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Trend to equilibrium solution for the discrete Safronov–Dubovskiĭ aggregation equation with forcing

Part of: Applications - Dynamical systems and ergodic theory General theory in ordinary differential equations Equations and inequalities involving nonlinear operators Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  16 November 2023

Arijit Das Open the ORCID record for Arijit Das [Opens in a new window]  and
Jitraj Saha Open the ORCID record for Jitraj Saha [Opens in a new window]
Arijit Das
Affiliation:
Department of Mathematics, National Institute of Technology Tiruchirappalli, Tamil Nadu 620015, India (arijitdasnitt@gmail.com;jitraj@nitt.edu)
Jitraj Saha
Affiliation:
Department of Mathematics, National Institute of Technology Tiruchirappalli, Tamil Nadu 620015, India (arijitdasnitt@gmail.com;jitraj@nitt.edu)
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Abstract

We consider the discrete Safronov-Dubovskiĭ aggregation equation associated with the physical condition, where particle injection and extraction take place in the dynamical system. In application, this model is used to describe the aggregation of particle-monomers in combination with sedimentation of particle-clusters. More precisely, we prove well-posedness of the considered model for a large class of aggregation kernel with source and efflux coefficients. Furthermore, over a long time period, we prove that the dynamical model attains a unique equilibrium solution with an exponential rate under a suitable condition on the forcing coefficient.

Keywords

Safronov–Dubovskiĭ aggregation equationforcing coefficientswell-posednessequilibrium solutionexponential convergence

MSC classification

Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 35Q70: PDEs in connection with mechanics of particles and systems 37N05: Dynamical systems in classical and celestial mechanics
Secondary: 47J35: Nonlinear evolution equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 3 , June 2025 , pp. 869 - 892
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

1. Introduction

Aggregation is a fundamental dynamic process to augment an animate or inanimate matter. Technically, the aggregation process describes an event where two smaller particles merge to form a large cluster. Generally, by this particulate process, although the number of daughter particles gradually decreases; however, the size of the new particle increases. In $1916$, M. von Smoluchowski proposed a mathematical structure for this particulate process, well known as Smoluchowski's aggregation equation (SAE) [Reference von Smoluchowski15]. In practical field of study, this equation is ubiquitously used to describe cloud physics [Reference Friedlander12], oceanography phenomena [Reference Boehm and Grant2] and different models in chemistry [Reference Anand, Mayya, Yu, Seipenbusch and Kasper1]. In this context, if $a_i(t)$, $i\in \mathbb {N}\backslash \{0\}$ denotes the concentration of $i$-clusters at the time $t\ge 0$ then the SAE reads as

(1.1)\begin{equation} \frac{\mathrm{d} a_i(t)}{\mathrm{d} t} = \mathcal{W}_i(\mathbf{a}(t)) \quad \text{with initial data}\quad a_i(0)=a_i^{in}\ge 0, \end{equation}

where $\mathcal {W}_i(\mathbf {a}(t)) := \sum _{j=1}^{i-1}\mathscr {K}_{i-j,j}a_{i-j}(t) a_j (t) - \sum _{j=1}^\infty \mathscr {K}_{i,j} a_i(t) a_j(t)$. The first and second terms in the right-hand side of $\mathcal {W}_i(\mathbf {a}(t))$ represent the birth and death coefficients, respectively. In general, $\mathscr {K}_{i,j}$ defines the aggregation kernel and is symmetric with respect to $i,\,j$, which physically indicates the rate at which $i$-mers merge with $j$-mers to form a large $(i+j)$-cluster. It is worth to note that due to SAE (1.1) the clusters can only grow in size while no smaller particles are created or inserted in the system. Therefore, existence of a equilibrium solution cannot be achieved from the original form of SAE (1.1). In their article, Carr and Costa [Reference Carr and Costa3] proved that for a certain class of aggregation kernel the gelation phenomena may occur at any time interval. However, there is a handful number of articles on various mathematical aspects such as existence, uniqueness and large-time analysis of the solution to the SAE (1.1) and its continuous form. In later years, the SAE was extended as in [Reference Dubovskiĭ11, Reference Ghosh, Paul and Kumar13] to achieve its stationary state by adding source and efflux term

(1.2)\begin{equation} \frac{\mathrm{d} a_i(t)}{\mathrm{d} t} = \mathcal{W}_i(\mathbf{a}(t)) + s_i - r_i a_i(t). \end{equation}

Here, $s_i$ represents the rate at which the cluster of $i$-mers is injected into the system. The third term is the removal term, which indicates the cluster of size $i$ is removed from the system at a rate $r_i a_i$. These two factors are together called forcing coefficients as they play a vital role to add or remove $i$-mers from the system. The existence of stationary solutions and their uniqueness is well studied in [Reference Crump and Seinfeld4, Reference White17]. In the recent years, for a large class of coagulation kernels, the well-posedness and convergence to the unique equilibrium to equation (1.2) have been studied in [Reference Kuehn and Throm14].

During 1990 s, renowned mathematician Pavel B. Dubovskiĭ has introduced a new mechanism of discrete aggregation process, which plays a significant role to describe several physical fields of studies such as astrophysics (cloud forming), cosmology (formation of planets and galaxies), astronomy (asteroid size distribution), etc.

In the past few decades [Reference Dubovski9, Reference Dubovski10]. In literature, this discrete aggregation model is referred as the discrete Safronov–Dubovskiĭ aggregation equation (SDAE) [Reference Das and Saha6, Reference Davidson8]. According to SDAE, for the above-mentioned discrete size distribution function $a_i(t)$, we consider that $a_0(t)$ is equal to zero at any time $t\ge 0$. With this consideration, the governing equation, which describes the time evolution dynamics of cluster growth, is given by

(1.3)\begin{equation} \frac{\mathrm{d} a_i(t)}{\mathrm{d} t} = a_{i-1}\left(t\right)\sum_{j=1}^{i-1} \beta_{i-1,j}ja_j\left(t\right)- a_{i} \left(t\right) \sum_{j=1}^{i} \beta_{i,j}ja_j\left(t\right) - a_{i} \left(t\right)\sum_{j=i}^{\infty}\beta_{i,j}a_j \left(t\right). \end{equation}

We now interpret the terms appearing in the right-hand side of equation (1.3). The aggregation kernel $\beta _{i,j}$ (with $i\ne j$) denotes the collision rate of $i$-mers with $j$-mers. In general, $\beta _{i,j}$ is a non-negative and symmetric function. The first term in the right-hand side of (1.3) represents the inclusion of $i$-mer particles into the system. This $i$-mer particles formed during the collision of $(i-1)$-mers with the monomers formed by the breakup of a $j$-mer into $j$ monomers. In the similar way, the second and third sum represents the removal or death of $i$-mers from the system due to the fusion of the monomers with $i$-mers and forming a larger $(i+1)$-mer. For a detailed interpretation of the second and third terms, readers can refer to [Reference Das and Saha6].

Since the pioneer studies of Dubovskiĭ [Reference Dubovski9, Reference Dubovski10], the discrete SDAE acquired a remarkable popularity in experimental research. In the literature of Safronov–Dubovskiĭ aggregation model, there are few articles available, which deal with the mathematical aspect such as existence, uniqueness, mass conservation, large time analysis or convergence of equilibrium solution to the discrete SDAE (1.3) [Reference Das and Saha6–Reference Davidson8, Reference Wattis16]. After the classical works of Dubovskiĭ, Wattis [Reference Wattis16] and Davidson [Reference Davidson8] reported some works on the mathematical aspect of discrete SDAE. Recently, Das and Saha [Reference Das and Saha6] established the existence of unique mass conserving solution for a large class of unbounded coagulation rate. Therefore, to summarize, most of the current research is devoted to the existence and uniqueness of solutions of the original Safronov–Dubovskiĭ aggregation equation. It turns out that achieving steady state becomes inevitable to stabilize several cosmological phenomena or to reduce the expense of some physical experiments. Like SAE, the discrete SDAE is pure aggregation equation, thus no smaller daughter particle can be generated by the system (1.3). Similar to the SAE with source and efflux term (1.2), it can be expected that the discrete SDAE with effect of external force approaches an equilibrium state as $t \to \infty$. In this regard, we consider the discrete Safronov–Dubovskiĭ aggregation equation with source and efflux term reads

(1.4)\begin{equation} \frac{{\mathrm{d}}a_i\left(t\right)}{{\mathrm{d}}t} = \mathcal{F}_i(\mathbf{a}(t)), \quad i\ge 1, \end{equation}

associated with the initial data

(1.5)\begin{equation} a_i(0) = a_i^{in}. \end{equation}

The operator $\mathcal {F}$ is defined as

(1.6)\begin{align} \mathcal{F}_i(\mathbf{a}(t)) & := a_{i-1}\left(t\right)\sum_{j=1}^{i-1} \beta_{i-1,j}ja_j\left(t\right)- a_{i} \left(t\right) \sum_{j=1}^{i} \beta_{i,j}ja_j\left(t\right)\nonumber\\ & \quad - a_{i}\left(t\right) \sum_{j=i}^{\infty} \beta_{i,j}a_j \left(t\right) + s_i - r_i a_i(t). \end{align}

The non-linear initial value problem (IVP) (1.4)–(1.5) represents the time evolution of the cluster growth dynamics under the effect of external force, in which the source function permits an external supply of particles into the system with rate $s_i$. On the other hand, the efflux term represents the rate $r_ia_i$, at which the particles are removed from the system. In application, this efflux term is often considered to define the sedimentation of particles due to gravity. We can retain the original SDAE (1.3) by setting $s_i = 0 = r_i$ in equation (1.4).

The objective of this article is devoted into two parts. First, we need to establish the well-posedness of the evolution equation (1.4) for a large class of aggregation kernels and removal rates which consolidate the model. This discussion is necessary to validate the model mathematically. Secondly, we show that for large time interval, the solutions converge to a unique equilibrium with exponential rate of convergence. This will be the first evidence in literature to discuss the stationary state solutions for the SDAE coupled with source and efflux components. Another underlying motivation for studying equation (1.4) is to work towards establishing a relation with additional differential equations, which is a common theme in the context of reaction–diffusion systems. Before the extension can be achieved, we should understand the motivation behind considering the injection and extraction terms in equation (1.4).

For particulate events, the moment functions take part an important role as some of them bear relation to significant physical entities. In this regard, for any $\mu \ge 0$, we define the $\mu -th$ order moment and the truncated moment of a solution to equation (1.4) as

(1.7)\begin{equation} \mathcal{M}_{\mu}\left(t\right) = \sum_{i=1}^{\infty}i^\mu a_i\left(t\right),\qquad \text{and}\qquad\mathcal{M}_{\mu}^m\left(t\right) = \sum_{i=1}^{m}i^\mu a_i\left(t\right) \quad\text{respectively}. \end{equation}

In general, zeroth-order moment denotes the total number and first-order moment represents the total mass of particles in the system. From the classical principle of conservation laws, mass can neither be created nor destroyed in any particulate system. Therefore, we can expect that the total mass $\mathcal {M}_{1}(t)= \sum _{i=1}^\infty i a_i(t)$ will remain unaltered by the original form of SDAE (1.3). In a current study, Das and Saha [Reference Das and Saha6] studied the mass conserving behaviour for a large class of aggregation rate

\[ \beta_{i,j} \le \beta_0 \left(1+i+j\right)^\alpha\quad {\text{with}\quad 0\le \alpha \le 1}. \]

Some more evidence on mass conservation of the SDAE can be found in [Reference Davidson8]. In contrast to the mass conserving behaviour, recently Das and Saha [Reference Das and Saha7] proved the occurrence of mass-loss phenomena at any time interval for the kinetic kernels satisfying the rate

\[ \beta_{\mathcal{L}} \left(i^\alpha j^\beta +i^\beta j^\alpha\right)\le \beta_{i,j} \le \beta_{\mathcal{U}} \left(1+i\right)^\omega \left(1+j\right)^\omega \quad \text{with}\quad 0 \le \alpha \le \beta~ \text{and}~ { \beta, \omega>1}. \]

Moreover, the authors also highlighted that the system (1.3) will be ill-posed for the aggregation kernel with growth rate $\beta _{i,j} = i^\lambda +j^\lambda$, with $\lambda >1$.

In this context, to attain the convergence to a unique equilibrium, we impose the additional force. More precisely, the current article adopts a removal rate with the coefficient $r_i$ such that the efflux term $r_ia_i$ causes the solution of (1.4) to exhibit analogous properties as a solution to the system with strong fragmentation regime. However, the consequence to inject or remove a particle externally is that, the mass conserving behaviour cannot be expected for (1.4). However, $\mathcal {M}_{1}$ in general satisfies the equation

\[ \frac{\mathrm{d} \mathcal{M}_{1}(t)}{\mathrm{d} t} = \sum_{i=1}^\infty i s_i - \sum_{i=1}^\infty i r_i a_i(t). \]

In spite of having high non-linearity due to aggregation coefficients in the considered model (1.4), our study successfully proves the existence of stationary state solutions as $t \to \infty$. In this regard, to establish well-posedness or achieve steady state, the growth rate of the source and efflux coefficients $s_i$ and $r_i$ play a significant role to attain the convergence to a unique equilibrium. As already mentioned, the removal coefficient $r_i$ usually sketches the effect of sedimentation. In this circumstance, we assume that all the clusters are spherical and represented by their concentration $i \in \mathbb {N}$ along with the fractal dimension $D(\gamma )$. Under these conditions, we can obtain the scaling $r_i \sim i^\gamma$ [Reference Boehm and Grant2, Reference Friedlander12].

On the other hand, concerning the source term $s_i$, a general assumption is that only monomers are injected, i.e. $s_i=0$ for $i>1$. In the present study, we allow the particle to be injected into the system at a more general rate. More precisely, we only require that, with the increment concentration of cluster, the injection rate $s_i$ decreases sufficiently fast.

The work is organized as follows. In § 2, we state some preliminary definitions on the solution and equilibrium solution of equation (1.4), which are essentially required in the subsequent discussion. Moreover, we brief all the conditions, which we assume on the kinetic coefficients. Based on these conditions, we estimate several higher-order moments of the solution to equation (1.4) in § 3. In the subsequent § 4, we establish the existence of solution to the IVP (1.4)–(1.5). In subsection 4.1, we introduce the truncated form of the IVP (1.4) followed by the proof of local existence of solutions to the truncated problem. Later in subsection 4.2, we prove the global existence theorem with the help of some strong convergence result like Arzelà–Ascoli theorem. We prove a contraction property in § 5, which plays a key role for existence and the rate of convergence to the equilibrium solution. Moreover, with this property, we discuss the uniqueness of solution to the IVP (1.4)–(1.5) in the same section. In the second part of this article, we achieve equilibrium solution with the help of previously obtained contraction property in § 6. Furthermore, we also prove that the solution converges to the steady state with exponential rate under a suitable smallness condition on the kinetic coefficient. Finally, we end our article by drawing some conclusions of the contribution in § 7.

2. Preliminaries: definition and hypothesis

For $\mu \ge 0$, let $\ell _{\mu }^1$ denote the weighted $\ell _{1}$ space of real sequence $\mathbf {a} := \{a_i\}_{i=1}^\infty$, defined as

\[ \ell_{\mu}^1 := \left\{\mathbf{a}=\{a_i\}_{i\in \mathbb{N}}~\left|~ a_i \in \left[0,\infty\right)\text{ for all }i \in \mathbb{N} \text{ and } \|\mathbf{a}\|_{\ell_{\mu}^1 } := \sum_{i=1}^{\infty}\right. i^\mu |a_i|<\infty \right\}. \]

Since the system of differential equation (1.4)–(1.5) is an infinite dimensional system, we need to define the solution.

Definition 2.1 Let $T>0$ be given. A solution of the Cauchy problem (1.4)–(1.5) on $[0,\, T)$ with the initial data $a_i(0) =: \mathbf {a}^{in} \in \ell _{1}^1$ is a continuous function $a_i(t) : [0,\, T)\to [0,\,\infty )$ for all $i \in \mathbb {N}$ such that

  1. (i) $a_i(0) = a^{in}_i$ for all $i \in \mathbb {N}$,

  2. (ii) for any $\mu \ge 1$, we have $\mathbf {a} \in L^\infty ([0,\,T),\, \ell _{1}^1)\cap C^1((0,\,T),\, \ell _{\mu }^1)$, and

  3. (iii) for each $i \in \mathbb {N}$ equation (1.4) satisfies for all $t \in (0,\,T)$.

The solution $\mathbf {a}$ will be global if $T=\infty$.

Definition 2.2 A solution $\mathbf {a}$ to the problem (1.4)–(1.5) is considered a stationary solution (global) if $\mathbf {a}$ remains independent of time, satisfying the condition:

\[ {\mathcal{F}_i\left(\mathbf{a}\right)=0, \quad i\ge 1.} \]

Also, $\mathbf {a}$ will be a equilibrium solution of the initial value problem (1.4)–(1.5) if it is a global stationary solution for the same problem.

In this article, we prove the results under the following assumptions on the coefficient of equation (1.4),

Hypothesis 2.1

  1. (H 1): There exist some constants $A_\ast >0$ such that $\min \{i,\,j\}\beta _{i,j} \le A_\ast (i^\alpha j^\beta +i^\beta j^\alpha )$ with $\alpha \le \beta$ and $\alpha,\, \beta \in [0,\,1]$.

  2. (H 2): The efflux coefficient has rapid growth for large cluster size, i.e. there exists a positive constant $R_\ast$ such that $r_i \ge R_\ast i^\gamma$, where $\gamma > \alpha +\beta$.

  3. (H 3): The source term has a fast decay rate, i.e. for each $\mu \ge 0$ there exists a positive constant $S_\ast ^\mu$ such that $\sum _{i=1}^\infty i^\mu s_i \le S_\ast ^\mu$.

In lieu of the above hypotheses, the aggregation kernel enjoys the unbounded kinetic rates at infinity. With assumption $(H_1)$, the aggregation rate covers the well-known diffusion-controlled growth kernel, $\beta _{i,j} = i^{-2/3}+j^{-2/3}$ (see [Reference Da Costa5]).

In the second hypothesis, the exponent $\gamma$ plays a crucial role in our study to obtain a suitable moment estimation. Moreover, this restriction on the efflux term includes a significant example of sedimentation of particles aggregating due to Brownian motion, i.e. $r_i= i^{2/3}$.

Lastly, hypothesis $(H_3)$ describes that the injected clusters gradually decrease with the size of the cluster. More precisely, the source term $s_i$ is permanently zero for large $i$-mers.

For notational convenience, we will use the following scaling factors;

(2.1)\begin{equation} \widehat A_\ast := \frac{A_\ast}{R_\ast} \quad \text{and} \quad \widehat S_\ast^\mu := \frac{S^\mu_\ast}{R_\ast}\quad \text{ for all }\mu \ge 0. \end{equation}

To estimate the moments, we note the following moment equation for a solution of equation (1.4).

Lemma 2.3 Let $\mathbf {a}$ be a solution to equation (1.4) and $\varphi = \{\varphi _i\}_{i\in \mathbb {N}}$ is a positive sequence of real numbers with at most polynomial growth, then

\[ \frac{{\mathrm{d}}}{{\mathrm{d}}t}\sum_{i=1}^\infty \varphi_i a_i = \sum_{i=1}^\infty \sum_{j=1}^i \left[ j \left(\varphi_{i+1}-\varphi_i\right)-\varphi_j\right]\beta_{i,j} a_i a_j +\sum_{i=1}^{\infty}\varphi_i s_i - \sum_{i=1}^\infty \varphi_i r_i a_i. \]

Proof. The proof of the lemma is straightforward.

3. Moment estimation

We obtain a priori estimates of several higher-order moments of solution to equation (1.4). These estimations are the primary requirement for the proofs of several theorems. In this regard, we now prove the following sequence of lemmas starting with the uniform-boundedness of the first moment, i.e. total mass. Based on this result, we derive a differential inequality for the higher-order moments, and by the application of Grönwall's inequality, we can obtain the estimation for all other moments.

Lemma 3.1 Assume ${(H_1)},\, {(H_2)}$ and ${(H_3)}$ hold; also consider that $\mathbf {a}$ be a solution to problem (1.4) with the initial data (1.5). Then the corresponding first moment $\mathcal {M}_{1}$ is uniformly bounded, that is, if we denote $\mathcal {M}_1^{in}:=\mathcal {M}_1 (0)$, we have

\[ \mathcal{M}_1 (t) \le \mathfrak{M}:= \max \left\{ \mathcal{M}_1^{in} , \widehat S_\ast^1 \right\}. \]

In particular, there exists a time $T>0$ such that $\mathcal {M}_1 (t) \le 2 \widehat S_\ast ^1$, for all $t \ge T$. Here, $T$ depends only on the constants $\mathcal {M}_1^{in},\, \widehat S_\ast ^1$ and $R_\ast$.

Proof. Choose $\varphi _i =i \chi _{\{i\le m\}}$ in lemma 2.3 and using hypotheses $(H_2)$ and $(H_3)$, we have

\begin{align*} \frac{{\mathrm{d}} \mathcal{M}_{1}^m(t)}{{\mathrm{d}}t} & = \sum_{i=1}^{m-1}\sum_{j=1}^{i}\left(i+1\right)j \beta_{i,j} a_i a_j - \sum_{i=1}^{m}\sum_{j=1}^{i}ij \beta_{i,j} a_i a_j -\sum_{i=1}^{m}\sum_{j=i}^{\infty}i \beta_{i,j} a_i a_j \\ & \quad + \sum_{i=1}^m i s_i - \sum_{i=1}^m i r_i a_i\\ & \le \sum_{i=1}^{m}\sum_{j=1}^{i}j \beta_{i,j} a_i a_j -\sum_{i=1}^{m}\sum_{j=i}^{m}i \beta_{i,j} a_i a_j + S_\ast^1 - R_\ast \mathcal{M}_{1}^m(t) \\ & = \sum_{i=1}^{m} \sum_{j=1}^{m} j \beta_{i,j} a_i a_j \!-\!\sum_{i=1}^{m}\sum_{j=i}^{m}\left(i\!+\!j\right) \beta_{i,j} a_i a_j \!+\! \sum_{\substack{i=1 \\ j=i}}^{m} i\beta_{i,i}a_i^2 \!+\! S_\ast^1 \!-\! R_\ast \mathcal{M}_{1}^m(t)\\ & = \frac{1}{2} \sum_{i=1}^{m}\sum_{j=1}^{m}\left(i+j\right)\beta_{i,j} a_i a_j -\sum_{i=1}^{m}\sum_{j=i}^{m}\left(i+j\right) \beta_{i,j} a_i a_j\\ & \quad + {\sum_{i=1}^{m} i\beta_{i,i}a_i^2}+ S_\ast^1 - R_\ast \mathcal{M}_{1}^m(t). \end{align*}

For first three sums, substitute $\phi _{i,j}:=(i+j)\beta _{i,j}a_{i} c_j$ in proposition $2.1$ of [Reference Das and Saha6], we can obtain

(3.1)\begin{equation} \frac{{\mathrm{d}} \mathcal{M}_{1}^m (t)}{{\mathrm{d}}t} + R_\ast \mathcal{M}_{1}^m (t) \!\le\! S_\ast^1 \quad \text{implies}\quad \mathcal{M}_{1}^m (t) \le \left(\mathcal{M}_{1}^{m}(0) + S_\ast^1 t\right) \!-\! R_\ast \int_{0}^{t}\mathcal{M}_{1}^m(\xi)\mathrm{d} \xi. \end{equation}

Using Grönwall's inequality to (3.1) and using relation (2.1), we get

(3.2)\begin{equation} \mathcal{M}_{1}^m (t) \le \left(\mathcal{M}_{1}^{m}(0) - \widehat S_\ast^1\right) \exp\left({-}R_\ast t\right)+ \widehat S_\ast^1. \end{equation}

Therefore, $\mathcal {M}_{1}^m (t) \le \max \left \{\mathcal {M}_{1}^{m}(0),\, \widehat S_\ast ^1\right \}$. Now using $\mathcal {M}_{1}^{m}(0) \le \mathcal {M}_{1}^{in}$ and taking the limit $m \to \infty$, we can get the first result.

The second claim will be obvious for $\mathcal {M}_{1}^{in} = 0$. If $\mathcal {M}_{1}^{in}>0$, the claim follows from relation (3.2) that it suffices to take $T > \max \left \{0,\, \frac {\log (\widehat S_\ast ^1/\mathcal {M}_{1}^{in})}{R_\ast }\right \}$.

We derive a differential inequality which will be used for the higher-order (greater than one) moment estimation of the corresponding solution to equation (1.4).

Lemma 3.2 Assume ${(H_1)}--{(H_3)}$ hold; also consider that $\mathbf {a}$ be a solution to problem (1.4) with the initial condition (1.5). Then for each $\mu >1$, we have the following differential inequality for the truncated moments corresponding to $\mathbf {a}$;

\[ \frac{{\mathrm{d}} \mathcal{M}_\mu^m(t)}{{\mathrm{d}}t} + \frac{ R_\ast}{2} \mathcal{M}_{\mu + \gamma}^m (t) \le \frac{\left(qR_\ast\right)^{1-p}}{2p}\left(2^{\mu+2}A_\ast\right)^p\left(\mathcal{M}_1^m(t)\right)^{1+p}+S_\ast^\mu, \]

where $p := \frac {\mu +\gamma -1}{\gamma -\alpha -\beta }$ and $q :=\frac {p}{p-1}$.

Proof. Taking the sum on equation (1.4) associated with the weight $\varphi _i = i^\mu \chi _{\{i\le m\}}$, we get

\begin{align*} {\frac{{\mathrm{d}} \mathcal{M}_\mu^m(t)}{{\mathrm{d}}t}} & = \sum_{i=1}^{m-1}\sum_{j=1}^{i}\left(i+1\right)^\mu j \beta_{i,j} a_i a_j\\ & \quad - \sum_{i=1}^{m} \left[\sum_{j=1}^{i} i^\mu j \beta_{i,j} a_i a_j + \sum_{j=i}^{\infty}i^\mu \beta_{i,j} a_i a_j - i^\mu s_i + i^\mu r_i a_i\right] \\ & \le \sum_{i=1}^{m} \left[\sum_{j=i}^{m} \left(\left(\left(j+1\right)^\mu - j^\mu\right)i - i^\mu\right) \beta_{i,j} a_i a_j + i^\mu s_i - i^\mu r_i a_i \right] \end{align*}

As $\mu >1$, we use the inequality $(i+j)^\mu \le 2^{\mu -1} (i^\mu +j^\mu )$ on the right-hand side of the above estimation

(3.3)\begin{equation} \begin{aligned} \frac{{\mathrm{d}} \mathcal{M}_\mu^m(t)}{{\mathrm{d}}t} & \le \sum_{i=1}^{m} \left[\sum_{j=i}^{m} \left(2^{\mu}i-1\right) \left(i^\mu + j^\mu\right) \beta_{i,j} a_i a_j + i^\mu s_i - i^\mu r_i a_i \right] \\ & \le \sum_{i=1}^{m}\left[\sum_{j=i}^{m} 2^{\mu}i j^{\mu} \beta_{i,j} a_i a_j + i^\mu s_i - i^\mu r_i a_i\right]. \end{aligned} \end{equation}

Recalling hypothesis $(H_1)$, that is $\beta _{i,j} \le A_\ast (i^\alpha j^\beta +i^\beta j^\alpha )$ together with $(H_2)$ and $(H_3)$ on inequality (3.3), we have

(3.4)\begin{equation} \frac{{\mathrm{d}} \mathcal{M}_{\mu}^m(t)}{{\mathrm{d}}t} \le 2^{\mu+1} A_\ast \mathcal{M}_{1}^m (t) \mathcal{M}_{\mu+\alpha+\beta}^m(t)+S_\ast^\mu - R_\ast \mathcal{M}_{\mu+\gamma}^m (t). \end{equation}

Thanks to Hölder's inequality, we can write

(3.5)\begin{equation} \mathcal{M}_{\mu+\alpha+\beta}^m (t)\le \left(\mathcal{M}_{1}^m(t)\right)^{\frac{1}{p}}\left(\mathcal{M}_{\mu+\gamma}^m(t)\right)^{\frac{1}{q}} \end{equation}

and thus inequality (3.4) is written as

(3.6)\begin{equation} \frac{{\mathrm{d}} \mathcal{M}_{\mu}^m(t)}{{\mathrm{d}}t} \le 2^{\mu+1} A_\ast\left(\mathcal{M}_{1}^m(t)\right)^{1+\frac{1}{p}} \left(\mathcal{M}_{\mu+\gamma}^m(t)\right)^{\frac{1}{q}} + S_\ast^\mu - R_\ast \mathcal{M}_{\mu+\gamma}^m(t). \end{equation}

Using Young's inequality (with $\epsilon$) yields that

(3.7)\begin{align} & 2^{\mu+1}A_\ast\left(\mathcal{M}_{1}^m(t)\right)^{1+\frac{1}{p}}\left(\mathcal{M}_{\mu+\gamma}^m(t)\right)^{\frac{1}{q}} \le \epsilon \left(\mathcal{M}_{\mu+\gamma}^m(t)\right)\nonumber\\ & \quad + \frac{\left(q \epsilon\right)^{1-p}}{p} \left(2^{\mu+1}A_\ast\right)^p \left(\mathcal{M}_{1}^m(t)\right)^{1+p}. \end{align}

Hence, combining both inequalities (3.6) and (3.7) for $\epsilon =\frac {R_\ast }{2}$, we can obtain the desired result.

We now obtain a generalized non-linear differential inequality which will be used to estimate the higher-order moments.

Lemma 3.3 Let $\zeta >0$ and $\phi \in C([0,\,\infty ),\,\mathbb {R}_{\ge 0})\cap C^1((t_0,\,\infty ))$ for some $t_0 \ge 0,$ satisfy

(3.8)\begin{equation} \frac{{\mathrm{d}} \phi (t) }{{\mathrm{d}}t} + \mathcal{A} \left[\phi (t)\right]^{1+\zeta} \le \mathcal{B}\quad \text{for all}\quad t \ge t_0 \end{equation}

with some constants $\mathcal {A}$ and $\mathcal {B}$, then

\[ \phi (t) \le \max \left\{\left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\zeta}}, \left(\frac{2}{\zeta \mathcal{A}}\right)^{\frac{1}{\zeta}}\left(t-t_0\right)^{-\frac{1}{\zeta}} \right\}\quad \text{for all}\quad t \ge t_0. \]

Proof. We claim that the set

\[ \left\{ t>t_0~ \bigg |~ \phi(t) \le \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\zeta}}\right\} \]

is non-empty. Otherwise, assume the contrary that

(3.9)\begin{equation} \phi (t) > \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\zeta}} \quad \text{for all}\quad t >t_0. \end{equation}

Combining inequalities (3.8) and (3.9), we get

(3.10)\begin{equation} \frac{{\mathrm{d}} \phi (t) }{{\mathrm{d}}t} \le - \mathcal{B} \quad\mbox{which implies}\quad \phi(t) \le \phi(t_0) -\left(t-t_0\right)\mathcal{B} \quad \text{for all}\quad t >t_0. \end{equation}

Thus, $\phi (t) <0$ whenever $t > t_0 + \frac {\phi (t_0)}{\mathcal {B}}$, which contradicts the non-negativity of $\phi$.

Next, assume that

\[ T:= \inf \left\{ t>t_0~ \bigg |~ \phi(t) \le \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\zeta}} \right\}. \]

If $T$ does not exist, then there exist a $t' \ge T$ and $\epsilon >0$ such that

(3.11)\begin{equation} \phi (t) > \phi(t') = \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\zeta}}\quad \text{for all}\quad t \in \left(t',t'+\epsilon\right). \end{equation}

Again by similar argument as (3.10), we get $\phi (t) < \phi (t')$ for $t \in (t',\,t'+\epsilon )$, which contradicts relation (3.11).

If $T= t_0$, then the proof is complete. Otherwise if $T> t_0$, we consider the interval $(t_0,\, T)$. The definition of $T$ gives $\mathcal {B} \le \frac {\mathcal {A}}{2}[\phi (t) ]^{1+\zeta }$. Using this bound on inequality (3.8), we get

\begin{align*} & \frac{{\mathrm{d}} \phi (t)}{{\mathrm{d}}t} + \frac{\mathcal{A}}{2} \left[\phi (t)\right]^{1+\zeta} \le 0 \quad \mbox{that is}\quad \phi (t) \\ & \quad \le \left(\frac{1}{\left(\phi(t_0)\right)^{-\zeta} + \frac{\mathcal{A}\zeta}{2}\left(t-t_0\right)}\right)^{\frac{1}{\zeta}} \quad \text{for all}\quad t \in \left(t_0, T\right). \end{align*}

Using the non-negativity of $\phi (t_0)$, we get

(3.12)\begin{equation} \phi(t) \le \left(\frac{2}{\zeta \mathcal{A}}\right)^{\frac{1}{\zeta}}\left(t-t_0\right)^{-\frac{1}{\zeta}} \quad \text{for all}\quad t \in (t_0,T). \end{equation}

Hence, the proof is completed for all $t \ge t_0$.

With the help of lemma 3.2 and lemma 3.3, we can estimate the higher-order moments of solution to (1.4).

Lemma 3.4 Assume ${(H_1)}--{(H_3)}$ hold; also consider that $\mathbf {a}$ be a solution to problem (1.4) with the corresponding first moment $\mathcal {M}_{1}$ and initial condition (1.5). Then for any $\mu >1$,

\[ \mathcal{M}_{\mu} (t) \le \max \left\{ \left(\frac{2^{1+p}\left(2^{\mu+1} \widehat A_\ast\right)^p q^{1-p}}{p} \mathfrak{M}^{1+p+\lambda_\mu} + 2 \widehat S_\ast^\mu \mathfrak{M} ^{\lambda_\mu} \right)^{\frac{1}{1+\lambda_\mu}}, \mathfrak{M}\left(\frac{4}{R_\ast \lambda_\mu t}\right)^{1/\lambda_\mu} \right\}, \]

where $\lambda _\mu := \frac {\gamma }{\mu -1}$ and $p,\,q$ defined in lemma 3.2. In particular, there exists a constant $\mathcal {C}_\mu$ such that $\mathcal {M}_{\mu } (t) \le \mathcal {C}_\mu (1+t^{-1/\lambda _\mu })$. Note that $\mathcal {C}_\mu$ depends only on the constants namely, $\alpha,\, \beta,\, \gamma,\, \widehat S_\ast ^1,\, \widehat S_\ast ^\mu,\, \widehat A_\ast$ and $R_\ast$.

Proof. Thanks to the Hölder's inequality, which gives the following result for the higher-order truncated moments

(3.13)\begin{equation} \left(\mathcal{M}_{\mu}^m(t)\right)^{\frac{\mu +\gamma -1}{\mu -1}} \le \left(\mathcal{M}_{\mu+\gamma}^m(t)\right) \left(\mathcal{M}_{1}^m(t)\right)^{\frac{\gamma}{\mu-1}}. \end{equation}

The uniform boundedness of the first truncated moment $\mathcal {M}_{1}^m$ yields

\[ \mathcal{M}_{\mu+\gamma}^m (t) \ge \left(\mathcal{M}_{\mu}^m(t)\right)^{\frac{\mu +\gamma -1}{\mu -1}} \mathfrak{M}^{-\frac{\gamma}{\mu-1}}. \]

Using the bound $\mathcal {M}_{1}^m(t) \le \max \left \{\mathcal {M}_{1}^{m}(0),\,~ \widehat S_\ast ^1\right \}$ and the above inequality on lemma 3.2, we have the following required differential inequality

(3.14)\begin{equation} \frac{{\mathrm{d}}\mathcal{M}_\mu^m(t)}{{\mathrm{d}}t} + \frac{R_\ast}{2} \mathfrak{M}^{-\frac{\gamma}{\mu-1}} \left(\mathcal{M}_{\mu}^m(t)\right)^{1+\lambda_\mu} \le \frac{\left(qR_\ast\right)^{1-p} \left(2^{\mu+2}A_\ast\right)^p}{2p} \mathfrak{M}^{1+p} + S_\ast^\mu. \end{equation}

Now, consider

\[ \mathcal{A} := \frac{R_\ast}{2} \mathfrak{M}^{-\frac{\gamma}{\mu-1}} \quad \text{and}\quad \mathcal{B} := \frac{\left(qR_\ast\right)^{1-p}\left(2^{\mu+2}A_\ast\right)^p}{2p}\mathfrak{M}^{1+p}+S_\ast^\mu \]

and apply lemma 3.3 for $t_0 =0$, we can get the claim. Finally, we can get the estimation $\mathcal {M}_{\mu } (t) \le \mathcal {C}_\mu (1+t^{-1/\lambda _\mu })$ by considering $\mathcal {C}_\mu := \max \left \{(\frac {2\mathcal {B}}{\mathcal {A}})^{\frac {1}{1+\lambda _\mu }},\, (\frac {2}{\zeta \mathcal {A}})^{\frac {1}{\lambda _\mu }} \right \}$.

The following lemma estimates the higher-order moments in which the estimates are free from the initial data.

Lemma 3.5 Assume ${(H_1)}--{(H_3)}$ hold; also consider that $\mathbf {a}$ be a solution to problem (1.4) with the corresponding first moment $\mathcal {M}_{1}$. If $\lambda _\mu = \frac {\gamma }{\mu -1}$ and $p,\,q$ defined in lemma 3.2, then for any $\mu >1$ there exists a time $T>0$ (depends on $\mathbf {a}$) such that

(3.15)\begin{equation} \mathcal{M}_{\mu} (t) \le \left\{\frac{2^{2+\lambda_\mu}}{p}\left(2^{\mu+3} \widehat A_\ast\right)^p q^{1-p} (\widehat S_\ast^1)^{1+p+\lambda_\mu} + 2^{2+\lambda_\mu}\widehat S_\ast^\mu \left(\widehat S_\ast^1\right)^{\lambda_\mu} \right\}^{\frac{1}{1+\lambda_\mu}} \end{equation}

and

(3.16)\begin{equation} \mathcal{M}_{\mu} (t) \le 4 \left\{ \frac{\left(2^{\mu+3} \widehat A_\ast\right)}{p}q^{1-p}\left(\widehat S_\ast^1\right)^{1+p} + \widehat S_\ast^\mu \right\} \quad \mbox{for all}\quad t \ge T. \end{equation}

Proof. From lemma 3.1, we get $T_1>0$ such that $\mathcal {M}_{1} (t) \le 2 S_\ast ^1$ for all $t \ge T_1$. Using this inequality on relation (3.13) and proceed similarly as lemma 3.3 to get

\begin{align*} & \frac{{\mathrm{d}}\mathcal{M}_\mu^m (t)}{{\mathrm{d}}t} + \frac{R_\ast \left(\widehat S_\ast^1\right)^{-\frac{\gamma}{\mu-1}}}{2^{1+\frac{\gamma}{\mu-1}}} \left(\mathcal{M}_{\mu}^m(t)\right)^{1+\lambda_\mu}\\ & \quad \le \frac{\left(qR_\ast\right)^{1-p}\left(2^{\mu+3}A_\ast\right)^p}{p} \left(\widehat S_\ast^1\right)^{1+p}+S_\ast^\mu \quad \text{for all}\quad t \ge T_1. \end{align*}

Setting $t_0 = T_1,\,~ \mathcal {A} := \frac {R_\ast (\widehat S_\ast ^1)^{-\frac {\gamma }{\mu -1}}}{{2^{1+\frac {\gamma }{\mu -1}}}},\,~ \mathcal {B} := \frac { (qR_\ast )^{1-p}(2^{\mu +3}A_\ast )^p}{p} (\widehat S_\ast ^1)^{1+p} + S_\ast ^\mu$ and applying lemma 3.3, we have

\[ \mathcal{M}_{\mu}^m (t) \le \max \left\{\left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\lambda_\mu}}, \left(\frac{2}{\zeta \mathcal{A}}\right)^{\frac{1}{\lambda_\mu}}\left(t-T_1\right)^{-\frac{1}{\lambda_\mu}} \right\}\quad \text{for all}\quad t \ge T_1. \]

Choose $T_2\ge T_1$, such that

(3.17)\begin{equation} \left(\frac{2}{\zeta \mathcal{A}}\right)^{\frac{1}{\lambda_\mu}}\left(t-T_1\right)^{-\frac{1}{\lambda_\mu}} \le \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\lambda_\mu}} \end{equation}

for all $t \ge T_2$. Now substituting the values of $\mathcal {A}$, $\mathcal {B}$ and taking limit $m \to \infty$, we obtain estimate (3.15) by recalling relation (2.1).

We now proceed to get inequality (3.16). Since $\gamma \ge 0$, we have the following results hold true for any higher-order moments

(3.18)\begin{equation} \mathcal{M}_{\mu}^m (t) \le \mathcal{M}_{\mu+\gamma}^m (t)\quad\text{and}\quad \mathcal{M}_{\mu} (t) \le \mathcal{M}_{\mu+\gamma} (t) \end{equation}

for all $t \ge 0$. Combining these results together with $\mathcal {M}_{1} (T) \le 2 S_\ast ^1$ for all $t \ge T_2$, lemma 3.2 generates

\[ \frac{{\mathrm{d}}\mathcal{M}_\mu^m (t)}{{\mathrm{d}}t} + \frac{ R_\ast}{2} \mathcal{M}_{\mu}^m (t) \le \mathcal{B}. \]

Direct application of Grönwall's inequality and estimation (3.15) together give

(3.19)\begin{equation} \mathcal{M}_\mu^m (t) \le \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\lambda_\mu}} \exp\left[-\frac{R_\ast}{2}\left(t-T_2\right)\right] + \left(1-\exp\left[-\frac{R_\ast}{2} \left(t-T_2\right) \right] \right)\frac{2 \mathcal{B}}{R_\ast}. \end{equation}

Suppose that $T \ge T_2$ such that

\[ \left(\frac{2\mathcal{B}}{\mathcal{A}}\right)^{\frac{1}{1+\lambda_\mu}} \exp\left[-\frac{R_\ast}{2} \left(t-T_2\right)\right] \le \frac{2 \mathcal{B}}{R_\ast}. \]

Setting the above inequality on (3.19) and passing the limit $m \to \infty$, we obtain estimate (3.16) for $t \ge T$ with the help of relation (2.1).

Remark 3.6 The above lemma 3.5 gives two different estimates of the moment function $\mathcal {M}_{\mu }(t)$. One of the estimates (3.15) depends on $\gamma$, and the second estimate (3.16) is independent of $\gamma$.

4. Existence of a solution

To prove the existence theorem, we adopt the similar approach which is frequently used for the discrete Safronov–Dubovskiĭ aggregation equation [Reference Das and Saha6]. We truncate all the coefficients $\beta,\,s,\,r$ and the initial data $\mathbf {a}^{in}$ of equation (1.4) as follows

(4.1)\begin{equation} \begin{aligned} & \beta_{i,j}^m := \left\{\begin{array}{@{}ll} \beta_{i,j}, & \mbox{when}\quad 1\le i,j \le m, \\ 0, & \mbox{otherwise,} \end{array}\right. \quad,\quad s_i^m := \left\{\begin{array}{@{}ll} s_i, & \mbox{when}\quad 1\le i \le m, \\ 0, & \mbox{otherwise}, \end{array}\right.\\ & r_i^m := \left\{\begin{array}{@{}ll} r_i, & \mbox{when}\quad 1\le i \le m, \\ 0, & \mbox{otherwise}, \end{array}\right. \quad\text{and}\quad a_i^{m,in} := \left\{\begin{array}{@{}ll} a_i^{in}, & \mbox{when}\quad 1\le i \le m, \\ 0, & \mbox{otherwise.} \end{array}\right. \end{aligned} \end{equation}

Therefore, the above truncations generate the following system of $m (\ge 2)$ ordinary differential equations of model (1.4);

(4.2)\begin{equation} \begin{array}{lll} \dfrac{{\mathrm{d}}a_i^m }{{\mathrm{d}}t} = & a_{i-1}^m \sum_{j=1}^{i-1} j\beta^m_{i-1,j} a_j^m \!-\! a_{i}^m \sum_{j=1}^{i} j\beta^m_{i,j} a_j^m \!-\! a_{i}^m \sum_{j=i}^{m} \beta^m_{i,j}a_j^m + s^m_i \!-\! r^m_i a^m_i, \\ & \hspace{1.4cm} \mbox{when} \quad i\le m, \\ a_i^m (t) = & 0, \hspace{1.1cm}\mbox{when} \quad i>m. \end{array} \end{equation}

Note that $\mathbf {a}^m= (a_i^m)_{i\in \mathbb {N}}$ is also a solution of equation (1.4) where all the coefficients are defined as in (4.1). Also from (4.1), it is clear that $\beta _{i,j}^m,\, s_i^m$ and $r^m_i$ satisfy the assumptions $(H_1),\, (H_2)$ and $(H_3)$, respectively. So, all the moment estimations derived in § 3 remain valid for the truncated solution $\mathbf {a}^m$.

4.1 Existence of solution for finite dimensional system

The following proposition proves the existence of unique global solution to the truncated system (4.2).

Proposition 4.1 Assume that $\mathbf {a}^{m,in} = \{a_i^{m,in}\}_{i\in \mathbb {N}} \in \ell _{1}^1$ and non-negative. Then system (4.2) has unique solution with $\mathbf {a}^m= (a_i^m)_{i\in \mathbb {N}} \in C^1 ([0,\,\infty ),\, \ell _{1}^1)$ for each $i \in \mathbb {N}$.

Proof. Existence and uniqueness of solution of finite dimensional system (4.2) follows from the classical argument from theory of ordinary differential equations. In this regard, we define a polynomial function $f_i:\mathbb {R}^m\to \mathbb {R}^m$ as

\[ f_i\left(x_1,\ldots,x_m\right) = x_{i-1} \sum_{j=1}^{i-1} j\beta_{i-1,j} x_j -x_i \sum_{j=1}^{i} j\beta_{i,j} x_j - x_i\sum_{j=i}^{m} \beta_{i,j}x_j+ s^m_i - r^m_i x_i \]

such that the truncated equation (4.2) is written as

\[ \frac{{\mathrm{d}}a_i^m (t)}{{\mathrm{d}}t} = f_i\left(a_1^m,\ldots,a_m^m\right),\quad \text{for all}\quad i=1,\ldots,m. \]

Therefore, each $f_i$ is a polynomial function of components $a_i$, moreover it is locally Lipschitz continuous. Therefore, the existence and uniqueness of local solution $\mathbf {a}^m$ to the Cauchy problem (2.1) follows from the standard Picard–Lindelöf existence theorem. Thus, there exists a maximal time interval $a_i^m \in C^1 ([0,\,T_\ast ))$ for all $i=1,\,\ldots,\,m$.

Again for the non-negativity of the solution, we consider that for arbitrary $\epsilon >0$, there exists a solution $a^\epsilon _i$ for the system, that is

\[ \frac{{\mathrm{d}}a_i^\epsilon (t)}{{\mathrm{d}}t} = f_i\left(a_1^\epsilon,\ldots,a_m^\epsilon\right)+\epsilon.\quad \text{for all}\quad i=1,\ldots,m. \]

Also consider, for some $t_0 > 0$ and $1 \le i \le m$, we have $a_i^\epsilon (t_0)> 0$ and $a_r^\epsilon (t_0)=0$ when $r\ne i$. Then, $\frac {{\mathrm {d}}a_r^\epsilon (t)}{{\mathrm {d}}t}= a^\epsilon _{r-1} (t) \sum _{j=1}^{r-1} j\beta _{r-1,j} a_j^\epsilon (t)+s_i+\epsilon >0$. Taking $\epsilon \longrightarrow 0$, we get the non-negativity result.

Finally, for the global existence of solution (i.e. $T_\ast =\infty$), we use lemma 3.1 as follows

(4.3)\begin{equation} 0\le a_i (t)\le {i^{{-}1}\sum_{i=1}^{m}ia_i}\le \mathfrak{M}. \end{equation}

So, the above result (4.3) shows that the solution cannot blow up on $[0,\,T_\ast )$, which implies the solution $\mathbf {a}^m \in C^1 ([0,\,\infty ),\, \ell _{1}^1)$.

4.2 Global existence theorem

In this section, we state and prove the main existence theorem of the global solution to the IVP (1.4)=(1.5). Before this, we will prove two important results which will be used to prove the compactness and passing the limit $n \to \infty$ on the local solution obtained from proposition 4.1.

Lemma 4.2 Assume conditions ${(H_1)}-{(H_3)}$ hold. Also consider that $\mathbf {a}^m$ be a solution to problem (4.2) with the corresponding initial first moment $\mathcal {M}_1^{in}:=\mathcal {M}_1 (0)$. Then for each fixed $i \in \mathbb {N}$, there exists a positive constant $\Re$ (depends on $i$ but not on $m$) such that

\[ \left\| \frac{{\mathrm{d}}a_i^m (t)}{{\mathrm{d}}t} \right\|_{L^\infty\left(0,\infty\right)} \le \Re \quad \mbox{for all}\quad m \ge i. \]

Proof. Using non-negativity of the coefficient $\beta,\,s$ and the solution $\mathbf {a}^m$, together with condition $(H_1)$ on equation (4.2), we get

(4.4)\begin{equation} \begin{aligned} \left|\frac{{\mathrm{d}}a_i^m (t)}{{\mathrm{d}}t}\right| & \le A_\ast \sum_{j=1}^{i-1} \left(\left(i-1\right)^\alpha j^\beta + \left(i-1\right)^\beta j^\alpha\right) a_{i-1}^ma_j^m +\nonumber\\& \quad A_\ast \sum_{j=1}^{i}\left(i^\alpha j^\beta + i^\beta + j^\alpha\right)a_{i}^ma_j^m \\ & \quad+ A_\ast \sum_{j=i}^{m}\left(i^\alpha j^\beta+i^\beta+j^\alpha\right)a_{i}^ma_j^m + s_i + r_ia_i^m. \end{aligned} \end{equation}

Applying the estimation $i^\mu a_i^m \le \| a^m\|_{\ell _{\mu }^1}$ on the above inequality, we have

\[ \left|\frac{{\mathrm{d}}a_i^m (t)}{{\mathrm{d}}t}\right| \le 6 A_\ast \| a^m\|_{\ell_{\alpha}^1}\| a^m\|_{\ell_{\beta}^1} + s_i \le 6A_\ast \left(\mathcal{M}_{1}^m(t)\right)^2 + S_\ast^1 + \frac{r_i}{i} \| a^m\|_{\ell_{1}^1}. \]

Applying lemma 3.1, we conclude that

\[ \left\| \frac{{\mathrm{d}}a_i^m (t)}{{\mathrm{d}}t} \right\|_{L^\infty\left(0,\infty\right)} \le 6 A_\ast \mathfrak{M}^2 + S_\ast^1 + \frac{r_i}{i}\mathfrak{M}.\]

Proposition 4.3 Let all the conditions ${(H_1)}-{(H_3)}$ hold. Also consider for the non-negative sequences $(\beta _{i,j}^n),\,(r^n)_{n\in \mathbb {N}}$, $(s^n)_{n\in \mathbb {N}}$ and $(\mathbf {a}^n)_{n\in \mathbb {N}} \in L^\infty ([0,\,\infty ),\,\ell _{1}^1)$ for each $n \in \mathbb {N}$, there exists some sequence $\mathbf {a} = (a_i)_{i\in \mathbb {N}},\, r = (r_i)_{i\in \mathbb {N}}$ and $s = (s_i)_{i\in \mathbb {N}}$ such that

\[ \beta_{i,j}^n \to \beta_{i,j},\quad r^n_i \to r_i, \quad s^n_i \to s_i\quad \text{as}\quad n \to \infty \quad \text{for each } i,j \in \mathbb{N}. \]

as well as $a^n_i (t) \to a_i(t)$ as $n \to \infty$ uniformly on compact subset of $[0,\,\infty )$. Moreover, there exists a constant $\Omega _1$ for $(\mathbf {a}^n)_{n\in \mathbb {N}}$ satisfying

\[ \mathcal{M}_{1}^n(t) := \sum_{i=1}^{\infty}ia^n_i(t) \le \Omega_1 \quad \text{for all}\quad t\ge 0 \]

and for all $\mu >1$ there exists some constant $\Omega _\mu$ and $\lambda \in [0,\,1)$ for $(\mathbf {a}^n)_{n\in \mathbb {N}}$ such that

\[ \mathcal{M}_{\mu}^n (t):= \sum_{i=1}^{\infty}ia^n_i(t) \le \Omega_\mu \left(1+t^{-\lambda}\right) \quad \text{for all}\quad t\ge 0. \]

Then

  1. (i) For each fixed $i \in \mathbb {N}$

    (4.5)\begin{align} \lim_{n \to \infty} & \left( \sum_{j=1}^{i-1} j\beta^n_{i-1,j} a_{i-1}^n a_j^n - \sum_{j=1}^{i} j\beta^n_{i,j} a_{i}^n a_j^n - \sum_{j=i}^{\infty} \beta^n_{i,j} a_{i}^na_j^n + s^n_i - r^n_i a^n_i\right)\nonumber\\ & = \sum_{j=1}^{i-1} j\beta_{i-1,j} a_{i-1} a_j - \sum_{j=1}^{i} j\beta_{i,j} a_{i} a_j - \sum_{j=i}^{\infty} \beta_{i,j} a_{i}a_j + s_i - r_i a_i \end{align}
    converges uniformly on each compact subset of $(0,\,\infty )$.

  2. (ii) For each fixed $i \in \mathbb {N}$ and for any $t >0$

    (4.6)\begin{align} \lim_{n \to \infty} & \left[ \int_{0}^{t} \left( \sum_{j=1}^{i-1} j\beta^n_{i-1,j}a_{i-1}^n (\xi) a_j^n (\xi) - \sum_{j=1}^{i} j\beta^n_{i,j} a^n_{i} (\xi) a_j^n (\xi) \right.\right.\nonumber\\& \left.\left.\quad - \sum_{j=i}^{\infty} \beta^n_{i,j} a^n_{i} (\xi) a^n_j (\xi)+ s^n_i - r^n_i a^n_i(\xi)\right)\right]\mathrm{d} \xi \nonumber\\ & = \int_{0}^{t} \left(\sum_{j=1}^{i-1} j\beta_{i-1,j}a_{i-1} (\xi) a_j (\xi)\right.\nonumber\\& \left. - \sum_{j=1}^{i} j\beta_{i,j} a_{i}(\xi) a_j (\xi) - \sum_{j=i}^{\infty} \beta_{i,j} a_{i} (\xi) a_j (\xi)+ s_i - r_i a_i(\xi) \right)\mathrm{d} \xi. \end{align}

Proof. With the help of Fatou's lemma and the assumption of the proposition, we can say that the limit sequence $(a_i)_{i\in \mathbb {N}}$ satisfies the estimates

(4.7)\begin{equation} \mathcal{M}_{1}(t) := \sum_{i=1}^{\infty}ia_i(t) \le \Omega_1 \quad\text{and}\quad \mathcal{M}_{\mu} (t):= \sum_{i=1}^{\infty}ia_i(t) \le \Omega_\mu \left(1+t^{-\lambda}\right). \end{equation}

We deduce result 4.5 by proving the convergence for each term separately, under the assumption $s^n_i \longrightarrow s_i$ as $n \longrightarrow \infty$ already assumed in the proposition. The locally uniform convergence of $a^n_i (t)$ to $a_i(t)$ on $[0,\,\infty )$ and finite number of terms in the following terms, we can get

(4.8)\begin{equation} \begin{aligned} & \sum_{j=1}^{i-1} j\beta^n_{i-1,j} a_{i-1}^n a_j^n \xrightarrow{n\to\infty} \sum_{j=1}^{i-1} j\beta_{i-1,j} a_{i-1} a_j, \\ & \quad\sum_{j=1}^{i} j\beta^n_{i,j} a_{i}^n a_j^n \xrightarrow{n\to\infty} \sum_{j=1}^{i} j\beta_{i,j} a_{i} a_j \quad\text{and}\\ & \quad r^n_i a^n_i \xrightarrow{n\to\infty} r_i a_i \end{aligned} \end{equation}

are also locally uniformly convergent on $[0,\,\infty )$. It remains to estimate the difference $\sum _{j=i}^{\infty } \beta ^n_{i,j} a_{i}^na_j^n - \sum _{j=i}^{\infty } \beta _{i,j} a_{i}a_j$. Choose some $Z \in \mathbb {N}$ (which we fix later) and rewrite the difference in the following way

(4.9)\begin{align} \left| \sum_{j=i}^{\infty} \beta^n_{i,j} a_{i}^n a_j^n - \sum_{j=i}^{\infty} \beta_{i,j} a_{i}a_j\right| & \le \left|a_{i}^n-a_{i}\right|\sum_{j=i}^{\infty}\beta^n_{i,j} a_j^n\nonumber\\ & \quad + a_i \sum_{j=i}^{Z-1}\left(\left|\beta^n_{i,j}-\beta_{i,j}\right|a_j^n+\beta_{i,j}\left|a_j^n-a_j\right|\right) \nonumber\\ & \quad+a_i \sum_{j=Z}^{\infty} \left(\beta^n_{i,j}a_j^n+\beta_{i,j}. a_j\right). \end{align}

Using condition $(H_1)$ in the first term on the right-hand side of (4.9), we get

(4.10)\begin{align} \left|a_{i}^n-a_{i}\right|\sum_{j=i}^{\infty}\beta^n_{i,j} a_j^n & \le A_\ast \left|a_{i}^n-a_{i}\right| \sum_{j=i}^{\infty}\left(i^\alpha j^\beta + i^\beta j^\alpha\right) a_j^n \nonumber\\ & \le 2 A_\ast i^\beta \left|a_{i}^n-a_{i}\right| \sum_{j=i}^\infty j a_j^n \le 2 A_\ast i^\beta \Omega_1 \left|a_{i}^n-a_{i}\right|. \end{align}

By using the assumption of proposition 4.1, we get the right-hand side of (4.10) that converges to zero as $n \longrightarrow \infty$ locally uniformly on $[0,\,\infty )$. To estimate the second term on the right-hand side of (4.9), we follow the same argument as (4.8). For the completeness, each term of the sum converges to zero locally uniformly on $[0,\,\infty )$ and since the sum contains a fixed number of terms,

(4.11)\begin{equation} \lim\limits_{n \to \infty}\left( a_i \sum_{j=i}^{Z-1}\left(\left|\beta^n_{i,j}-\beta_{i,j}\right|a_j^n+\beta_{i,j}\left|a_j^n-a_j\right|\right)\right) =0 \end{equation}

converges locally uniformly on $[0,\,\infty )$. Finally, to estimate the third term on the right-hand side of (4.9), we use the relation $\beta _{i,j} \le A_\ast (i^\alpha j^\beta + i^\beta j^\alpha ) \le 2 A_\ast i^\beta j^\beta$ (since $\alpha \le \beta < \mu$) and $\lambda \in [0,\,1)$ we get

(4.12)\begin{align} a_i(\xi) \sum_{j=Z}^{\infty} \left(\beta^n_{i,j}a_j^n(\xi)+\beta_{i,j} a_j(\xi)\right)& \le 2A_\ast Z^{\beta-\mu}i^\beta a_i(\xi)\sum_{j=Z}^{\infty} j^\mu \left(a_j^n(\xi)+ a_j(\xi)\right)\nonumber\\ & \le 2A_\ast \Omega_1 Z^{\beta-\mu} \left(\mathcal{M}_{\mu}^n(\xi)+\mathcal{M}_{\mu}(\xi)\right)\nonumber\\ & \le 2A_\ast \Omega_1 \Omega_\mu \left(1+\xi^{-\lambda}\right) Z^{\beta-\mu}\longrightarrow 0\quad\text{as}\quad Z \longrightarrow 0 \end{align}

converges locally uniformly on $[0,\,\infty )$. Since the right-hand side of (4.11) is independent of $n$, the right-hand side of (4.9) is arbitrarily small by taking $n \longrightarrow \infty$ and then $Z \longrightarrow \infty$. This estimation together with (4.8), we get the convergence result (4.5).

Moreover, since all the estimations (4.8), (4.10) and (4.11) are uniform with respect to $\xi$ and since $\int _{0}^{t}(1+\xi ^{-\lambda }) \mathrm {d} \xi = t+ t^{1-\lambda }/(1-\lambda )$, the convergence result (4.6) follows directly.

We are now at the stage where we can prove the global existence theorem for the IVP (1.4).

Theorem 4.4 Global existence theorem

Consider that $\mathbf {a}^{in}\in \ell _{1}^1$. Under the assumptions $(H_1)-(H_3)$, there exists at least one global solution $\mathbf {a}$ to (1.4)–(1.5) with initial condition $\mathbf {a}^{in}$.

Proof. Let $(a^m)_{m\in \mathbb {N}}$ be the solution of $m$-dimensional system (4.2) obtained from proposition 4.1 and lemma 3.1 states that, $\mathbf {a}^m$ have a uniformly bounded first moment, i.e. $\mathcal {M}_{1}^m (t) \le \mathfrak {M}$. Therefore, lemma 4.2 guarantees that $(a^m)_{m\in \mathbb {N}}$ is uniformly bounded on $C^{0,1}(0,\,T)$ for all fixed $T\in (0,\,\infty )$. Thus, by Arzelà–Ascoli theorem we ensure that there exists a subsequence of $(a_i^{m})_{m \in \mathbb {N}}$ (for notational convenience, we will not relabel) and a continuous sequence $\mathbf {a}= (a_i)_{i\in \mathbb {N}}$ such that

(4.13)\begin{equation} a_i^{m} \longrightarrow a_i \quad \text{as}\quad m \longrightarrow \infty \end{equation}

converges locally uniformly on $[0,\,\infty )$. Moreover, with the help of lemmas 3.1, 3.4 and Fatou's lemma, we have for some $\mu >1$ and $\lambda \in [0,\,1)$ that

(4.14)\begin{equation} \mathcal{M}_{1}(t) \le \mathfrak{M}\quad \text{and}\quad \mathcal{M}_{\mu} (t)\le \Omega_\mu \left(1+t^{-\lambda}\right). \end{equation}

Since $(a_i^m)_{m \in \mathbb {N}}$ is a solution of (4.2), we have

(4.15)\begin{align} {a_i^m (t) - a_i^{m,in}}& =\int_{0}^{t} \left(a_{i-1}^m (\xi) \sum_{j=1}^{i-1}\right. j\beta^m_{i-1,j} a_j^m (\xi) - a_{i}^m (\xi) \sum_{j=1}^{i} j \beta^m_{i,j} a_j^m (\xi) \nonumber\\ & \left.- a_{i}^m (\xi)\sum_{j=i}^{\infty} \beta^m_{i,j}a_j^m (\xi)+ s^m_i - r^m_i a^m_i(\xi)\right) \mathrm{d} \xi. \end{align}

Applying proposition 4.1 by taking the limit $m \longrightarrow \infty$ on equation (4.15), we get

(4.16)\begin{align} {a_i (t) - a_i^{in}}& =\int_{0}^{t} \left(a_{i-1} (\xi) \sum_{j=1}^{i-1}\right. j\beta_{i-1,j} a_j (\xi) - a_{i} (\xi) \sum_{j=1}^{i} j \beta_{i,j} a_j (\xi) \nonumber\\ & \left.- a_{i} (\xi)\sum_{j=i}^{\infty} \beta_{i,j}a_j (\xi)+ s_i - r_i a_i(\xi)\right) \mathrm{d} \xi. \end{align}

Now, consider (4.14) with the convergence (4.13), we get $\mathbf {a} \in L^\infty ([0,\,\infty ),\, \ell _{1}^1)\cap C^1((0,\,\infty ),\, \ell _{\mu }^1)$. Moreover, continuity of the initial data and equation (4.16) implies that $\mathbf {a} \in C^1([0,\,\infty ))$. So, by the help of Leibnitz rule differentiating (4.16) with respect to $t$ establishes that $\mathbf {a}(t)$ solves (1.4). Hence, the proof of theorem 4.4 is completed.

5. A contraction property

This property plays a key roll to prove the existence and the rate of convergence to the equilibrium solution. Moreover, relying on this contraction property, we will also prove the uniqueness of the solution to equation (1.4).

Lemma 5.1 Suppose the assumptions $(H_1)-(H_3)$ hold, also $\mathbf {c} = (c_i)_{i\in \mathbb {N}}$ and $\mathbf {d}= (d_i)_{i\in \mathbb {N}}$ be two solutions to equation (1.4). Then for $\mu \ge 1$ there exists a constant $k_\mu$, such that

\[ \frac{{\mathrm{d}} }{{\mathrm{d}}t} \sum_{i=1}^\infty i^\mu \left|c_i - d_i\right| \le \left( 2A_\ast \left(2^\mu +2+k_\mu\right)\sum_{j=1}^\infty j^{\mu+\beta}\left(c_j+d_j\right)-R_\ast\right)\sum_{i=1}^\infty i^\mu \left|c_i - d_i\right|. \]

Proof. Consider lemma 2.3 for the solutions $\mathbf {c} = (c_i)_{i\in \mathbb {N}}$, $\mathbf {d}= (d_i)_{i\in \mathbb {N}}$ and taking the difference by setting $\varphi _i = i^\mu \operatorname {sgn} (c_i - d_i)$, we get

(5.1)\begin{align} \frac{{\mathrm{d}}}{{\mathrm{d}}t}\sum_{i=1}^{\infty} i^\mu \left|c_i - d_i\right| & = \sum_{i=1}^\infty \sum_{j=i}^\infty \left[ i \left(\varphi_{j+1}\!-\!\varphi_j\right)-\varphi_i\right]\beta_{i,j} \left(c_ic_j - d_i d_j\right) \!-\! \sum_{i=1}^{\infty}\varphi_i \left(c_i-d_i\right) r_i \nonumber\\ & = \sum_{i=1}^\infty \sum_{j=i}^\infty \left[ i \left(\varphi_{j+1}-\varphi_j\right)-\varphi_i\right]\beta_{i,j} \left[c_j\left(c_i-d_i\right)+d_i\left(c_i-d_i\right)\right]\nonumber\\ & \quad - \sum_{i=1}^{\infty}\varphi_i \left(c_i-d_i\right) r_i \nonumber\\ & = \sum_{i=1}^\infty \sum_{j=i}^\infty \left[ i \left(\varphi_{j+1}-\varphi_j\right)-\varphi_i\right]\beta_{i,j}c_j\left(c_i-d_i\right)\nonumber\\ & \quad + \sum_{i=1}^\infty \sum_{j=i}^\infty \left[ i \left(\varphi_{j+1}-\varphi_j\right)-\varphi_i\right]\beta_{i,j}d_i\left(c_i-d_i\right)\nonumber\\ & \quad - \sum_{i=1}^{\infty}\varphi_i \left(c_i-d_i\right) r_i. \end{align}

Putting the explicit form of $\varphi _i$, we can get the following estimations

(5.2)\begin{align} & \left[ i \left(\varphi_{j+1}-\varphi_j\right)-\varphi_i\right] \left(c_i-d_i\right)\nonumber\\ & \quad = \left(c_i-d_i\right) \left[ i \left(\left(j+1\right)^\mu \operatorname{sgn} \left(c_{j+1}-d_{j+1}\right) -j^\mu \operatorname{sgn} \left(c_j-d_j\right)\right)- i^\mu \operatorname{sgn} \left(c_i - d_i\right)\right] \nonumber\\ & \quad= \left(c_i-d_i\right) \left[ i \left(j+1\right)^\mu \operatorname{sgn} \left(c_{j+1}-d_{j+1}\right) -ij^\mu \operatorname{sgn} \left(c_j-d_j\right)\right] - i^\mu \left|c_i-d_i\right|\nonumber\\ & \quad \le \left|c_i-d_i\right| \left[i\left(1+j\right)^\mu +ij^\mu -i^\mu\right] \nonumber\\ & \quad \le \left(2^\mu+1\right) i j^\mu\left|c_i-d_i\right| \end{align}
(5.3)\begin{align} & \left[ i \left(\varphi_{j+1}-\varphi_j\right)-\varphi_i\right] \left(c_j-d_j\right)\nonumber\\ & \quad= \left(c_j - d_j\right) \left[ i \left(j+1\right)^\mu \operatorname{sgn} \left(c_{j+1}-d_{j+1}\right) -i^\mu \operatorname{sgn} \left(c_i-d_i\right)\right] -ij^\mu \left|c_j-d_j\right| \nonumber\\ & \quad\le \left|c_j-d_j\right| \left[i\left(1+j\right)^\mu +i^\mu -ij^\mu\right] \nonumber\\ & \quad\le \left|c_j-d_j\right| \left[k_\mu ij^{\mu-1} +i^\mu\right] \end{align}

for some constant $k_\mu$. Use estimations (5.2) and (5.3) on (5.1), to get

\begin{align*} \frac{{\mathrm{d}}}{{\mathrm{d}}t}\sum_{i=1}^{\infty} i^\mu \left|c_i - d_i\right| & \le \left(2^\mu+1\right)\sum_{i=1}^\infty \sum_{j=i}^\infty i j^\mu\beta_{i,j} c_j \left|c_i-d_i\right|\nonumber\\ & \quad + \sum_{i=1}^\infty \sum_{j=i}^\infty \left[k_\mu i j^{\mu-1} +i^\mu\right]\beta_{i,j} d_i \left|c_j-d_j\right| - \sum_{i=1}^{\infty}i^\mu \left|c_i-d_i\right| r_i\nonumber\\ & \le 2A_\ast\left[\left(2^\mu+1\right)\sum_{i=1}^\infty \sum_{j=i}^\infty i^\beta j^{\mu+\beta} c_j \left|c_i-d_i\right|\right.\nonumber\\ & \left.\quad + \left(k_\mu +1\right) \sum_{i=1}^\infty \sum_{j=i}^\infty i^\beta j^{\mu+\beta-1} d_i \left|c_j-d_j\right|\right]\nonumber\\ & \quad - \sum_{i=1}^{\infty}i^\mu \left|c_i-d_i\right| r_i\nonumber\\ & \le 2A_\ast \left( k_\mu +2^\mu +2\right) \sum_{i=1}^\infty \sum_{j=i}^\infty \left(i^\beta j^{\mu+\beta} c_j + j^\beta i^{\mu+\beta-1} d_j\right) \left|c_i-d_i\right|\nonumber \\ & \quad - R_\ast \sum_{i=1}^{\infty}i^\mu \left|c_i-d_i\right| \nonumber \\ & \le \left(2A_\ast \left( k_\mu +2^\mu +2\right) \sum_{j=1}^\infty j^{\mu+\beta}\left(c_j +d_j\right) - R_\ast\right) \sum_{i=1}^{\infty} i^\mu \left|c_i-d_i\right|.\nonumber \end{align*}

Lemma 5.2 Suppose assumptions $(H_1)-(H_3)$ hold and we redefine $p,\, q$ and $\lambda$ as

\[ p= \frac{\mu+\gamma-1}{\gamma-\alpha-\beta}, \quad q=\frac{p}{p-1} \quad \text{and}\quad \lambda_{\mu+\beta}=\frac{\gamma}{\mu+\beta-1} \quad \text{respectively.} \]

Then for any pair of solutions $\mathbf {c} = (c_i)_{i\in \mathbb {N}}$ and $\mathbf {d}= (d_i)_{i\in \mathbb {N}}$, there exists a time $T_\ast$ such that

\[ \frac{{\mathrm{d}}}{{\mathrm{d}}t} \sum_{i=1}^{\infty} i^\mu \left|c_i - d_i\right| \le - \eta \sum_{i=1}^{\infty} i^\mu \left|c_i - d_i\right| \qquad \text{for almost every} \quad t \ge T_\ast, \]

where $\eta := \max \{\eta _1,\,\eta _2\}$ with $\eta _1$ and $\eta _2$ defined as

\begin{align*} \eta_1 & := R_\ast \\ & - 4A_\ast \left( k_\mu +2^\mu +2\right) \bigg\{\frac{q^{1-p}2^{2+\lambda_{\mu+\beta}}}{p}\left(2^{\mu+\beta+3} \widehat A_\ast\right)^p (\widehat S_\ast^1)^{1+p+\lambda_{\mu+\beta}}\\ & \quad + 2^{2+\lambda_{\mu+\beta}}\widehat S_\ast^{\mu+\beta} \left(\widehat S_\ast^1\right)^{\lambda_{\mu+\beta}} \bigg\}^{\frac{1}{1+\lambda_{\mu+\beta}}}\\ & >0 \end{align*}

and

\[ \eta_2 := R_\ast{-} 16A_\ast \left( k_\mu +2^\mu +2\right) \bigg\{ \frac{\left(2^{\mu+\beta+3} \widehat A_\ast\right)}{p}q^{1-p}\left(\widehat S_\ast^1\right)^{1+p} + \widehat S_\ast^{\mu+\beta} \bigg\}>0. \]

Proof. We need to show that there exists a time $T_\ast$ such that

(5.4)\begin{equation} 2A_\ast \left( k_\mu +2^\mu +2\right) \sum_{j=1}^\infty j^{\mu+\beta}\left(c_j +d_j\right) - R_\ast \le - \eta \qquad \text{for almost every} \quad t \ge T_\ast. \end{equation}

According to lemma 3.5, for a large time $T_\ast$ the left-hand side of the inequality (5.4)

\begin{align*} & 2A_\ast \left( k_\mu +2^\mu +2\right) \sum_{j=1}^\infty j^{\mu+\beta}\left(c_j +d_j\right) - R_\ast\\ & \quad\le 4A_\ast \left( k_\mu +2^\mu +2\right) \min \Bigg \{4\left( \frac{\left(2^{\mu+\beta+3} \widehat A_\ast\right)}{p}q^{1-p}\left(\widehat S_\ast^1\right)^{1+p} + \widehat S_\ast^{\mu+\beta} \right) , \\ & \quad\left(\frac{1}{p}2^{2+\lambda_{\mu+\beta}}\left(2^{\mu+\beta+3} \widehat A_\ast\right)^p q^{1-p} (\widehat S_\ast^1)^{1+p+\lambda_{\mu+\beta}}\right.\\ & \left. \quad + 2^{2+\lambda_{\mu+\beta}}\widehat S_\ast^{\mu+\beta} \left(\widehat S_\ast^1\right)^{\lambda_{\mu+\beta}} \right)^{\frac{1}{1+\lambda_{\mu+\beta}}}\Bigg\}- R_\ast \end{align*}

for all $t \ge T_\ast$. Now, using the definition of $\eta _1$ and $\eta _2$ and take $\eta := \max \{\eta _1,\,\eta _2\}$, we can conclude the proof.

The uniqueness of the solution to equation (1.4) is obtained through lemma 5.1.

Proposition 5.3 Uniqueness of global solution

Let all the conditions of theorem 4.4 hold. Then there exists atmost one global solution to equation (1.4).

Proof. Consider $\mathbf {c}= (c_i)$ and $\mathbf {d}=(d_i)$ be two solutions to equation (1.4). Set $\mu =1$ in lemma 5.1, to get

\[ \frac{{\mathrm{d}}}{{\mathrm{d}}t}\sum_{i=1}^{\infty} i \left|c_i - d_i\right| \le \left( 10A_\ast \sum_{j=1}^\infty j^{\beta+1}\left(c_j+d_j\right)-R_\ast\right)\sum_{i=1}^\infty i \left|c_i - d_i\right|. \]

Recalling lemma 3.4, for $R_\ast >0$ the above assumption can take the form

\[ \frac{{\mathrm{d}}}{{\mathrm{d}}t}\sum_{i=1}^{\infty} i \left|c_i - d_i\right| \le \left(20A_\ast \mathcal{C}_{1+\beta}\left(1+t^{-\beta/\gamma}\right) \right)\sum_{i=1}^\infty i \left|c_i - d_i\right|. \]

The fact $\beta < \gamma$ gives $t \to (1+ t^{-\beta /\gamma })$ is integrable at zero. Thus, due to the Grönwall's inequality, we can obtain $\mathbf {c} = \mathbf {d}$.

6. Existence and convergence of equilibrium solution

In this section, we will establish the existence of a unique equilibrium solution. Moreover, with the help of lemma 5.2, we will show that any solution to equation (1.4) converges to this equilibrium with an exponential rate.

Theorem 6.1 Let conditions $(H_1)-(H_3)$ hold. Moreover, for each $\mu \ge 1$ either

\begin{align*} & 2A_\ast \left( k_\mu +2^\mu +2\right) \bigg\{\frac{1}{p}2^{2+\lambda_{\mu+\beta}}\left(2^{\mu+\beta+3} \widehat A_\ast\right)^p q^{1-p} (\widehat S_\ast^1)^{1+p+\lambda_{\mu+\beta}}\\ & \quad + 2^{2+\lambda_{\mu+\beta}}\widehat S_\ast^{\mu+\beta} \left(\widehat S_\ast^1\right)^{\lambda_{\mu+\beta}} \bigg\}^{\frac{1}{1+\lambda_{\mu+\beta}}} - R_\ast{<}0 \end{align*}

or

\[ 8A_\ast \left( k_\mu +2^\mu +2\right) \bigg\{ \frac{\left(2^{\mu+\beta+3} \widehat A_\ast\right)}{p}q^{1-p}\left(\widehat S_\ast^1\right)^{1+p} + \widehat S_\ast^{\mu+\beta} \bigg\}- R_\ast{<}0, \]

where $p= \frac {\mu +\gamma -1}{\gamma -\alpha -\beta },\, \quad q=\frac {p}{p-1} \quad \text {and}\quad \lambda _{\mu +\beta }=\frac {\gamma }{\mu +\beta -1}$. Then

  1. (i) there exists a unique stationary solution $Q= (Q_i)_{i \in \mathbb {N}} \in \ell _{\mu }^1$ for all $\mu \ge 1$,

  2. (ii) for each solution $\mathbf {a}$, there exists some constant $K,\, \eta >0$ (independent of $\mathbf {a}$) and a time $T_c >0$ such that

    \[ \|\mathbf{a}(t) - Q\|_{\ell_{\mu}^1} \le K \exp[-\eta t] \quad \text{for all} \quad t \ge T_c. \]

We will prove the following lemma, which is a direct consequence of lemma 5.2.

Lemma 6.2 Let all the conditions of theorem 6.1 hold. Then for any solution $\mathbf {a} = (a_i)_{i\in \mathbb {N}}$ to equation (1.4), we have

\[ \lim\limits_{t \to \infty} \left|\frac{\mathrm{d}}{\mathrm{d} t}a_i(t)\right| =0 \quad \text{for all}\quad i\in \mathbb{N}. \]

Proof. For any $h >0$, we introduce a shifted (along time) sequence $\mathbf {a}^h= (a^h_i)_{i\in \mathbb {N}}$, defined as $a_i^h(t) =a_i(t+h)$. Therefore, $\mathbf {a}^h$ is again a solution to equation (1.4). Applying lemma 5.2 with these two solutions $\mathbf {a}$ and $\mathbf {a}^h$ yields

(6.1)\begin{equation} \frac{{\mathrm{d}}}{{\mathrm{d}}t} \sum_{i=1}^{\infty} i^\mu \left|a_i(t+h) - a_i(t)\right| \le - \eta \sum_{i=1}^{\infty} i^\mu \left|a_i(t+h) - a_i(t)\right| \qquad \text{for all} \quad t \ge T_\ast. \end{equation}

Integrating

\begin{align*} & \sum_{i=1}^{\infty} i^\mu \left|a_i(t+h) - a_i(t)\right| \le \sum_{i=1}^{\infty} i^\mu \left|a_i(T_\ast{+}h)\right.\\ & \left.\quad - a_i(T_\ast)\right| \exp\left[-\eta\left(t-T_\ast\right)\right] \qquad \text{for all} \quad t \ge T_\ast. \end{align*}

Since $\mathbf {a} \in C^1((0,\,\infty ),\, \ell _{\mu }^1)$, so taking the limit $h \to 0$ yields

\[ \Big\|\frac{{\mathrm{d}}}{{\mathrm{d}}t} a_i(t)\Big\|_{\ell_{\mu}^1} \le\Big \|\frac{{\mathrm{d}}}{{\mathrm{d}}t} a_i(T_\ast)\Big\|_{\ell_{\mu}^1} \exp\left[-\eta\left(t-T_\ast\right)\right] \qquad \text{for all} \quad t \ge T_\ast. \]

Further taking limit $t \to \infty$ on the above estimation, we obtain

\[ \lim\limits_{t \to \infty} \Big\|\frac{{\mathrm{d}}}{{\mathrm{d}}t} a_i(t)\Big\|_{\ell_{\mu}^1} =0. \]

Which completes the proof.

Now we have reached the point where we can prove the existence and uniqueness of the equilibrium solution. Consequently, we will prove exponential convergence to the equilibrium solution.

Proof of theorem 6.1 We will prove the theorem in three steps.

Existence of equilibrium solution: Let $\mathbf {a} = (a_i)_{i \in \mathbb {N}}$ be a solution to equation (1.4) associated with some initial data $\mathbf {a}^{in}$. According to lemmas 3.1 and 3.5, for each $\mu >1$ we have

(6.2)\begin{equation} \mathcal{M}_{1} (t) \le 2 \widehat S_\ast^1 \quad \text{and}\quad \mathcal{M}_{\mu} (t) \le \Omega_\mu \left(1+t^{-\lambda_\mu}\right)\quad \text{for sufficiently large}~t. \end{equation}

This yields the existence of a sequence $(t_n)$ which satisfies $t_n \to \infty$ as $n \to \infty$ and non-negative sequence $(Q_i)$ such that

\[ \mathbf{a}(t_n) \longrightarrow Q \quad\text{in } \ell_{\mu}^1 \quad \text{for all } \mu \ge 1\quad \text{as }n \longrightarrow \infty. \]

In particular, with the help of Fatou's lemma and estimation (6.2), the sequence $Q$ also satisfies

\[ \sum_{i=1}^\infty i Q_i \le 2 \widehat S_\ast^1 \quad \text{and}\quad \sum_{i=1}^\infty i^\mu Q_i \le \Omega_\mu \quad \text{for all}\quad\mu >1. \]

It is now a straightforward application of proposition 4.1 to the stationary sequence $\mathbf {a}(t_n)$ and taking into account lemma 5.2, we get

\[ \mathcal{F}_i(Q) = \lim\limits_{n \to \infty} \mathcal{F}_i(\mathbf{a}(t_n))= \lim\limits_{n \to \infty}\frac{\mathrm{d}}{\mathrm{d} t}a_i(t_n)=0. \]

This proves that $Q$ is an equilibrium solution to equation (1.4).

Uniqueness of equilibrium solution: On the contrary, we assume that there exist two equilibrium solutions $Q^1$ and $Q^2$ to equation (1.4). Since $Q^1$ and $Q^2$ both are solutions of equation (1.4), applying lemma 6.2 we can obtain

\[ 0= \frac{{\mathrm{d}}}{{\mathrm{d}}t} \sum_{i=1}^{\infty} i^\mu \left|Q^1_i - Q^2_i\right| \le - \eta \sum_{i=1}^{\infty} i^\mu \left|Q^1_i - Q^2_i\right| \qquad \text{for all }\mu \ge 1, \]

from which the uniqueness follows.

Convergence to equilibrium: Applying lemma 5.2 for any solution $\mathbf {a}$ and equilibrium solution $Q_i$, we have

\[ \frac{{\mathrm{d}}}{{\mathrm{d}}t} \sum_{i=1}^{\infty} i^\mu \left|a_i - Q_i\right| \le - \eta \sum_{i=1}^{\infty} i^\mu \left|a_i - Q_i\right| \qquad \text{for all }t \ge T_\ast. \]

By integrating the above differential inequality, we can obtain

(6.3)\begin{equation} \sum_{i=1}^\infty i^\mu \left|a_i(t) - Q_i \right| \le \sum_{i=1}^\infty i^\mu \left|a_i(T_\ast) - Q_i \right| \exp[-\eta (t-T_\ast)] \quad \text{for all} \quad t \ge T_\ast. \end{equation}

According to lemma 3.5, there exists a constant $\tilde {K}$ which is independent of $\mathbf {a}$ and a time $\tilde {T}$ such that

(6.4)\begin{equation} \sum_{i=1}^\infty i^\mu \left|a_i(T_\ast) - Q_i \right| \le \sum_{i=1}^\infty i^\mu \left(a_i + Q_i\right) \le 2 \tilde{K} \quad \text{for all} \quad t \ge \tilde{T}. \end{equation}

Finally, take $T_c=\max \{T_\ast,\,\tilde {T}\}$, $K:= 2\tilde {K} \exp (T_\ast )$ and using esimation (6.4) on (6.3) we get

\[ \sum_{i=1}^\infty i^\mu \left|a_i(t) - Q_i \right| \le K \exp[-\eta t] \quad \text{for all} \quad t \ge T_c, \]

which concludes the proof.

7. Concluding remarks

In this article, we consider an extension of the Safronov–Dubovskiĭ aggregation equation, where particles input and output can take place. A complete theoretical investigation for well-posedness and convergence to the steady-state solution to the IVP (1.4)–(1.5) has been done. Here all the associated kinetic coefficients are unbounded and cover a large class of physical kinetic rates. Initially, we truncated system (1.4) in a finite-dimensional system and with the help of standard Picard–Lindelöf theorem, we have proved the local existence theorem. After that, a strong convergence result, which is known as Arzelà–Ascoli theorem, ensures the convergence of the sequence of solution whose limit function is proved to be a solution of the IVP (1.4)–(1.5). In the subsequent sections, by proving some contraction property of the solution, for a long time limit, we established that the solution converges to a unique equilibrium solution with the exponential convergence rate.

Acknowledgements

A. D. thanks the Ministry of Education (MoE), Govt. of India for their funding support during their PhD programme. J. S. thanks NITT for their support through seed grant (file no.: NITT / R & C / SEED GRANT / 19 - 20 / P - 13 / MATHS / JS / E1) during this work.

Competing interest

None.

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