statement of Theorem 6.1 then follows by Proposition 5.14 and Proposition 5.17 and by the fact that when γ + λ = 1 we are assuming that there exists a δ > 0 such that Φ((0, δ)) = 0 .Proposition 6.2 (Support of the self-similar solution).Let K be a homogeneous symmetric coagulation kernel, of homogeneity γ, satisfying (1.5), (1.6), with γ, λ such that (1.7) holds and such that γ + 2λ > 1.Let Φ ∈ M + (R * ) be a self-similar profile as in Definition 3.1.Then Φ((0, ξ)) = 0 for ξ := min 1 M γ+λ (Φ) Proof.We start explaining the proof in a heuristic manner without entering in the technical details that will be explained later.Equation (1.19) can be rewritten in the following flux form where J φ is given by (5.44).For x smaller than ξ we have that x γ+λ M γ+λ − x ≥ 0 and hence the function is increasing and right-continuous at x = 0.This implies that lim Since for every x ≤ ξ we also have that x γ+λ+1 M γ+λ − 2x 2 1−γ ≥ 0 we deduce that L ≥ 0. We make a scaling argument to identify the value of L. To this end we notice that the units of the flux are given by [J Φ (x)] = [Φ] ].
Due to (6.12) we have that H satisfies Expression (6.15) for H implies that H is negative for z ∈ (δ, ξ).On the other hand, if we assume that φ is different from zero in (0, ξ) we deduce that H is positive on z ∈ (δ, ξ), by definition.This contradiction implies that H(z) = 0 for z ∈ (δ, ξ).
Notice that this is true by assumption when γ + λ = 0, therefore we restrict the attention to the case γ + λ < 0. To this end we adapt the argument used in the proof of Proposition 5.15, we refer there for the technical aspects, and prove that R * x γ+λ Φ(dx) < ∞ implies Hence M 2(γ+λ)+δ−1 < ∞ and the desired conclusion follows.
As a consequence, taking n sufficiently large, we deduce that M 0 (Φ) < ∞.We can therefore consider the test function ϕ ≡ 1 in equation (3.3).This implies which is a contradiction.