Mass-conserving self-similar solutions to coagulation-fragmentation equations

TL;DR

This paper proves the existence of mass-conserving self-similar solutions for certain coagulation-fragmentation equations with small total mass, using a dynamical and compactness approach.

Contribution

It introduces a novel method combining dynamical systems and compactness techniques to establish solutions for a class of coagulation-fragmentation equations.

Findings

Existence of solutions for small total mass

Mass conservation in self-similar solutions

Application of dynamical and compactness methods

Abstract

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularised coagulation-fragmentation equation in scaling variables and a compactness method.

Equations

$${\partial }_{t}f(t,x)$$

$$f(0,x)$$

$$\mathcal{C}f(x):=\frac{1}{2}{\int }_0^{x}K(y,x-y)f(x-y)f(y)\mathrm{d}y-{\int }_0^{\infty }K(x,y)f(x)f(y)\mathrm{d}y,x\in (0,\infty ),$$

$$\mathcal{F}f(x):=-a(x)f(x)+{\int }_x^{\infty }a(y)b(x,y)f(y)\mathrm{d}y,x\in (0,\infty ),$$

$${\int }_0^{y}xb(x,y)\mathrm{d}x=y,y>0,\text{ and }b(x,y)=0,x>y>0;$$

$$M_1(f(t)):={\int }_0^{\infty }xf(t,x)\mathrm{d}x=\varrho =M_1(f^{in}):={\int }_0^{\infty }x{f}^{in}(x)\mathrm{d}x,t\ge 0.$$

$$M_1(f(t))<M_1(f^{in}),t>T_l.$$

$$K(x,y)=K_0\left(x^{\alpha }{y}^{\lambda -\alpha }+x^{\lambda -\alpha }{y}^{\alpha }\right),(x,y)\in (0,\infty {)}^2,$$

$$a(x)=a_0{x}^{\gamma },b(x,y)=b_{\nu }(x,y):=(\nu +2)x^{\nu }{y}^{-\nu -1},0<x<y,$$

$$\alpha =1,\lambda =2,\gamma =1,\nu =0.$$

$$f_r(t,x):=r^{2}f(r^{1-\lambda }t,rx),(t,x)\in [0,\infty )\times (0,\infty ),$$

$$f(t,x)=t^{2/(\lambda -1)}f\left(1,x{t}^{1/(\lambda -1)}\right),(t,x)\in (0,\infty {)}^2,$$

$$(t,x)⟼t^{2/(\lambda -1)}\psi \left(x{t}^{1/(\lambda -1)}\right),(t,x)\in (0,\infty {)}^2.$$

$$\alpha =1,\lambda =2,\gamma =1,\nu =-1,$$

$$\lambda \in (1,2],\gamma :=\lambda -1\in (0,1],\alpha \in \left[\max \left\{\frac{1}{2},\lambda -1\right\},\frac{\lambda }{2}\right],$$

$$a(x)$$

$$K(x,y)$$

$$b(x,y)=\frac{1}{y}B\left(\frac{x}{y}\right),0<x<y,$$

$$B\ge 0\text{ a.e. in }(0,1),B\in L^1((0,1),z\mathrm{d}z),{\int }_0^{1}zB(z)\mathrm{d}z=1,$$

$${\mathfrak{b}}_{m,p}:={\int }_0^1{z}^{m}B(z{)}^p\mathrm{d}z<\infty$$

$${\mathcal{A}}_{\nu }:=\{(m,p)\in (-1,\infty )\times [1,\infty ):m+p\nu >-1\}.$$

$$(m,1)\in {\mathcal{A}}_{\nu }\text{ for all }m>-\nu -1.$$

$$(m,p)\in {\mathcal{A}}_{\nu }\text{ for all }p\in \left[1,\frac{m+1}{|\nu |}\right).$$

$$-\nu -1<\alpha .$$

$${\int }_0^{1}z|\ln z|B(z)\mathrm{d}z\le \underset{z\in (0,1)}{\sup }\left\{z^{(2+\nu )/2}|\ln z|\right\}{\int }_0^1{z}^{-\nu /2}B(z)\mathrm{d}z=\frac{2{\mathfrak{b}}_{-\nu /2,1}}{e(\nu +2)},$$

$${\mathfrak{b}}_{\ln }:={\int }_0^{1}z|\ln z|B(z)\mathrm{d}z<\infty .$$

$${\varrho }_{\star }:=\frac{a_0{\mathfrak{b}}_{\ln }}{2K_0\ln 2}.$$

$$X_m:=L^1((0,\infty ),x^m\mathrm{d}x),M_m(h):={\int }_0^{\infty }{x}^{m}h(x)\mathrm{d}x.$$

$$m_0\in (-\nu -1,\alpha )\cap [0,1),m_1:=\max \{m_0,2-\lambda \}\in (0,1).$$

$$\phi \in X_1^{+}\cap L^{q_1}((0,\infty ),x^{m_1}\mathrm{d}x)\cap \bigcap _{m\ge m_0}{X}_m,M_1(\phi )=\varrho ,$$

$${\int }_0^{\infty }\left[\vartheta (x)-x{\partial }_x\vartheta (x)\right]\phi (x)\mathrm{d}x$$

$$-{\int }_0^{\infty }a(y)N_{\vartheta }(y)\phi (y)\mathrm{d}y$$

$${\Theta }_1:=\left\{\vartheta \in W^{1,\infty }(0,\infty ):\vartheta (0)=0\right\},$$

$${\chi }_{\vartheta }(x,y)$$

$$N_{\vartheta }(y)$$

$$F_S(t,x):=s_{\lambda }(t{)}^2\phi \left(x{s}_{\lambda }(t)\right),(t,x)\in [0,\infty )\times (0,\infty ),$$

$$F_S\in C([0,T],X_{m_1,w})\cap C([0,T],X_{1,w})\cap L^{\infty }((0,T),X_{m_0})$$

Full text

1. Introduction

Coagulation-fragmentation equations are mean-field models describing the time evolution of the size distribution function $f$ of a system of particles varying their sizes due to the combined effect of binary coalescence and multiple breakage. The dynamics of the size distribution function $f(t,x)$ of particles of size $x\in(0,\infty)$ at time $t>0$ is governed by the nonlinear integral equation

[TABLE]

account for the coagulation and fragmentation processes, respectively. In (1.1c), the coagulation kernel $K$ is a non-negative and symmetric function defined on $(0,\infty)^{2}$ and $K(x,y)=K(y,x)$ is the rate at which two particles of respective sizes $x$ and $y$ collide and merge. In (1.1d), $a(x)$ is the overall fragmentation rate of particles of size $x$ and the distribution of the sizes of fragments resulting from the splitting of a particle of size $y$ is the daughter distribution function $x\mapsto b(x,y)$. Since we discard the possibility of loss of matter during breakup, $b$ is assumed to satisfy

[TABLE]

that is, the fragmentation of a particle of size $y$ only produces particles of smaller sizes and no matter is lost. Coagulation being also a mass-conserving process, we expect that matter is conserved throughout time evolution; that is,

[TABLE]

Breakdown in finite time of the identity (1.3) may actually occur; that is, there is $T_{l}\in[0,\infty)$ such that

[TABLE]

This feature is due, either to a runaway growth generated by a coagulation kernel increasing rapidly for large sizes, a phenomenon known as gelation [25, 26, 27], or to the appearance of dust resulting from an overall fragmentation rate $a$ which is unbounded as $x\to 0$, a phenomenon referred to as shattering [13, 28]. Loosely speaking, for the coagulation and fragmentation coefficients given by

[TABLE]

with $\gamma\in\mathbb{R}$, $\nu\in(-2,\infty)$, and $a_{0}>0$, gelation after a finite time occurs when $\alpha>1/2$ in (1.4a) and $\gamma\in(0,\lambda-1)$ in (1.4b) [11, 10, 18, 20, 25, 27], while shattering is observed when $\gamma<0$ in (1.4b) and there is no coagulation $(K_{0}=0)$ [3, 13, 28]. In contrast, mass-conserving solutions to (1.1) satisfying (1.3) for all $t\geq 0$ exist when, either $\lambda\in[0,1]$ and $\gamma\geq 0$, or $\lambda\in(1,2]$ and $\gamma>\lambda-1$ [2, 4, 5, 6, 7, 9, 10, 12, 21, 23, 34, 35, 38]. The previous discussion reveals that the value $\gamma=\lambda-1>0$ is a borderline case with respect to the occurrence of the gelation phenomenon. Indeed, on the one hand, when $\lambda\in(1,2]$, $\gamma=\lambda-1$, and $\alpha>-\nu-1$ in (1.4), mass-conserving solutions to (1.1) on $[0,\infty)$ exist when $M_{1}(f^{in})$ is sufficiently small [19], which is in accordance with numerical simulations performed in [33] for the particular choice

[TABLE]

On the other hand, gelation (in finite time) takes place when $\alpha=1$, $\lambda=2$, $\gamma=1$, $\nu>-1$, and $M_{1}(f^{in})$ is large enough [7, 33, 37].

Besides, the choice $\gamma=\lambda-1>0$ in (1.4) has another interesting feature. Indeed, in this case, equation (1.1a) satisfies a scale invariance which complies with the conservation of matter (1.3). More precisely, if $f$ is a solution to (1.1a) and $r>0$, then the function $f_{r}$ defined by

[TABLE]

is also a solution to (1.1a) and $M_{1}(f_{r}(t))=M_{1}(f(r^{1-\lambda}t))$ for $t\geq 0$. We then look for particular solutions to (1.1a) which are left invariant by the transformation (1.6), that is, $f_{r}=f$ for all $r>0$; that is, according to (1.6), $r^{2}f(r^{1-\lambda}t,rx)=f(t,x)$ for all $(r,t,x)\in(0,\infty)^{3}$. The choice $r=t^{1/(\lambda-1)}$ in the previous identity gives

[TABLE]

and raises the question of the existence of mass-conserving self-similar solutions of the form

[TABLE]

In (1.7), the profile $\psi$ is yet to be determined and is requested to have a finite total mass $M_{1}(\psi)=\varrho\in(0,\infty)$. According to the numerical simulations performed in [33], such solutions exist for sufficiently small values of $\varrho$ and are expected to describe the long term dynamics of mass-conserving solutions to (1.1) with the same total mass $\varrho$. Thus, the existence, uniqueness, and properties of mass-conserving self-similar solutions to (1.1a) of the form (1.7) are of high interest.

The purpose of this paper is to provide one step in that direction and figure out whether self-similar solutions to (1.1a) of the form (1.6) do exist when $\gamma=\lambda-1>0$ in (1.4). Such a quest is not hopeless. Indeed, on the one hand, when the parameters in (1.4) are given by (1.5), their existence is supported by numerical simulations performed in [33], which indicate that there exist mass-conserving self-similar solutions to (1.1a) of the form (1.7) with $M_{1}(\psi)=\varrho$, provided the ratio $a_{0}/(\varrho K_{0})$ is large enough. On the other hand, if

[TABLE]

then, for any $\varrho>0$, the existence of a unique mass-conserving self-similar solution to (1.1a) of the form (1.7) with $M_{1}(\psi)=\varrho$ is shown in [24] and this particular solution is a global attractor for the dynamics of (1.1) when the initial condition $f^{in}$ satisfies $M_{1}(f^{in})=\varrho$. The approach developed in [24] heavily relies on the specific structure of (1.1a) for the choice of parameters (1.8), which allows us to use the Laplace transform, and is thus not likely to be adapted to the more general setting considered herein. Instead, we first construct mass-conserving self-similar solution to (1.1a) of the form (1.7) for a restricted class of daughter distribution functions $b$ by a dynamical approach and carefully keep track of the dependence of the estimates on the various parameters involved in $K$, $a$, and $b$. We next use a compactness method to extend the existence result to a broader class of $b$.

Specifically, we consider

[TABLE]

Observe that $\mathcal{A}_{\nu}$ is non-empty since

[TABLE]

We finally assume that the small size behaviour of the coagulation kernel $K$ is related to the possible singularity of $B$ for small sizes and require

[TABLE]

Since $(-\nu/2,1]\in\mathcal{A}_{\nu}$ by (1.10), we infer from (1.9f) and the inequality

[TABLE]

that

[TABLE]

We then set

[TABLE]

For $m\in\mathbb{R}$, we define the weighted $L^{1}$-space $X_{m}$ and the moment $M_{m}(h)$ of order $m$ of $h\in X_{m}$ by

[TABLE]

We also denote the positive cone of $X_{m}$ by $X_{m}^{+}$, while $X_{m,w}$ denotes the space $X_{m}$ endowed with its weak topology.

For the above described class of coagulation and fragmentation coefficients, the main result of this paper guarantees the existence of at least one mass-conserving self-similar solution to (1.1a) of the form (1.7) (up to a rescaling, see Remark 1.2 below) with a sufficiently small total mass $\varrho$.

Theorem 1.1**.**

Consider coagulation and fragmentation coefficients $K$, $a$, and $b$ satisfying (1.9) and fix two auxiliary parameters

[TABLE]

Let $\varrho\in(0,\varrho_{\star})$.

• (a)

There are $q_{1}\in(1,2)$ (defined in (2.9) below) and a non-negative profile

[TABLE]

such that $(m_{1},q_{1})\in\mathcal{A}_{\nu}$ and

[TABLE]

for all $\vartheta\in\Theta_{1}$, where

[TABLE]

and

[TABLE]

• (b)

The function $F_{S}$ defined by

[TABLE]

with $s_{\lambda}(t):=(1+(\lambda-1)t)^{1/(\lambda-1)}$, $t\geq 0$, is a mass-conserving weak solution to (1.1) on $[0,\infty)$ with initial condition $f^{in}=\varphi$ in the following sense: for any $T>0$,

[TABLE]

and satisfies

[TABLE]

for all $t\in(0,\infty)$ and $\vartheta\in\Theta_{m_{1}}$, where $\Theta_{0}:=L^{\infty}(0,\infty)$ and

[TABLE]

Remark 1.2**.**

The self-similar ansatz (1.7) differs slightly from that of $F_{S}$ in Theorem 1.1, see (1.20). However, they can both be mapped to each other, up to an $X_{1}$-invariant dilation of the profile. Indeed, if $F_{S}(t,x)=s_{\lambda}(t)^{2}\varphi\left(xs_{\lambda}(t)\right)$, $(t,x)\in(0,\infty)^{2}$, is a mass-conserving self-similar solution to (1.1a) of the form (1.20), then it is actually well-defined for $(t,x)\in(-1/(\lambda-1),\infty)\times(0,\infty)$. Combining this property with the autonomous character of the coagulation-fragmentation equation (1.1a) implies that $\tilde{F}_{S}(t,x):=F_{S}(t-(\lambda-1)^{-1},x)$, $(t,x)\in(0,\infty)^{2}$, is also a solution to (1.1a) and satisfies

[TABLE]

with $\psi(y)=(\lambda-1)^{-2/(\lambda-1)}\varphi\left(y(\lambda-1)^{-1/(\lambda-1)}\right)$, $y>0$. In other words, $\tilde{F}_{S}$ is a mass-conserving self-similar solution to (1.1a) of the form (1.7) and it has total mass $\varrho$, since $M_{1}(\varphi)=M_{1}(\psi)=\varrho$ by (1.15).

On the one hand, Theorem 1.1 and Remark 1.2 provide the existence of mass-conserving self-similar solutions to (1.1) of the form (1.7) with a sufficiently small total mass for the parameters given by (1.5), which is in perfect agreement with the numerical simulations performed in [33]. It is yet unclear whether $\varrho_{\star}$ is the largest value of $\varrho$ for which a mass-conserving self-similar solution to (1.1) of the form (1.7) with total mass $\varrho$ exists. However, Theorem 1.1 cannot be valid for any $\varrho>0$ in general. Indeed, when the parameters in (1.9) are given by (1.5), gelation occurs for sufficiently large mass, as indicated by explicit computations performed in [33, 37] and proved in [7] when $a_{0}/(\varrho K_{0})<1$. On the other hand, Theorem 1.1 provides the existence of mass-conserving self-similar solutions to (1.1) of the form (1.7) with a sufficiently small total mass for the parameters given by (1.8), a result which is far from optimal, since such a solution exists for any value of the total mass, according to [24]. A possible explanation for this discrepancy is that the absence of a threshold mass is due to the non-integrability as $x\to 0$ of the daughter distribution function $b_{-1}$, which is not really exploited in the proof of Theorem 1.1 below.

Let us now describe the approach we use in this paper to prove Theorem 1.1. Owing to the homogeneity of $K$, $a$, and $B$, inserting the ansatz (1.20) in (1.1a) implies that $\varphi$ solves the integro-differential equation

[TABLE]

Unfortunately, the equation (1.22) seems hardly tractable as an initial value problem with initial condition at $y=0$. Indeed, on the one hand, the right hand side of (1.22) depends not only on the past $(0,y)$ of $y$ but also on its future $(y,\infty)$. On the other hand, the left hand side is degenerate, as the factor $y$ in front of $\mathrm{d}\varphi/\mathrm{d}y$ vanishes at $y=0$. Assuming further that $y^{2}\varphi(y)\to 0$ as $y\to 0$, one can get rid of the derivative in (1.22) and show that $\varphi$ also satisfies the nonlinear integral equation

[TABLE]

for $y\in(0,\infty)$, see [13, 36]. It is however unclear whether this alternative formulation is more helpful than (1.22) to investigate the existence issue, though it has been extensively used to determine the behaviour for small and large sizes of the profile of mass-conserving self-similar solutions to the coagulation equation [16, 26, 31, 32, 36]. We thus employ a different approach here, which has already proved successful for the coagulation equation [12, 15, 32] and the fragmentation equation [12, 29]. It relies on the construction of a convex and compact subset of $X_{1}$ which is left invariant by the evolution equation associated to (1.22). This evolution equation is actually obtained from (1.1) by using the so-called scaling or self-similar variables. More precisely, recalling that $s_{\lambda}(t)=(1+(\lambda-1)t)^{1/(\lambda-1)}$, $t\geq 0$, we introduce the scaling variables

[TABLE]

and the rescaled size distribution function

[TABLE]

Equivalently,

[TABLE]

Now, if $f$ is a solution to (1.1), then $g$ solves

[TABLE]

Comparing (1.22) and (1.26a), we readily see that $\varphi$ is a stationary solution to (1.26a), so that proving Theorem 1.1 amounts to find a steady-state solution to (1.26a). To this end, we shall use a consequence of Schauder’s fixed point theorem which guarantees the existence of a steady state for a dynamical system defined in a closed subset $Y$ of a Banach space $X$ which leaves invariant a convex and compact subset of $Y$, see [1, Proposition 22.13] and [17, Proof of Theorem 5.2] (see also [12, Theorem 1.2] for the extension of this result to a Banach space endowed with its weak topology). Applying the just mentioned result requires identifying a suitable functional framework in which, not only (1.26) is well-posed, but also leaves invariant a convex and compact subset of the chosen function space. To achieve this goal, the assumption (1.9f) for any $(m,p)\in\mathcal{A}_{\nu}$ does not seem to be sufficient and we first construct a family $(b_{\varepsilon},B_{\varepsilon})_{\varepsilon\in(0,1)}$ of approximations of $(b,B)$, which satisfy not only (1.9d) and (1.9e), but also (1.9f) for any $(m,p)\in\mathcal{A}_{0}$ and $B_{\varepsilon}\in W^{1,1}(0,1)$. We then prove that the corresponding rescaled coagulation-fragmentation equation (1.26) is well-posed in $X_{1}$ for initial conditions $f^{in}\in X_{m_{0}}^{+}\cap X_{1+\lambda}$ satisfying $M_{1}(f^{in})=\varrho\in(0,\varrho_{\star})$. We also show the existence of an invariant convex and compact subset $\mathcal{Z}_{\varepsilon}$ of $X_{1}$ for the associated dynamical system. According to the above mentioned result, this analysis guarantees the existence of a stationary solution $\varphi_{\varepsilon}\in X_{1}^{+}$ to (1.26a) satisfying $M_{1}(\varphi_{\varepsilon})=\varrho$. Moreover, it turns out that there is a convex and sequentially weakly compact subset $\mathcal{Z}$ of $X_{1}$ such that $\mathcal{Z}_{\varepsilon}\subset\mathcal{Z}$ for all $\varepsilon\in(0,1)$. Consequently, $(\varphi_{\varepsilon})_{\varepsilon\in(0,1)}$ is relatively sequentially weakly compact in $X_{1}$ and the information derived from $\mathcal{Z}$ allows us to prove that cluster points in $X_{1,w}$ of $(\varphi_{\varepsilon})_{\varepsilon\in(0,1)}$ as $\varepsilon\to 0$ solve (1.22), thereby completing the proof of Theorem 1.1.

Remark 1.3**.**

In the companion paper [19], we prove that, given an initial condition $f^{in}\in X_{m_{0}}^{+}\cap X_{2\lambda-\alpha}$ satisfying $M_{1}(f^{in})=\varrho\in(0,\varrho_{\star})$, the coagulation-fragmentation equation (1.1) has a unique mass-conserving weak solution on $[0,\infty)$ under the same assumptions (1.9) on the coagulation and fragmentation coefficients. This result is perfectly consistent with the numerical simulations performed in [33], as is Theorem 1.1.

2. Self-similar solutions: a regularised problem

In this section, we assume that $K$, $a$, and $b$ are coagulation and fragmentation coefficients satisfying (1.9) and we fix $\varrho\in(0,\varrho_{\star})$.

As already mentioned, two steps are needed to prove Theorem 1.1 and this section is devoted to the first step; that is, the proof of Theorem 1.1 for a family $(b_{\varepsilon})_{\varepsilon>0}$ of approximations of the daughter distribution function $b$. We begin with the construction of a suitably regularised version of the daughter distribution function $b$. To this end, we fix a non-negative function $\zeta\in C_{0}^{\infty}(\mathbb{R})$ such that

[TABLE]

and set $\zeta_{\varepsilon}(z):=\varepsilon^{-2}\zeta(z\varepsilon^{-2})$ for $z\in\mathbb{R}$ and $\varepsilon\in(0,1)$. For $\varepsilon\in(0,1)$, we define

[TABLE]

As we shall see below, see (2.2b), the parameter $\beta_{\varepsilon}$ is positive for $\varepsilon>0$ sufficiently small, so that $B_{\varepsilon}$ is well-defined for such values of $\varepsilon$. Indeed, thanks to (1.9e), (1.9f), and the properties of $\zeta$,

[TABLE]

An obvious consequence of (2.2c) is that

[TABLE]

where

[TABLE]

Recalling that $1+\mathfrak{b}_{1+\lambda-\alpha,1}>2\mathfrak{b}_{1+\lambda-\alpha,1}$ due to $1+\lambda-\alpha>1$, it follows from (2.2b) and (2.3) that there is $\varepsilon_{0}\in(0,1)$ such that, for $\varepsilon\in(0,\varepsilon_{0})$,

[TABLE]

An immediate consequence of (2.4) is that, for $\varepsilon\in(0,\varepsilon_{0})$,

[TABLE]

Morever,

[TABLE]

and

[TABLE]

Remark 2.1**.**

In fact, if the function $B$ in (1.9e) satisfies (1.9f) for any $(m,p)\in\mathcal{A}_{0}$, as well as $B(0)=0$ and $B\in W^{1,1}(0,1)$, then we may take $B_{\varepsilon}=B$. This is true in particular for the parabolic daughter distribution function corresponding to $B(z)=12z(1-z)$, $z\in(0,1)$.

Next, since $\varrho\in(0,\varrho_{\star})$, we infer from (2.3) that there is $\varepsilon_{\varrho}\in(0,\varepsilon_{0})$ such that

[TABLE]

Finally, since $m_{1}+\lambda-1\in(m_{0},\lambda)$ by (1.9a) and $(m_{1},1)\in\mathcal{A}_{\nu}$ by (1.10a), we may fix

[TABLE]

The main result of this section is then the following:

Proposition 2.2**.**

Let $\varepsilon\in(0,\varepsilon_{\varrho})$. There is

[TABLE]

such that $M_{1}(\varphi_{\varepsilon})=\varrho$ and

[TABLE]

for all $\vartheta\in\Theta_{1}$, where $\Theta_{1}$ is defined in (1.17) and

[TABLE]

Moreover,

[TABLE]

• (a)

There is $\ell>0$ depending only on $\lambda$, $\alpha$, $K_{0}$, $a_{0}$, $B$, $\nu$, $m_{0}$, $m_{1}$, $q_{1}$, and $\varrho$ such that

[TABLE]

• (b)

For all $m\geq 1+\lambda$, there is $L(m)>0$ depending only on $\lambda$, $\alpha$, $K_{0}$, $a_{0}$, $B$, $\nu$, $m_{0}$, $m_{1}$, $q_{1}$, $\varrho$, and $m$ such that

[TABLE]

The main steps in the proof of Proposition 2.2 are the derivation of (2.11a) and (2.11c). The former is inspired from [12, Lemma 4.2] and combines a differential inequality for a superlinear moment, involving here the weight $x\mapsto x\ln{x}$, and a differential inequality for a sublinear moment. The validity of (2.11a) requires the smallness condition $\varrho\in(0,\varrho_{\star})$, the value of $\varrho_{\star}$ being prescribed by an algebraic inequality established in [19, Lemma 2.3], see (2.20) below. As for (2.11c), it relies on the monotonicity of $x\mapsto x^{m_{1}}K(x,y)$ to handle the contribution of the coagulation term, similar arguments being used in [7, 8, 22, 30] to derive $L^{p}$-estimates for solutions to coagulation-fragmentation equations.

2.1. Scaling variables and well-posedness

Let $\varepsilon\in(0,\varepsilon_{\varrho})$. We begin with the existence and uniqueness of a mass-conserving weak solution to

[TABLE]

where $\mathcal{F}_{\varepsilon}$ denotes the fragmentation operator with $b$ replaced with $b_{\varepsilon}$.

Proposition 2.3**.**

Consider an initial condition $f^{in}\in X_{1}^{+}\cap X_{m_{0}}\cap X_{2\lambda-\alpha}$ such that

[TABLE]

There is a unique mass-conserving weak solution $g_{\varepsilon}$ to (1.1) on $[0,\infty)$ satisfying

[TABLE]

[TABLE]

and

[TABLE]

for all $s\in(0,\infty)$ and $\vartheta\in\Theta_{m_{1}}$, where $\Theta_{0}:=L^{\infty}(0,\infty)$ and

[TABLE]

We recall that $N_{\vartheta,\varepsilon}$ in (2.15) is defined in Proposition 2.2,

Proof.

Owing to (1.9a), (1.9b), (1.9c), (2.1c), (2.2a), and the integrability properties of $B_{\varepsilon}$, we are in a position to apply [19, Theorem 1.2], which guarantees the existence and uniqueness of a mass-conserving weak solution $f_{\varepsilon}$ to the coagulation-fragmentation equation

[TABLE]

which satisfies

[TABLE]

for any $T>0$ and $M_{1}(f_{\varepsilon}(t))=\varrho$ for $t\geq 0$. Setting

[TABLE]

completes the proof of Proposition 2.3. ∎

The next results are devoted to the derivation of a series of estimates satisfied by the weak solutions to (2.12) provided by Proposition 2.3, except for Lemma 2.12 where the continuous dependence of $\Psi_{\varepsilon}(\cdot;f^{in})$ in $X_{1}$ with respect to the initial condition is established.

Throughout the remainder of this section, $\kappa$ and $(\kappa_{i})_{i\geq 1}$ are positive constants depending only on $\lambda$, $\alpha$, $K_{0}$, $a_{0}$, $B$, $\nu$, $m_{0}$, $m_{1}$, $q_{1}$, and $\varrho$. Dependence upon additional parameters is indicated explicitly.

2.2. Moment Estimates

We begin with the derivation of estimates for moments of order $m\in[m_{1},1]$, the parameter $m_{1}$ being defined in (1.14).

Lemma 2.4**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ such that $M_{1}(f^{in})=\varrho$ and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). For $m\in[m_{1},1)$, there is $\kappa_{\ref{kap1}}(m)>0$ depending on $m$ such that, for $t\geq 0$,

[TABLE]

Proof.

Let $s\geq 0$ and consider $m\in[m_{1},1)$. Then

[TABLE]

and

[TABLE]

Consequently, we infer from (1.9b), (2.5), (2.15) (with $\vartheta(x)=x^{m}$, $x>0$), and the non-negativity of $g_{\varepsilon}$ and $K$ that

[TABLE]

Observing that $m+\lambda-1\in[1,\lambda)$, it follows from (2.14) and Hölder’s inequality that

[TABLE]

We combine the previous two inequalities and use Young’s inequality (since $m+\lambda-2<\lambda-1$) to obtain

[TABLE]

with

[TABLE]

We next set $\bar{\vartheta}(x)=x\ln{x}$ for $x\geq 0$ and recall the inequality

[TABLE]

established in [19, Lemma 2.3], along with the following consequence of (1.9a), (1.9c), and Young’s inequality

[TABLE]

Also, by (2.1c) and (2.2a),

[TABLE]

We then infer from (1.9b), (1.9c), (2.8), (2.14), and (2.15) (with $\vartheta=\bar{\vartheta}$) that

[TABLE]

the parameter $\delta_{\varrho}$ being defined in (2.19). Combining (2.18) and the previous inequality, we find

[TABLE]

where

[TABLE]

Since

[TABLE]

there holds

[TABLE]

Consequently, setting $\kappa_{\ref{kap2}}(m):=\min\{1-m,\delta_{\varrho}\}$ and using once more (2.14), we obtain

[TABLE]

Integrating with respect to $s$ gives

[TABLE]

for $s\geq 0$ and Lemma 2.4 follows with $\kappa_{\ref{kap1}}(m):=\kappa_{\ref{kap2.5}}(m)/\kappa_{\ref{kap2}}(m)$. ∎

From now on, we assume that $f^{in}$ satisfies

[TABLE]

A straightforward consequence of Lemma 2.4 is the following estimate.

Corollary 2.5**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ satisfying (2.22) and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). There is $\kappa_{\ref{kap3}}>0$ such that

[TABLE]

Proof.

Let $s\geq 0$. Since

[TABLE]

it follows from (2.22) and Lemma 2.4 (with $m=m_{1}$) that

[TABLE]

from which Corollary 2.5 follows. ∎

Thanks to Corollary 2.5, we may derive additional information on the behaviour of $g_{\varepsilon}$ for large sizes.

Lemma 2.6**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ satisfying (2.22) and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). Assume also that $f^{in}\in X_{m}$ for some $m>1+\lambda-\alpha$. Then there is $\kappa_{\ref{kap4}}(m)>0$ depending on $m$ such that

[TABLE]

Proof.

Let $s\geq 0$. We infer from (2.2) and (2.15) that

[TABLE]

with

[TABLE]

On the one hand, since $\lambda>1$, it follows from (2.14), (2.22), and Hölder’s inequality that

[TABLE]

Equivalently,

[TABLE]

In addition, by (2.5),

[TABLE]

Consequently,

[TABLE]

with

[TABLE]

On the other hand, to estimate the contribution of the coagulation term, we argue as in [19, Lemma 2.6]. Since $m>1$, there is $c_{m}>0$ depending only on $m$ such that

[TABLE]

and it follows from (1.9c) and the previous inequality that

[TABLE]

Owing to (1.9a) and $m>1+\lambda-\alpha\geq 1+\alpha$, both $m+\lambda-\alpha-1$ and $m+\alpha-1$ belong to $[1,m]$ and we deduce from (2.14), (2.22), and Hölder’s inequality that

[TABLE]

Also, introducing

[TABLE]

and noticing that $1<1+\alpha\leq 1+\lambda-\alpha<m$, we infer from (2.14), (2.22), and Hölder’s inequality that, for $R>1$,

[TABLE]

and

[TABLE]

Collecting the above estimates, we find

[TABLE]

for $R>1$. Owing to Corollary 2.5,

[TABLE]

Introducing $R_{m}>1$ defined by

[TABLE]

and taking $R=R_{m}$ in the previous estimate on $P_{m,\varepsilon}(s)$ give

[TABLE]

Since $m+\alpha-2<m+\lambda-2$ and $m+\lambda-\alpha-2<m+\lambda-2$, we apply Young’s inequality to obtain

[TABLE]

We now combine (2.23), (2.24), and (2.26) and obtain

[TABLE]

Hence, using once more Young’s inequality,

[TABLE]

with $\kappa_{\ref{kap4}}(m):=(\kappa_{\ref{kap6}}(m)/\delta_{\varrho,m})^{(m-1)/(m+\lambda-2)}$. Lemma 2.6 is then a consequence of the comparison principle. ∎

We finally return to the behaviour for small sizes.

Lemma 2.7**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ satisfying (2.22) and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). For $m\in[m_{0},m_{1})$, there is $\kappa_{\ref{kap7}}(m)>0$ depending on $m$ such that, if $f^{in}\in X_{m}$, then

[TABLE]

where

[TABLE]

Proof.

We first note that $\mathcal{M}_{1+\lambda,\varepsilon}$ is indeed finite according to Lemma 2.6. Next, let $s\geq 0$. As at the beginning of the proof of Lemma 2.4, we infer from (2.2), (2.5), and (2.12) that

[TABLE]

Since $m+\lambda-1\in(m,1+\lambda)$, we deduce from Hölder’s inequality that

[TABLE]

Consequently,

[TABLE]

with $\kappa_{\ref{kap7}}(m):=(a_{0}(1+\mathfrak{b}_{m_{0},1})/(1-m))^{(1+\lambda-m)/(\lambda-1)}$. Lemma 2.7 follows from the above differential inequality and the comparison principle. ∎

Up to now, we have derived estimates which do not depend on $\varepsilon\in(0,\varepsilon_{\varrho})$ and which will thus be of utmost importance in the next section to take the limit $\varepsilon\to 0$. However, these estimates do not provide enough control on the behaviour for small sizes for the proof of Proposition 2.2, for which the next result is required.

Lemma 2.8**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ satisfying (2.22) and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). For $m\in(-1,0]$, there is $\kappa_{\ref{kap100}}(m,\varepsilon)>0$ depending on $m$ and $\varepsilon$ such that

[TABLE]

Proof.

The proof is exactly the same as that of Lemma 2.7 with the only difference that $\mathfrak{b}_{m,1,\varepsilon}$ cannot be bounded from above by a constant which does not depend on $\varepsilon$ for all $m\in(-1,0]$, though it is finite due to (2.6). ∎

2.3. Weighted $L^{q_{1}}$-estimate

The last estimate which does not depend on $\varepsilon\in(0,\varepsilon_{\varrho})$ is the following weighted $L^{q_{1}}$-estimate, the exponent $q_{1}$ being defined in (2.9).

Lemma 2.9**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ satisfying (2.22) and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). If $f^{in}$ also belongs to $L^{q_{1}}((0,\infty),y^{m_{1}}\mathrm{d}y)$, then there is $\kappa_{\ref{kap8}}>0$ such that

[TABLE]

where $\mu_{1}:=(m_{1}+1+q_{1}(\lambda-2))/q_{1}>m_{0}$ and

[TABLE]

Proof.

We first observe that, as $\mu_{1}\in(m_{0},\lambda)$ by (2.9), Lemma 2.6, Lemma 2.9, and Hölder’s inequality imply that $\mathcal{M}_{\mu_{1},\varepsilon}$ is finite. We next set

[TABLE]

and infer from (2.12) that

[TABLE]

On the one hand, we use a monotonicity argument as in [8, 19, 22, 30] to estimate the contribution of the coagulation term. More precisely, thanks to the symmetry of $K$ and the subadditivity of $x\mapsto x^{m_{1}}$,

[TABLE]

We now use Young’s inequality to obtain

[TABLE]

Owing to the monotonicity of $x\mapsto x^{m_{1}}K(x,y)$ for all $y\in(0,\infty)$, the right hand side of the previous inequality is non-positive. Consequently,

[TABLE]

On the other hand, it follows from (1.9b), (2.1c), and Fubini’s theorem that

[TABLE]

Since

[TABLE]

by (2.4) and Hölder’s inequality, we conclude that

[TABLE]

Collecting (2.27), (2.28), and (2.29), we end up with

[TABLE]

with $\kappa_{\ref{kap8}}=(a_{0}q_{1})^{q_{1}}(1+\mathfrak{b}_{m_{1},q_{1}})/(2q_{1}-m_{1}-1)^{q_{1}}$. Lemma 2.9 follows from the above differential inequality by the comparison principle. ∎

2.4. $W^{1,1}$-estimate

It turns out that the weighted $L^{q_{1}}$-estimate derived in Lemma 2.9, though at the heart of the proof of Theorem 1.1, is not sufficient to prove Proposition 2.2, and the final estimate needed for the proof of Proposition 2.2 is the following $W^{1,1}$-estimate which depends strongly on $\varepsilon\in(0,\varepsilon_{\varrho})$.

Lemma 2.10**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in X_{0}^{+}\cap X_{1+\lambda}$ satisfying (2.22) and let $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$ be given by (2.17). Assume also that $f^{in}\in X_{\lambda-2}\cap W^{1,1}(0,\infty)$. Then there is $\kappa_{\ref{kap9}}(\varepsilon)>0$ depending on $\varepsilon$ such that

[TABLE]

where

[TABLE]

Proof.

We first note that $\mathcal{M}_{\lambda-2,\varepsilon}$ is finite according to Lemma 2.8, as $\lambda-2\in(-1,0)$ by (1.9a). Introducing $G_{\varepsilon}:=\partial_{x}g_{\varepsilon}$, $\Sigma_{\varepsilon}:=\mathrm{sign}(G_{\varepsilon})$, and using that $K(x,0)=0$, it follows from (2.12a) that $G_{\varepsilon}$ solves

[TABLE]

for $(s,x)\in(0,\infty)^{2}$, where $\partial_{1}K$ and $\partial_{1}b_{\varepsilon}$ denote the partial derivatives with respect to the first variable of $K$ and $b_{\varepsilon}$, respectively.

Let $s\geq 0$. We multiply (2.30) by $\Sigma_{\varepsilon}$, integrate with respect to $x$ over $(0,\infty)$ and then infer from (1.9b), (2.1c), and Fubini’s theorem that

[TABLE]

Setting

[TABLE]

which is finite according to (2.6), and observing that

[TABLE]

due to (1.9a) and (1.9c), we end up with

[TABLE]

We next infer from (1.9a) and Hölder’s inequality that

[TABLE]

so that, by (2.14) and (2.22),

[TABLE]

Collecting the above inequalities and using (2.7), we conclude that

[TABLE]

with $\kappa_{\ref{kap9}}(\varepsilon):=\left(a_{0}\bar{B}_{\varepsilon}+3\varrho K_{0}\right)/2$. Integrating the previous differential inequality gives Lemma 2.10. ∎

2.5. Invariant Set

The analysis performed in the previous three sections now allows us to construct a compact and convex subset of $X_{1}$ which is left invariant by (2.12). Let us first recall that, owing to (2.9), the parameter $\mu_{1}$ (defined in Lemma 2.9) satisfies

[TABLE]

For $\varepsilon\in(0,\varepsilon_{\varrho})$, we define the subset $\mathcal{Z}_{\varepsilon}$ of $X_{1}^{+}$ as follows: $h\in\mathcal{Z}_{\varepsilon}$ if and only if $h$ satisfies the following conditions:

[TABLE]

Note that we may assume that $E_{\varrho}:x\mapsto\varrho e^{-x}$ belongs to $\mathcal{Z}_{\varepsilon}$, after possibly taking larger constants in (2.32) without changing their dependence with respect to the involved parameters. In particular, $\mathcal{Z}_{\varepsilon}$ is non-empty.

As we shall see now, the outcome of the analysis performed in the previous sections provides the invariance of $\mathcal{Z}_{\varepsilon}$ for the dynamics of (2.12) when $\varepsilon\in(0,\varepsilon_{\varrho})$.

Lemma 2.11**.**

Consider $\varepsilon\in(0,\varepsilon_{\varrho})$ and $f^{in}\in\mathcal{Z}_{\varepsilon}$. Then $\Psi_{\varepsilon}(s;f^{in})\in\mathcal{Z}_{\varepsilon}$ for all $s\geq 0$. Furthermore, $\mathcal{Z}_{\varepsilon}$ is a non-empty, convex, and compact subset of $X_{1}$.

Proof.

Let $f^{in}\in\mathcal{Z}_{\varepsilon}$. Setting $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$, see (2.17), it satisfies (2.14) by Lemma 2.4, from which we readily obtain that $g_{\varepsilon}(s)\in X_{1}^{+}$ and $M_{1}(g_{\varepsilon}(s))=\varrho$ for all $s\geq 0$.

Next, let $s\geq 0$. We infer from (2.32b) and Lemma 2.4 (with $m=m_{1}$) that $g_{\varepsilon}(s)$ satisfies (2.32b). Also, since $f^{in}$ satisfies (2.22) according to (2.32b), we are in a position to apply Lemma 2.6 for $m\geq 1+\lambda>1+\lambda-\alpha$ and deduce from (2.32c) for $f^{in}$ that (2.32c) is satisfied by $g_{\varepsilon}(s)$ for any $m\geq 1+\lambda$. This property (with $m=1+\lambda$) along with Lemma 2.7 (with $m=m_{0}$) guarantees that $g_{\varepsilon}(s)$ satisfies (2.32d). We further use (2.32c) (with $m=1+\lambda$) and (2.32d) that we just established for $g_{\varepsilon}$ together with (2.31) and Hölder’s inequality to obtain

[TABLE]

Hence, $g_{\varepsilon}(s)$ satisfies (2.32e) for $s\geq 0$. We now combine the just established property (2.32e) for $g_{\varepsilon}$ with Lemma 2.10 and realize that $g_{\varepsilon}(s)$ satisfies (2.32f) for $s\geq 0$. Finally, since $f^{in}$ satisfies (2.32g) and (2.32h), it follows at once from the already proved property (2.32c) for $g_{\varepsilon}$ (for $m=1+\lambda$), Lemma 2.8, and Lemma 2.10 that $g_{\varepsilon}(s)$ also satisfies (2.32g) and (2.32h). Summarizing, we have shown that $g_{\varepsilon}(s)\in\mathcal{Z}_{\varepsilon}$ for all $s\geq 0$.

Next, the set $\mathcal{Z}_{\varepsilon}$ is convex and its compactness in $X_{1}$ follows from its boundedness in $X_{\lambda-2}\cap X_{1+\lambda}$, the compactness of the embedding of $W^{1,1}(1/R,R)$ in $L^{1}(1/R,R)$, which holds true for all $R>1$, and Vitali’s theorem [14, Theorem 2.24]. ∎

To complete the proof of Proposition 2.2, the missing tile is the continuity of weak solutions to (2.12) with respect to the initial condition which we establish now.

Lemma 2.12**.**

Let $\varepsilon\in(0,\varepsilon_{\varrho})$.

• (a)

For $s\geq 0$, the map $f^{in}\longmapsto\Psi_{\varepsilon}(s;f^{in})$, defined in (2.17), is continuous from $\mathcal{Z}_{\varepsilon}$ endowed with the norm topology of $X_{1}$ to itself.

• (b)

For $f^{in}\in\mathcal{Z}_{\varepsilon}$, the map $s\longmapsto\Psi_{\varepsilon}(s;f^{in})$ belongs to $C([0,\infty),X_{1})$.

In other words, $\Psi_{\varepsilon}:[0,\infty)\times\mathcal{Z}_{\varepsilon}\longrightarrow\mathcal{Z}_{\varepsilon}$ is a dynamical system for the norm topology of $X_{1}$.

Proof of Lemma 2.12 (a).

Consider $(f_{1}^{in},f_{2}^{in})\in\mathcal{Z}_{\varepsilon}^{2}$ and put $g_{i,\varepsilon}:=\Psi_{\varepsilon}(\cdot;f_{i}^{in})$, $i=1,2$. Arguing as in the proof of [19, Theorem 1.2 (c)], it follows from (2.12) that, for $s\geq 0$,

[TABLE]

where $W(x)=x^{\alpha}+x^{\lambda}$, $x\geq 0$, and

[TABLE]

Since both $f_{1}^{in}$ and $f_{2}^{in}$ belong to $\mathcal{Z}_{\varepsilon}$, so do $g_{1,\varepsilon}(s)$ and $g_{2,\varepsilon}(s)$ for all $s\geq 0$ by Lemma 2.11. Consequently, as $m_{0}<\alpha<2\lambda-\alpha\leq 1+\lambda$ by (1.9a) and (1.14),

[TABLE]

In addition,

[TABLE]

by (1.9a) and we infer from (2.5) and the previous differential inequality that, for $s\geq 0$,

[TABLE]

with $\kappa_{\ref{kap10}}(\varepsilon):=1+9K_{0}V_{\varepsilon}+a_{0}\mathfrak{b}_{\alpha,1,\varepsilon}$.

Now, $W(x)\geq x$ for $x\geq 0$ as $\alpha\leq 1<\lambda$, while, for $R>1$, it follows from (1.9a) and (1.14) that

[TABLE]

the last inequality relying on the property $f_{i}^{in}\in\mathcal{Z}_{\varepsilon}$, $i=1,2$. Combining (2.33) and the previous inequalities gives, for $s\geq 0$,

[TABLE]

with

[TABLE]

Since $\omega(r)\longrightarrow 0$ as $r\to 0$, the claimed continuity follows. ∎

Proof of Lemma 2.12 (b).

Set $g_{\varepsilon}=\Psi_{\varepsilon}(\cdot;f^{in})$. Let $s\geq 0$. We infer from (1.9a), (1.9b), (1.9c), (1.14), (2.1c), (2.7), (2.12a), (2.14), (2.22), and Hölder’s inequality that

[TABLE]

Since $g_{\varepsilon}(s)\in\mathcal{Z}_{\varepsilon}$ by Lemma 2.11, we further obtain

[TABLE]

Hence, for $s_{2}>s_{1}\geq 0$ and $R\geq 1$,

[TABLE]

Choosing $R=(s_{2}-s_{1})^{-1/(\lambda+2)}$ if $s_{2}-s_{1}<1$ and $R=1$ otherwise in the previous inequality, we are led to

[TABLE]

which provides the claimed continuity. ∎

We have now established all the properties required to prove Proposition 2.2.

Proof of Proposition 2.2.

Let $\varepsilon\in(0,\varepsilon_{\varrho})$. Owing to Lemma 2.11 and Lemma 2.12, $\Psi_{\varepsilon}$ is a dynamical system on $\mathcal{Z}_{\varepsilon}$ endowed with the norm topology of $X_{1}$ and $\mathcal{Z}_{\varepsilon}$ is a non-empty, convex, and compact subset of $X_{1}$, which is additionally left positively invariant by $\Psi_{\varepsilon}$. A consequence of Schauder’s fixed point theorem, see [1, Proposition 22.13] or [17, Proof of Theorem 5.2], implies that there is $\varphi_{\varepsilon}\in\mathcal{Z}_{\varepsilon}$ such that $\Psi_{\varepsilon}(s;\varphi_{\varepsilon})=\varphi_{\varepsilon}$ for all $s\geq 0$. In other words, $\varphi_{\varepsilon}$ is a stationary solution to (2.12a), from which we deduce that it satisfies (2.10). Also, since $\varphi_{\varepsilon}$ lies in $\mathcal{Z}_{\varepsilon}$, it has the properties (2.11) due to (2.32b), (2.32c), (2.32d), and (2.32f). ∎

3. Self-similar solutions

In this section, we assume that $K$, $a$, and $b$ are coagulation and fragmentation coefficients satisfying (1.9) and we fix $\varrho\in(0,\varrho_{\star})$. For $\varepsilon\in(0,\varepsilon_{\varrho})$, it follows from Proposition 2.2 that there is

[TABLE]

satisfying (2.10),

[TABLE]

and

[TABLE]

for all $m\geq 1+\lambda$. Since $q_{1}>1$ and $m_{1}<1$, we infer from (3.1), (3.2), the reflexivity of $L^{q_{1}}((0,\infty),x^{m_{1}}\mathrm{d}x)$, and Dunford-Pettis’ theorem that there are $\varphi\in X_{m_{1}}\cap L^{q_{1}}((0,\infty),x^{m_{1}}\mathrm{d}x)$ and a subsequence $(\varphi_{\varepsilon_{n}})_{n\geq 1}$ of $(\varphi_{\varepsilon})_{\varepsilon\in(0,\varepsilon_{\varrho})}$ such that

[TABLE]

Combining (3.2), (3.3), and (3.4), we further obtain that $\varphi\in X_{m_{0}}$ and

[TABLE]

Since the positive cone $X_{1}^{+}$ of $X_{1}$ is weakly closed in $X_{1}$, we infer from (3.1) and (3.5) (with $m=1$) that

[TABLE]

We are left with taking the limit $\varepsilon\to 0$ in (2.10). To this end, consider $\vartheta\in\Theta_{1}$, the space $\Theta_{1}$ being defined in (1.17), and note that

[TABLE]

Then $x\mapsto\vartheta(x)/x$ belongs to $L^{\infty}(0,\infty)$ and it readily follows from (3.5) (with $m=1$) that

[TABLE]

Similarly, $\chi_{\vartheta}\in L^{\infty}((0,\infty)^{2})$ and we argue as in [34], see also [7], to deduce from (1.9a), (1.9c), (1.14), and (3.5) (with $m=\alpha$ and $m=\lambda-\alpha$) that

[TABLE]

Finally, by (1.9a), (1.9b), and (3.5) (with $m=\lambda$),

[TABLE]

while (2.1), (2.2a), and (3.7) entail, for $y\in(0,\infty)$,

[TABLE]

Using once more (3.7), we obtain, for $y\in(0,\infty)$,

[TABLE]

Hence, thanks to (2.2c) (with $(m,p)=(1,1)$),

[TABLE]

which implies, in turn,

[TABLE]

Due to (3.10), (3.11), and (3.12), we are in a position to apply [14, Proposition 2.61] (which is a consequence of Dunford-Pettis’ and Egorov’s theorems) and conclude that

[TABLE]

Having established (3.8), (3.9), and (3.13), we may take the limit $\varepsilon\to 0$ in (2.10) and deduce that $\varphi$ satisfies (1.16), thereby completing the proof of Theorem 1.1.