2.1. Well-posedness

We state the main results on existence and uniqueness of solutions under some assumptions on the kinetic coefficients, mainly on the coagulation ones, and on the initial data. Let $\alpha \in [0,1]$ .

1. (H1) There exists $a \geq 0$ such that $0 \leq a_{i} \leq a i^\alpha$ and $b_i \geq 0$ for all $i\geq 1$ .

In the classical proof of existence from Ball, Carr and Penrose [Reference Ball, Carr and Penrose7] or for the general coagulation-fragmentation equations from [Reference Ball and Carr6] or [Reference Laurençot34], the natural space for the existence is $X_1$ since the equations are mass preserving. Here, equations (1.1) do not preserve mass; therefore, we can consider a larger space for the existence, in which mass does not necessarily have meaning or may blow up. We have the following existence results in $X_\alpha$ for $\alpha \in [0,1]$ .

Theorem 2.3 (Existence and moment spreading). Let $C^{init} \in X_\alpha ^+$ . Under the hypothesis (H1), there exists at least one solution $C \in \mathcal{C}^0([0,+\infty ),X_\alpha ^+)$ to ( 1.1 ) with initial data $C(0) = C^{init}$ . Moreover, if there exists $\mu \geq \alpha$ such that

\begin{equation*} \sum _{i=1}^{+\infty } i^\mu C_i^{init} \lt +\infty , \end{equation*}

then any solution satisfies for all $T\gt 0$

\begin{equation*} \sup _{t\in [0,T]}\sum _{i=1}^{+\infty } i^\mu C_i(t) \lt +\infty . \end{equation*}

We actually prove a more general result including more than just power law moments but a wider class, see Section 3. This theorem is inspired by [Reference Laurençot34, Theorem 2.5], and it generalises [Reference Carr and da Costa18, Theorem 3.3], which proves moment spreading in coagulation-fragmentation equations.

Theorem 2.4 (Partial uniqueness). Let $C^{init}\in X_\alpha ^+$ and $T\in (0,+\infty ]$ . Under the hypothesis (H1), there exists at most one solution $C$ to ( 1.1 ) on $[0,T)$ with initial condition $C(0)=C^{init}$ such that

(2.3) \begin{equation} C \in \mathrm{L}^\infty ([0,t],X_\alpha ^+) \text{ and } \sum _{i=1}^{+\infty } i^{2\alpha } C_i \in \mathrm{L}^1(0,t) \text{ for all } t \in (0,T). \end{equation}

Theorem 2.5 (Well-posedness). Let $C^{init}\in X_{2\alpha }^+$ and $T\in (0,+\infty ]$ . Under the hypothesis (H1), there exists a unique solution $C$ to ( 1.1 ) on $[0,T)$ with initial condition $C(0)=C^{init}$ . Moreover, this solution satisfies

\begin{equation*} C \in \mathcal{C}^0([0,t],X_\alpha ^+) \text{ and } \sum _{i=1}^{+\infty } i^{2\alpha } C_i \in \mathrm{L}^1(0,t) \text{ for all } t \in (0,T). \end{equation*}

Theorem2.5 is an immediate consequence of Theorems2.4 and 2.3. The partial uniqueness result Theorem2.4 is inspired by [Reference Ball and Carr6, Theorem 4.2] and [Reference Laurençot34, Proposition5.1] and state uniqueness under moments control. Laurençot and Mischler [Reference Laurençot and Mischler35, Theorem 2.1] gave another proof of uniqueness, on the Becker–Döring equations, without the assumption on moments control, but with an assumption on coefficients $(b_i)_{i\geq 1}$ . Their proof is based on estimates of distribution tails and mass conservation. The proof of Theorem2.4 is given in Section 4.

In the remainder, we will always assume that the kinetics coefficients satisfy the condition (H1) for some suitable $\alpha \in [0,1]$ .

2.2. Long-time behaviour

2.2.1. Uniform-in-time bound

The first uniform-in-time bound is an upper bound on $C_1(t)$ . We obtain directly from the equation on $C_1$ that for all $t\in [0,+\infty )$ , we have

(2.4) \begin{equation} C_1(t) \leq \kappa (\lambda ,C^{init}) \,:\!= \max \left \{C_1^{init},\sqrt {\frac {\lambda }{2a_1}}\right \}\!. \end{equation}

We prove an upper uniform-in-time bound on exponential moments under strong fragmentation, and this will allow us to prove the local exponential stability of a steady state. The following proposition is inspired by a similar result in [Reference Jabin and Niethammer32, Lemma 3.3].

Proposition 2.6 (Exponential moments uniform-in-time bound). Assume that the kinetics coefficients satisfy the following conditions:

(2.5) \begin{equation} \forall i\geq 1, \,a_i \geq \underline {a} \gt 0. \end{equation}

Let $\lambda \geq 0$ and $C^{init}$ small enough such that

(2.6) \begin{equation} \kappa (\lambda ,C^{init}) \lt \inf _{i\geq 1} \frac {b_i}{a_i}, \end{equation}

and there exists $\nu \gt 0$ such that

(2.7) \begin{equation} \sum _{i=1}^{+\infty } e^{\nu i}C_i^{init} \lt +\infty . \end{equation}

Then, there exists a constant $\Xi _1 \gt 0$ such that for all $t\geq 0$ ,

(2.8) \begin{equation} \sum _{i=1}^{+\infty } e^{\nu i}C_i(t) \leq \Xi _1, \end{equation}

where $\Xi _1$ depends only on $\lambda$ , $(a_i)_{i\geq 1}$ , $(b_i)_{i\geq 1}$ , $\nu$ and initial data.

We also prove, under some hypotheses on the kinetic coefficients, an upper bound on the mass and a lower bound on $C_1$ . Details and proofs are presented in Section 5.1.

2.2.2. Steady states

We define the detailed balance coefficients $Q_i$ by

(2.9) \begin{equation} Q_1 \,:\!= 1,\,\, Q_i \,:\!= \prod \limits _{j=2}^i \frac {a_{j-1}}{b_j}, \text{ pour } i\geq 2. \end{equation}

These coefficients play a significant role in the long-term behaviour of the solutions. It is from these coefficients that two significant quantities are defined. The first quantity is the critical monomer density, denoted $z_s$ , which is defined as the radius of convergence of the power series $\sum a_iQ_iz^i$ . This quantity is also called monomer saturation density, hence the subscript. The second quantity is the critical production threshold, denoted $\lambda _s$ , which is defined as follows:

(2.10) \begin{equation} \lambda _s \,:\!= \sup _{z\leq z_s} \sum _{j=1}^{+\infty } a_jQ_jz^{j+1} + a_1 z^2. \end{equation}

We emphasise that both $z_s$ and $\lambda _s$ are completely determined by the coefficients $a_i$ and $b_i$ .

Remark 2.7. As the function $z\mapsto a_jQ_jz^{j+1}$ is increasing, the critical production threshold can be rewritten, when the series converges, as follows:

\begin{equation*} \lambda _s = \sum _{j=1}^{+\infty } a_jQ_jz_s^{j+1} + a_1 z_s^2. \end{equation*}

In the following, we add two hypotheses that can be readily verified in the case of constant coefficients, certain power laws or Niethammer’s coefficients [Reference Niethammer40, (1.5) and (1.6)]. These conditions are as follows.

1. (H2) There exists $\underline {a}\gt 0$ and $\underline {b}\gt 0$ such that $\inf \limits _{i\geq 1}a_i=\underline {a}$ and $\inf \limits _{i\geq 1}b_i=\underline {b}$ ,

2. (H3) There exists $l\gt 0$ such that $\lim \limits _{i\to +\infty }\dfrac {Q_{i+1}}{Q_i}=\dfrac {1}{l z_s}$ ,

3. (H4) $\lim \limits _{i\to +\infty }\dfrac {a_{i+1}}{a_i} = \lim \limits _{i\to +\infty }\dfrac {b_{i+1}}{b_i} = l$ .

Remark 2.8. Hypotheses (H3) and (H4) imply that $\lim \limits _{i\to +\infty }\frac {b_i}{a_i} = z_s$ .

Whether or not a steady state exists depends on the critical production rate $\lambda _s$ and the production rate $\lambda$ and, more particularly, on their comparison. By steady state, we understand a constant solution $C$ , in particular, $\sum _{i=1}^{+\infty } a_iC_i \lt \infty$ .

Theorem 2.9 (Steady states). Under the assumptions (H2)–(H4), we have the two following possibilities:

1. 1. (Sub-critical) If $\lambda \leq \lambda _s$ , then there exists a unique $z\leq z_s$ verifying

   (2.11) \begin{equation} \lambda = \sum _{j=1}^{+\infty } a_jQ_jz^{j+1} + a_1z^2 \end{equation}
   such that $(Q_iz^i)_{i\geq 1}$ is a steady state. Moreover, if $b_i\leq ia_i$ for $i\gg 1$ , then $(Q_iz^i)_{i\geq 1}$ is the unique steady-state or if $b_i \gt i^\nu a_i$ for $i\gg 1$ and $\nu \gt 1$ , then there is an infinity of steady state.

2. 2. (Super-critical) Else $\lambda \gt \lambda _s$ , and then there exists no steady state.

2.2.3. Local exponential stability

The objective is to determine the long-term behaviour of the solution. We do not find a Lyapunov functional; consequently, we used the linearised system of equations. We analyse the spectral properties of the linearised operator, which can be represented as a perturbation of the one studied in the work of Cañizo and Lods [Reference Cañizo and Lods15].

In what follows, we assume the additional hypothesis:

1. (H5) $\exists \,\beta \in [0,1], \exists \,b \gt 0, \,\forall i\geq 1, \,b_i\leq bi^\beta$ .

Remark 2.10. We point out that Remark 2. 8 with $z_s \neq 0$ and (H2) implies that $\inf _{i\geq 1}\frac {b_i}{a_i}$ is a positive constant.

Theorem 2.11 (Local exponential convergence). Assume (H1)–(H5). Let $\lambda \lt \lambda _s$ and $C^{init}$ such that

(2.12) \begin{equation} \kappa (\lambda ,C^{init}) \lt \inf _{i\geq 1}\frac {b_i}{a_i} \end{equation}

and that ( 2.7 ) holds for some $\nu \gt 0$ . Let $C=(C_i)_{i\geq 1}$ a solution of ( 1.1 ) with initial condition $C^{init}$ . Take $z$ satisfying ( 2.11 ). Then, there exists some $\eta \in (0,\nu )$ , some $K,\varepsilon \gt 0$ and some $\mu _\ast \gt 0$ such that if

(2.13) \begin{equation} \frac {8b}{\sqrt {z}}\sqrt {\sum _{i=1}^{+\infty }i^{2\beta }Q_iz^i}\sup _{k\geq 1} \left (\sum _{j=k+1}^{+\infty } Q_jz^j\right )\left (\sum _{j=1}^k\frac {1}{a_jQ_jz^j}\right )\lt 1 \end{equation}

and

(2.14) \begin{equation} \sum _{i=1}^{+\infty } \exp\!(\eta i)|C_i^{init}-Q_iz^i| \leq \varepsilon , \end{equation}

then

(2.15) \begin{equation} \sum _{i=1}^{+\infty }\exp\!(\eta i)|C_i(t)-Q_iz^i| \leq K\exp\!(\!-\mu _\ast t),\, \forall t\geq 0. \end{equation}

Remark 2.12. Condition ( 2.13 ) is a sufficient condition. One might ask whether this condition is ever met. To address this, we give the following examples.

Example 2.13 (Constant coefficients). Constant kinetics coefficients $a_i = a$ and $b_i = b$ satisfy hypotheses (H1)–(H5) for some $a,b$ well choose. The condition ( 2.12 ) becomes either $C_1^{init} \lt \frac {b}{a}$ or $\lambda \lt 2\frac {b^2}{a}$ . The condition ( 2.13 ) becomes

\begin{equation*} \frac {(z_s-z)^{5/2}}{8zz_s^{5/2}} \gt 1. \end{equation*}

Since the left-hand side goes to $+\infty$ when $z$ goes to $0$ , this condition holds for $\lambda$ small enough. Indeed, recalling the definition of $z$ ( 2.11 ), we see that the smaller $\lambda$ is, the smaller $z$ is too.

Example 2.14 (Linear coefficients). Linear coefficients $a_i = ai$ and $b_i = bi$ satisfy hypotheses (H1)–(H5) for some $a,b$ well choose. The condition ( 2.13 ) becomes for $z$ small enough

\begin{equation*} \frac {32z}{\sqrt {z_s}(z_s-z)}\lt 1. \end{equation*}

As before, since the left-hand side goes to $0$ when $z$ goes to $0$ , this condition holds for $\lambda$ small enough.

Details for Examples2.13 and 2.14 are, respectively, presented in Appendices A.1 and A.2.

2.3. Numerical simulations

To simulate our equations numerically, we need to truncate our system, since we obviously cannot keep an infinite number of ordinary differential equations. We, therefore, have different ways of truncating our system. Generally speaking, for coagulation-fragmentation equations, there are two ‘natural’ truncations: conservative truncation and non-conservative truncation. Conservative truncation means that clusters of the maximum size cannot grow in size through the coagulation process, and those clusters are not the result of fragmentation; we have the following system of equations:

(2.16) \begin{equation} \left \{ \begin{aligned} \frac {d}{dt}C_1(t) &= \lambda - \sum _{j=1}^{n-1}a_{j}C_1(t)C_j(t) - a_1C_1(t)^2, & i=1, \\ \frac {d}{dt}C_i(t) &= J_{i-1}(t) - J_i(t), & 1\lt i\lt n, \\ \frac {d}{dt}C_n(t) &= J_{n-1}(t), & i=n. \end{aligned} \right . \end{equation}

This truncation corresponds to $J_i = 0$ for $i\geq n$ . Another truncation exists, taking $J_n = a_nC_1C_n$ , referred to as non-conservative truncation, which means that at the size of truncation, we are allowing this maximum size to coagulate and as before no aggregates of this size are the result of depolymerisation.

To numerically simulate these previous systems, we can just use standard Ordinary Differential Equation (ODE) schemes (e.g. Runge-Kutta 4); however, choosing a large truncation $n$ may involve very long computation time. Therefore, the goal is to develop a coarse-grain scheme, consisting of sub-sampling the clusters at some given size, preserving the asymptotics of the system.

The main idea is to see our truncation (2.16) as some discretisation of a particular Partial Differential Equation (PDE) for some size step $\Delta x$ , in which the size of the aggregates $x$ is a continuous variable. Let $n\geq 2$ be an integer, the aforementioned PDE is the following:

(2.17) \begin{equation} \frac {\partial C(t,x)}{\partial t} = -\frac {\partial J(t,x)}{\partial x}, \text{ in } [0,T]\times (1,n), \end{equation}

with

(2.18) \begin{equation} J(t,x) = -a(x)Q(x)C(t,1)^{x+1/2}\frac {\partial }{\partial x}\left (\frac {C(t,x)}{Q(x)C(t,1)^x}\right )\!, \end{equation}

where the functions $a(x),b(x)$ , and $Q(x)$ are the step functions of the discrete quantities $a_i,b_i$ , and $Q_i$ . Formally, (2.18) is obtained through the following discretisation with $\Delta x = 1$

(2.19) \begin{align} J_i(t) & = \frac {1}{\Delta x}\left (a_{i-1}C_1C_{i-1} - a_{i-1}\frac {Q_{i-1}}{Q_i}C_1\right ) \nonumber \\ &= -a_{i-1}Q_{i-1}C_1^i \frac {1}{\Delta x}\left (\frac {C_i}{Q_iC_1^i}-\frac {C_{i-1}}{Q_{i-1}C_1^{i-1}}\right ) \approx J\left (t,i-\frac {1}{2}\right )\!. \end{align}

To which we add boundary conditions, the right one depends on the truncation that we are considering, and the left one comes from $C_1$ . This idea is, in particular, presented by Duncan and Soheili [Reference Duncan and Soheili26]. Thus, taking another discretisation of the PDE, for example, with a non-uniform mesh, and simulating the PDE on this mesh should approach the desired solution. To that end, we are going to use the idea presented by Goudon and Monasse [Reference Goudon and Monasse29]. We mention the work of Bolton and Wattis [Reference Bolton and Wattis12], in which they develop a coarse-grain scheme on the Becker–Döring equations. Details are presented in Section 6.

2.4. Open questions

We highlight some questions that remain open. The uniqueness of a solution without the moment control, as in [Reference Laurençot and Mischler35, Theorem 2.1], is open since no quantity is preserved. The long-time behaviour of the solution, except for our local stability result, remains open even in the sub-critical case, and especially when there exists an infinity of steady states. The behaviour of the solution in the super-critical case is even more unclear since no steady state exists.

3.1. Preliminary results

As in many works on similar coagulation-fragmentation equations (e.g. [Reference Ball and Carr6, Reference Ball, Carr and Penrose7, Reference Carr17] or [Reference Carr and da Costa18]), the existence of a solution to (1.1) satisfying $C(0)=C^{init}$ comes by taking the limit of solutions to a finite number system of differential equations obtained by truncating our system of equations (1.1). More precisely, let $n\geq 3$ , and recall the system of $n$ ordinary differential equations (2.16), to which we still add the initial condition

(3.4) \begin{equation} C_i^n(0) = C_i^{init} \geq 0, \, 1\leq i\leq n. \end{equation}

As in the infinite system, we can write the weak form of this system of equations. It reads as follows:

(3.5) \begin{equation} \frac {d}{dt}\sum _{i=1}^{n} \varphi _iC_i = \lambda \varphi _1 - \varphi _2b_2C_2 + \sum _{i=1}^{n-1} \left [\varphi _{i+1}-\varphi _i-\varphi _1\right ]a_iC_1C_i + \sum _{i=3}^{n} \left [\varphi _{i-1}-\varphi _i\right ]b_iC_i. \end{equation}

To start, we prove the well-posedness of the truncated system of equations (2.16).

Proposition 3.3. Let $n\geq 3$ and $C^{init}\in X_\alpha ^+$ . There exists a unique solution $C^n \in \mathcal{C}^1([0,+\infty ),\mathbb{R}^n)$ to ( 2.16 ) satisfying $C_i^n(0)=C_i^{init}$ for $i=1,\ldots ,n$ . This solution satisfies $C_i^n(t) \geq 0$ for all $t\geq 0$ and $i=1,\ldots ,n$ and

(3.6) \begin{equation} C_i^n(t) \leq \sum _{i=1}^n i^\alpha C_i^n(t) \leq \lambda T + \|C^{init}\|_{X_\alpha ^+} \end{equation}

for all $T\in (0,+\infty )$ and $i=1,\ldots ,n$ .

Proof. The proof is a straightforward application of the Cauchy–Lipschitz theorem. The non-negativity follows from [Reference Ducrot, Griette, Liu and Magal25, Theorem 7.1 (Positivity)]. The fact that

\begin{equation*} \frac {d}{dt}\sum _{i=1}^n C_i^n(t) \leq \lambda \end{equation*}

entails non-explosion in finite time. Moreover, using (3.5) with $\varphi _i = i^\alpha$ and since $\alpha \in [0,1]$ , we obtain

\begin{align*} \frac {d}{dt}\sum _{i=1}^n i^\alpha C_i^n &= \lambda - 2^\alpha b_2C_2^n + \sum _{i=1}^{n-1}\left [(i+1)^\alpha -i^\alpha -1\right ]a_iC_i^nC_1^n + \sum _{i=3}^n\left [(i-1)^\alpha -i^\alpha \right ]b_iC_i^n \leq \lambda , \end{align*}

which entails (3.6).

Our objective is now to show that we can extract from the sequence $(C^n)_{n\geq 1}$ , even if we take $C_i^n = 0$ for $i\gt n$ , a sub-sequence, which converges, and the limit is a solution, i.e. satisfies the Definition2.1. In order to do that, we need to establish the following two lemmas.

Lemme 3.4. Let $T\in (0,+\infty )$ and $U\in \mathcal{K}_1\cup \mathcal{K}_2$ . There exists a constant $\gamma _T$ depending only on $a, U, C^{init}$ , and $T$ such that for all $n\geq 3$ and $t \in [0,T]$ , we have

(3.7) \begin{equation} \sum _{i=1}^n U(i^\alpha )C_i^n(t) \leq \gamma _T \sum _{i=1}^n U(i^\alpha )C_i^{init} + \lambda U(1)T\gamma _T \end{equation}

Proof. Let $n\geq 3$ and $M_U^n(t) \,:\!= \sum _{i=1}^n U(i^\alpha )C_i^n(t)$ . We have for $t\geq 0$

\begin{align*} \frac {d}{dt}M_U^n(t) &\leq \lambda U(1) - U(2^\alpha )b_2C_2^n(t) + \sum _{i=3}^{n} \left [U((i-1)^\alpha )-U(i^\alpha )\right ]b_{i}C_{i}^n(t) \\ &\quad + \sum _{i=1}^{n-1}\left [U((i+1)^\alpha )-U(i^\alpha )-U(1)\right ]a_{i}C_i^n(t)C_1^n(t) \end{align*}

Noticing, for all $i\geq 1$ and $\alpha \in [0,1]$ , $ (i+1)^\alpha \leq i^\alpha + 1$ , and as $U \in \mathcal{K}_1\cup \mathcal{K}_2$ , $U$ is increasing, thus $U((i+1)^\alpha ) \leq U(i^\alpha + 1)$ for all $i\geq 1$ . Moreover, from [Reference Laurençot34, Lemma 3.2], there exists a constant $m_U$ depending only on $U$ such that for all $i\geq 1$

(3.8) \begin{equation} (i^\alpha + 1)(U(i^\alpha + 1)-U(i^\alpha )-U(1)) \leq m_U(i^\alpha U(1) + U(i^\alpha )). \end{equation}

Therefore, by (H1) and (3.8), we obtain

\begin{align*} \frac {d}{dt}M_U^n(t) &\leq \lambda U(1) - U(2^\alpha )b_2C_2^n(t) + \sum _{i=3}^{n} \left [U((i-1)^\alpha )-U(i^\alpha )\right ]b_{i}C_{i}^n(t) \\ &\quad + am_U\sum _{i=1}^{n-1}(i^\alpha U(1)+U(i^\alpha ))C_i^n(t)C_1^n(t), \end{align*}

with $a$ , respectively, $m_U$ , the constant from (H1), respectively (3.8). As $U$ is increasing, $U((i-1)^\alpha )-U(i^\alpha ) \leq 0$ for all $i\geq 1$ , thus

\begin{align*} \frac {d}{dt}M_U^n(t) &\leq \lambda U(1) + 2 a m_U M_\alpha ^n(t) M_U^n(t) \\ &\leq \lambda U(1) + 2a m_U\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)M_U^n(t) \end{align*}

Therefore, using Grönwall’s lemma, we conclude

\begin{equation*} M_U^n(t) \leq \big(M_U^n(0) + \lambda U(1)T\big)e^{2a m_U\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)T} \end{equation*}

which ends the proof.

Lemme 3.5. Let $T \in (0,+\infty )$ . For each $i\geq 1$ , there exists a constant $\Gamma _i(T)$ depending only on $a, i, \|C^{init}\|_{X_\alpha }$ et $T$ such that, for all $n\geq i$ ,

(3.9) \begin{equation} \left \| \frac {d}{dt}C_i^n \right \|_{\mathrm{L}^1(0,T)} \leq \Gamma _i(T). \end{equation}

Proof. From Proposition3.3, there exists a constant $K_T$ such that $\left | C_i^n(t)\right | \leq K_T$ for all $i\geq 1$ . Hence, for $i\geq 2$ , $\left |\dfrac {d}{dt}C_i^n(t)\right |$ is uniformly bounded in $\mathrm{L}^1(0,T)$ according to $n$ , thanks to equations (2.16). It remains to bound $\left |\dfrac {d}{dt}C_1^n(t)\right |$ , which follows from the bound

\begin{equation*} \int _0^T \sum _{i=1}^{n-1}a_iC_1^nC_i^n \leq K_Ta \int _0^T\sum _{i=1}^n i^\alpha C_i^n \leq TaK_T^2. \end{equation*}

3.2. Proof of existence theorem

We are now in a position to prove Theorem3.2. However, let us not forget a refined version of the de La Vallée–Poussin theorem for integrable functions from [Reference Lê38, Proposition I.1.1], which we are going to need. That is the key point of Laurençot’s proof in [Reference Laurençot34, Theorems 2.3 and 2.5] that enables him to provide a different kind of proof without the need for cleverly decomposing the sums as in [Reference Ball and Carr6, Theorems 2.4 and 2.5].

Theorem 3.6 (de La Vallée-Poussin). Let $\left (\Omega ,\mathcal{B},\mu \right )$ a measured space, and let $w\in \mathrm{L}^1(\Omega ,\mathcal{B},\mu )$ . Then, there exists a function $V \in \mathcal{K}_{1,\infty }$ such that

\begin{equation*} V(|w|) \in \mathrm{L}^1(\Omega ,\mathcal{B},\mu ). \end{equation*}

Proof of Theorem 3.2. First, we want to apply Theorem3.6 with $\Omega = \mathbb{N}^\ast$ . To that end, we define the measure $\mu$ by

\begin{equation*} \mu (I) = \sum _{i\in I} C_i^{init},\,\, I \subset \mathbb{N}\backslash \left \{0\right \}\!. \end{equation*}

The condition $C^{init} \in X_\alpha ^+$ ensures that the function $x\mapsto x^\alpha$ belongs to $\mathrm{L}^1(\Omega ,\mathcal{B},\mu )$ . Therefore, using Theorem3.6, there exists a function $U_0 \in \mathcal{K}_{1,\infty }$ such that $x\mapsto U_0(x^\alpha )$ belongs to $\mathrm{L}^1(\Omega ,\mathcal{B},\mu )$ , i.e.

(3.10) \begin{equation} \mathcal{U}_0 \,:\!=\, \sum _{i=1}^{+\infty } U_0(i^\alpha )C_i^{init} \lt \infty . \end{equation}

Moreover, for $n\geq i$ ,

(3.11) \begin{equation} C_i^n(t) \leq M_\alpha ^n(t) \leq \lambda T + \|C^{init}\|_{X_\alpha },\text{ on } [0,T], \end{equation}

and by Lemma3.5, the sequence $(C_i^n)_{n\geq i}$ is bounded in $\mathrm{L}^\infty (0,T)\cap \mathrm{W}^{1,1}(0,T)$ for each $i\geq 1$ and $T\in (0,\infty )$ . Therefore, using Helly’s theorem [Reference Kolmogorov and Fomin33, Théorème 5 (p. 372)], there exists a subsequence of $(C_i^n)_{n\geq i}$ , still denoted $(C_i^n)_{n\geq i}$ , and a sequence $C$ of bounded variation functions such that

(3.12) \begin{equation} \lim _{n\to \infty } C_i^n(t) = C_i(t),\,\,\forall i\geq 1,\,\forall t\geq 0. \end{equation}

Thus, using (3.11) and (3.12), we have $C\in X_\alpha ^+$ , with

(3.13) \begin{equation} \|C(t)\|_{X_\alpha ^+} \leq \lambda T + \|C^{init}\|_{X_\alpha }. \end{equation}

It now remains to show that the potential candidate $C$ is indeed a solution of (1.1), satisfying $C(0) = C^{init}$ . To that end, we need to show two estimates. First, as $U_0 \in \mathcal{K}_{1,\infty }$ , using Lemma3.4 and (3.10), we have for all $T\geq 0$ , and $n\geq 3$ ,

(3.14) \begin{equation} \sum _{i=1}^n U_0(i^\alpha )C_i^n(t) \leq \xi (T),\,\, t\in [0,T], \end{equation}

where the constant $\xi (T)$ depend only on $a, C^{init}, U_0$ , and $T$ .

Let $T\in (0,+\infty )$ and $m\geq 2$ , considering $C_i^n = 0$ if $i\geq n$ , the previous estimate becomes, for $n\geq m+1$ ,

\begin{equation*} \sum _{i=1}^m U_0(i^\alpha )C_i^n(t) \leq \xi (T),\,\, t\in [0,T]. \end{equation*}

Thanks to (3.12), we can pass to the limit when $n\to +\infty$ in the previous estimates; thus, it stays true substituting $C_i^n$ by $C_i$ . We then pass to the limit on $m\to +\infty$ , and we obtain

(3.15) \begin{equation} \sum _{i=1}^{+\infty } U_0(i^\alpha )C_i(t) \leq \xi (T),\,\, t\in [0,T]. \end{equation}

Second, using (H1) and (3.13), we obtain for all $i\geq 1$

(3.16) \begin{equation} \sum _{j=1}^{+\infty }a_{j}C_j \in \mathrm{L}^1(0,T). \end{equation}

Now, we state that, for all $i\geq 1$ , we have

(3.17) \begin{align} \lim _{n\to +\infty }\left \| C_1^n\sum _{j=1}^{n-1} a_{j}C_j^n - C_1 \sum _{j=1}^{+\infty } a_{j}C_j \right \|_{\mathrm{L}^1(0,T)} = 0,\end{align}

(3.18) \begin{align} \lim _{n\to +\infty }\left \| a_iC_i^nC_1^n - a_iC_iC_1 \right \|_{\mathrm{L}^1(0,T)} = 0,\end{align}

(3.19) \begin{align} \lim _{n\to +\infty } \left \| b_iC_i^n - b_iC_i \right \|_{\mathrm{L}^1(0,T)} = 0. \end{align}

The last two assertions (3.18) and (3.19) come from (3.12), (3.13), (3.6) and from Lebesgue-dominated convergence theorem. It remains to prove the first assertion (3.17). Let $i\geq 1$ fixed, and $m\geq 2$ . As before, using (3.12), (3.13) and (3.6) and Lebesgue-dominated convergence theorem, we obtain for $m\leq n-1$ ,

(3.20) \begin{equation} \lim _{n\to +\infty } \left \| \sum _{j=1}^m a_{j}\big(C_1^nC_j^n-C_1C_j\big) \right \|_{\mathrm{L}^1(0,T)} = 0. \end{equation}

Moreover, for $n-1\geq m+1$ , and using (H1), (3.13) and (3.15), we obtain

(3.21) \begin{align} \left \|\sum _{j=m+1}^{+\infty }a_{j}C_1C_j\right \|_{\mathrm{L}^1(0,T)} &\leq a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)\left \|\sum _{j=m+1}^{+\infty }j^\alpha C_j\right \|_{\mathrm{L}^1(0,T)} \nonumber \\[4pt] &\leq a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big) \sup _{j\geq m+1} \frac {j^\alpha }{U_0(j^\alpha )}\left \|\sum _{j=m+1}^{+\infty } U_0(j^\alpha ) C_j\right \|_{\mathrm{L}^1(0,T)} \nonumber \\[4pt] &\leq a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)T\xi (T) \sup _{j\geq m+1} \frac {j^\alpha }{U_0(j^\alpha )}. \end{align}

Likewise, using (H1), (3.6) and (3.14), we obtain

(3.22) \begin{align} \left \|\sum _{j=m+1}^{n-i}a_{j}C_1^nC_j^n\right \|_{\mathrm{L}^1(0,T)} &\leq a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)\left \|\sum _{j=m+1}^{+\infty }j^\alpha C_j^n\right \|_{\mathrm{L}^1(0,T)} \nonumber \\ &\leq a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big) \sup _{j\geq m+1} \frac {j^\alpha }{U_0(j^\alpha )}\left \|\sum _{j=m+1}^{+\infty } U_0(j^\alpha ) C_j^n\right \|_{\mathrm{L}^1(0,T)} \nonumber \\ &\leq a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)T\xi (T) \sup _{j\geq m+1} \frac {j^\alpha }{U_0(j^\alpha )}. \end{align}

Therefore, combining (3.20)–(3.22), we obtain that for all $m\geq 2$ ,

\begin{equation*} \limsup _{n\to +\infty } \left \| \sum _{j=1}^{n-1} a_{j}C_1^nC_j^n - \sum _{j=1}^{+\infty } a_{j}C_1C_j \right \|_{\mathrm{L}^1(0,T)} \leq 2a\big(\lambda T + \|C^{init}\|_{X_\alpha }\big)T\xi (T) \sup _{j\geq m+1}\frac {j^\alpha }{U_0(j^\alpha )}. \end{equation*}

As $U_0$ belongs to $\mathcal{K}_{1,\infty }$ , we have that the right-hand side of the above inequality tends to $0$ when $m\to +\infty$ , thus we obtain (3.17).

Now, thanks to (3.12) and (3.17)–(3.19), passing to the limit in $n$ we obtain $(ii)$ of the Definition2.1. By (3.16), we recover the continuity of $C_i$ for all $i\geq 1$ . Therefore, we have shown that $C=(C_i)_{i\geq 1} \in X_\alpha ^+$ is a solution of (1.1) on $[0,T]$ satisfying $C(0)=C^{init}$ . Note that the continuity of $C({\cdot})$ comes from the uniform convergence thanks to Dini’s theorem of the sequence of continuous functions $\left (\sum \limits _{i=1}^n i^\alpha C_i({\cdot})\right )_{n\geq 1}$ on $[0,T]$ .

To conclude, we only need to show that the solution constructed in the previous proof enjoys the additional property (3.3). As $U \in \mathcal{K}_1\cup \mathcal{K}_2$ , (3.3) follows from Lemma3.4 and the convergence (3.12).

5.1. Uniform-in-time bound

We start by proving Proposition2.6.

Proof of Proposition 2.6. Let $M\gt 1$ , for convenience, we rewrite condition (2.6) as

(5.1) \begin{equation} \frac {b_i}{a_i} \geq M\kappa \text{ for all } i\geq 1. \end{equation}

Let $t\geq 0$ and $\mu \in (1,M)$ . We have the following formal computations:

\begin{align*} \frac {d}{dt}\sum _{i=1}^{+\infty }\mu ^iC_i(t) &= \mu \left (\lambda - \sum _{j=1}^{+\infty }a_jC_j(t)C_1(t)-a_1C_1(t)^2\right ) + \sum _{i=2}^{+\infty }\mu ^i(J_{i-1}(t)-J_i(t)) \\ &= \mu \left (\lambda - \sum _{j=1}^{+\infty }a_jC_j(t)C_1(t)-a_1C_1(t)^2\right ) + (\mu - 1)\sum _{i=1}^{+\infty }\mu ^iJ_i + \mu (a_1C_1(t)^2-b_2C_2(t)) \\ &\leq \mu \lambda - \mu a_1C_1(t)^2 + (\mu -1)\sum _{i=1}^{+\infty }\mu ^i(a_iC_i(t)C_1(t)-b_{i+1}C_{i+1}(t)) + \mu (a_1C_1(t)^2-b_2C_2(t)) \\ &= \mu \lambda + (\mu -1)\sum _{i=1}^{+\infty }\mu ^{i-1}a_i\left (\mu C_1(t)-\frac {b_i}{a_i}\right )C_i(t) +(\mu -1)b_1C_1(t) - \mu b_2C_2(t) \\ &= \mu \lambda + \frac {(\mu -1)}{\mu }\sum _{i=1}^{+\infty }\mu ^{i}a_i\left (\mu C_1(t)-\frac {b_i}{a_i}\right )C_i(t) +(\mu -1)b_1C_1(t) - \mu b_2C_2(t) \\ &\leq \mu \lambda +(\mu -1)b_1C_1(t) + \frac {(\mu -1)}{\mu }\sum _{i=1}^{+\infty }\mu ^{i}a_i\left (\mu C_1(t)-\frac {b_i}{a_i}\right )C_i(t). \end{align*}

Then, using (2.4) and (5.1) and denoting $\varepsilon = (M-\mu )\kappa \gt 0$ , we have

\begin{align*} \frac {d}{dt}\sum _{i=1}^{+\infty }\mu ^iC_i(t) &\leq \mu \lambda + (\mu -1)b_1\kappa - \varepsilon \frac {\mu -1}{\mu }\sum _{i=1}^{+\infty }a_i\mu ^iC_i(t) \\ &\leq \mu \lambda + (\mu -1)b_1\kappa - \varepsilon \frac {\mu -1}{\mu }\underline {a}\sum _{i=1}^{+\infty }\mu ^iC_i(t). \end{align*}

Therefore, using Grönwall’s lemma, we obtain for all $t\geq 0$ and $\mu \in (1,M)$ ,

\begin{equation*} \sum _{i=1}^{+\infty }\mu ^iC_i(t) \leq \sum _{i=1}^{+\infty }\mu ^iC_i^{init} + \frac {\lambda \mu +(\mu -1)b_1\kappa }{\varepsilon \frac {\mu -1}{\mu }\underline {a}} \,:\!=\, \Xi _1, \end{equation*}

where $\Xi _1$ depends only on $\lambda$ , $(a_i)_{i\geq 1}$ , $(b_i)_{i\geq 1}$ , $\mu$ and the initial condition. Taking $\nu = \ln (\mu )$ , we have the desired estimation (2.8).

The rigorous proof can be left to the reader, as it consists in reproducing these estimates on the truncated system (2.16), (3.4) and then letting $n\to +\infty$ using $C_i^n \to C_i$ and $\sum _{i=1}^n \mu ^iC_i^n \to \sum _{i=1}^{+\infty } \mu ^iC_i$ .

As mention in the beginning, we have an upper bound on the mass and lower bound on $C_1(t)$ .

Proposition 5.1 (Mass uniform-in-time bound). Let $b\gt 0$ . Assume that $C^{init} \in X_1^+$ and that the kinetics coefficients satisfy the following conditions:

(5.2) \begin{equation} \forall i\geq 1,\, b_i\geq bi. \end{equation}

Then, for all $t\geq 0$ ,

(5.3) \begin{equation} \sum _{i=1}^{+\infty } iC_i(t) \leq \max \left \{\sum _{i=1}^{+\infty } iC_i^{init}, \frac {\lambda + b_1\kappa }{b}\right \}\!. \end{equation}

Proof of Proposition 5.1. Recalling (2.1) and using the assumption (5.2) and the uniform-in-time bound of $C_1$ (2.4), we have for all $t\geq 0$ ,

\begin{align*} \frac {d}{dt}\sum _{i=1}^{+\infty }iC_i(t) &= \lambda -b_2C_2 - \sum _{i=2}^{+\infty }b_iC_i \\ &\leq \lambda -2bC_2+b_1C_1-b\sum _{i=1}^{+\infty }iC_i \\ &\leq \lambda + b_1\kappa - b\sum _{i=1}^{+\infty } iC_i. \end{align*}

Therefore, for all $t\geq 0$ ,

\begin{equation*} \sum _{i=1}^{+\infty }iC_i(t) \leq \max \left \{\sum _{i=1}^{+\infty } iC_i^{init}, \frac {\lambda + b_1\kappa }{b}\right \}\!. \end{equation*}

Proposition 5.2 ( ${{C}}_{1}$ uniform-in-time positivity). Let $a,b\gt 0$ . Assume that the kinetics coefficients satisfy the following conditions:

(5.4) \begin{equation} \forall i\geq 1,\, a_i\leq ai \,\text{ and }\, b_i\geq bi. \end{equation}

Then, there exists a constant $\eta \gt 0$ and $t_1 \geq 0$ such that for all $t\geq t_1$ , we have

(5.5) \begin{equation} C_1(t) \geq \eta . \end{equation}

Proof of Proposition 5.2. First, using (5.4) and the uniform-in-time bound of the mass (5.3), we have for all $t\geq 0$ ,

\begin{equation*} \sum _{i=1}^{+\infty }a_iC_i(t) \leq a \sum _{i=1}^{+\infty }iC_i(t) \leq a \max \left \{\sum _{i=1}^{+\infty }iC_i^{init},\frac {\lambda +b_1\kappa }{b}\right \}\!. \end{equation*}

Then, for all $t\geq 0$ , we have

\begin{align*} \frac {d}{dt}C_1(t) &= \lambda - \sum _{j=1}^{+\infty }a_jC_j(t)C_1(t) - a_1C_1(t)^2 \\ &\geq \lambda - a \max \left \{\sum _{i=1}^{+\infty }iC_i^{init},\frac {\lambda +b_1\kappa }{b}\right \}C_1(t) - a_1C_1(t)^2. \end{align*}

Therefore, as long as $C_1 \leq \min \left \{ 1,a \max \left \{\sum \limits _{i=1}^{+\infty }iC_i^{init},\frac {\lambda +b_1\kappa }{b}\right \} +a \right \}$ , we have

\begin{align*} \frac {d}{dt}C_1(t) &\geq \lambda - \left [ a \max \left \{\sum _{i=1}^{+\infty }iC_i^{init},\frac {\lambda +b_1\kappa }{b}\right \} + a_1\right ] C_1(t)\\ &\geq \lambda - \left [ a \max \left \{\sum _{i=1}^{+\infty }iC_i^{init},\frac {\lambda +b_1\kappa }{b}\right \} + a\right ] C_1(t) \geq 0. \end{align*}

Hence, there exists $t_1 \geq 0$ such that for all $t\geq t_1$ ,

\begin{equation*} C_1(t) \geq \frac {1}{2} \min \left \{ 1, a \max \left \{\sum _{i=1}^{+\infty }iC_i^{init},\frac {\lambda +b_1\kappa }{b}\right \} + a \right \} \,=\!:\, \eta . \end{equation*}

5.2. Steady states

In this subsection, we prove Theorem2.9. Let $C=(C_i)_{i\geq 1}$ a constant solution. From (1.1), we obtain

\begin{equation*} J_{i-1} = J_i, \,\forall i \geq 2. \end{equation*}

Therefore, there exists $J \in \mathbb{R}$ such that

(5.6) \begin{equation} J_i = J, \,\forall i\geq 1. \end{equation}

We then have three possibilities: either $J=0$ , or $J \gt 0$ or $J\lt 0$ . We immediately obtain that there is no steady state with a positive flux $J\gt 0$ . Indeed, from (H1) and the positivity of solutions, we have

\begin{equation*} a_iC_1C_i = J + b_{i+1}C_{i+1} \geq J. \end{equation*}

Therefore,

\begin{equation*} \sum _{i=1}^{+\infty }a_iC_1C_i \geq \sum _{i=1}^{+\infty }J = +\infty , \end{equation*}

which means that there is no steady state when $J\gt 0$ due to the equation on $C_1$ .

The case $J=0$ gives the steady-state $(Q_iz^i)_{i\geq 1}$ . Assuming that a steady state exists, we denote it $C^{eq}$ , the equation (5.6) entails the following relation:

\begin{equation*} a_iC_1^{eq}C_i^{eq} = b_{i+1}C_{i+1}^{eq}, \,\forall i\geq 1. \end{equation*}

Thus,

\begin{equation*} C_i^{eq} = \left (\prod _{j=2}^{i} \frac {a_{j-1}}{b_j}\right )\big(C_1^{eq}\big)^i = Q_i\big(C_1^{eq}\big)^i, \,\forall i\geq 1. \end{equation*}

The steady state is therefore entirely determined by $C_1^{eq}$ ; to determine this value, we use the first equation of the system (1.1), which has not yet been used. We, therefore, have that $C_1^{eq}$ is a solution of the following equation in $z$

(5.7) \begin{equation} \lambda = \sum _{j=1}^{+\infty }a_{j}Q_jz^{j+1} + a_1 z^2. \end{equation}

By comparing with the critical production threshold $\lambda _s$ defined in (2.10), the equation (5.7) admits a unique solution when $\lambda \leq \lambda _s$ , which verifies $z \leq z_s$ .

The last case is the negative flux $J\lt 0$ . Using equation (5.6), we have

(5.8) \begin{equation} J(z) \,:\!=\, a_izg_i(z) - b_{i+1}g_{i+1}(z), \,\, \forall i\geq 1, \end{equation}

where $z$ and functions $g_i(z)$ designate a constant solution. In other words, we search for a steady state

\begin{equation*} C_i = g_i(z), \end{equation*}

for all $i\geq 1$ , with $g_1(z) = z$ .

We now want to determine $J$ and the $g_i$ functions. To do this, we start from (5.8), divide the two members of this equality by $a_iQ_iz^{i+1}$ , and using (2.9), we obtain

(5.9) \begin{equation} \frac {J(z)}{a_iQ_iz^{i+1}} = \frac {a_izg_i(z)}{a_iQ_iz^{i+1}} - \frac {b_{i+1}g_{i+1}(z)}{a_iQ_iz^{i+1}} = \frac {g_i(z)}{Q_iz^i} - \frac {g_{i+1}(z)}{Q_{i+1}z^{i+1}} \end{equation}

Summing from $i=1$ to $k-1$ , we obtain

(5.10) \begin{equation} J(z)\sum _{i=1}^{k-1} \frac {1}{a_iQ_iz^{i+1}} = 1 - \frac {g_k(z)}{Q_kz^k}. \end{equation}

Rewriting (5.10), we obtain the expression of $g_k(z)$ for all $k\geq 2$ :

(5.11) \begin{equation} g_k(z) = Q_kz^k\left [ 1 - J(z)\sum _{i=1}^{k-1}\frac {1}{a_iQ_iz^{i+1}} \right ]\!, \end{equation}

and we also find $g_1(z) = z$ .

In order for the sequence $(g_k(z))_{k\geq 1}$ to be a steady state, it needs to be summable against $(a_k)_{k\geq 1}$ . We have

\begin{equation*} \sum _{k=2}^{+\infty }a_kg_k(z) = \sum _{k=2}^{+\infty } a_kQ_kz^k - J(z)\sum _{k=2}^{+\infty }a_kQ_kz_k\left (\sum _{i=1}^{k-1}\frac {1}{a_iQ_iz^{i+1}}\right )\!. \end{equation*}

From the definition of $z_s$ as the radius of convergence of a power series, we observe that if $z\gt z_s$ , then the series $\sum a_kQ_kz^k$ is divergent; therefore, there is no steady state in the super-critical case. We now suppose that $z\leq z_s$ . We have for all $k\geq 1$ ,

(5.12) \begin{align} a_kg_k(z) = a_kQ_kz^k\left [ 1 - J(z) \sum _{i=1}^{k-1}\frac {1}{a_iQ_iz^{i+1}} \right ] &\gt |J(z)|a_kQ_kz_k\sum _{i=1}^{k-1}\frac {1}{a_iQ_iz^{i+1}} \nonumber\\ &= |J(z)| \frac {a_k}{b_k} + \sum _{i=1}^{k-2}\frac {a_kQ_k}{a_iQ_i}z^{k-(i+1)} \end{align}

Therefore, using (H3) and (H4), if $z_s\lt +\infty$ , then $a_kg_k(z)$ does not converge towards 0, and therefore, the series diverge. This implies that there is no steady state in the super-critical case, and no other steady state in the sub-critical case when $z_s$ is finite.

It remains to prove the uniqueness of the steady state or the existence of an infinite number of steady-state depending on the comparison of the kinetics rates.

First case: $b_i \leq i a_i$ for $i \gg 1$ . Recalling the expression (5.11), we show that in this case they are not summable against $(a_k)_{k\geq 1}$ . Let $h(x)=\sum _{k=1}^{+\infty } a_kQ_kx^k$ for $x\in [0,z_s)$ ; noticing that all derivatives of $h$ are increasing in $x$ and using (2.9), we have

\begin{align*} \sum _{k=2}^{+\infty } a_kg_k(z) &\geq |J|\sum _{k=2}^{+\infty }a_kQ_kz^k\sum _{i=1}^{k-1}\frac {1}{a_iQ_iz^{i+1}} = |J|\sum _{i=2}^{+\infty } \frac {1}{a_iQ_iz^{i+1}}\sum _{k=i+1}^{+\infty }a_kQ_kz^k \\ &= |J|\sum _{i=2}^{+\infty } \frac {1}{a_iQ_iz^{i+1}}\int _0^z h^{(i+1)}(t)\frac {(z-t)^i}{i!}dt \geq \frac {|J|}{z} \sum _{i=2}^{+\infty } \frac {1}{a_iQ_iz^i}\frac {h^{(i+1)}(0)}{i!} \int _0^z (z-t)^i dt \\ &= \frac {|J|}{z}\sum _{i=2}^{+\infty } \frac {(i+1)a_{i+1}}{b_{i+1}} \int _0^z \left (1-\frac {t}{z}\right )^i dt = |J|\sum _{i=2}^{+\infty } \frac {a_{i+1}}{b_{i+1}} \\ &\geq |J|\sum _{i=2}^{+\infty } \frac {1}{i} = +\infty . \end{align*}

Which proves that in this case, there is no other steady state.

Second case: $b_i \gt i^\nu a_i$ for $i\gg 1$ with $\nu \gt 1$ . Using the relation (5.6), we obtain for $J\lt 0$

\begin{equation*} C_{i+1} = \frac {a_izC_i+|J|}{b_{i+1}} \,\forall i\geq 1 \text{ and } C_1 = z. \end{equation*}

Solving the recurring sequence, we obtain for $i\geq 2$

\begin{equation*} C_i = |J|\sum _{k=0}^{i-2} \frac {Q_i}{a_{i-1-k}Q_{i-1-k}}z^k + Q_iz^i. \end{equation*}

First, we prove that this sequence is summable against $(a_i)_{i\geq 1}$ . Since $\sum a_iQ_iz^i$ is finite for $z\lt z_s$ , it remains to prove that the first term is summable against $(a_i)_{i\geq 1}$ . We have

\begin{equation*} \sum _{i=1}^{+\infty }a_i\sum _{k=0}^{i-2} \left (\frac {Q_i}{a_{i-1-k}Q_{i-1-k}}\right )z^k = \sum _{k=0}^{+\infty } \sum _{i=k+2}^{+\infty } \frac {a_iQ_i}{a_{i-1-k}Q_{i-1-k}}z^k = \sum _{k=0}^{+\infty } \underbrace {\left [\sum _{j=2}^{+\infty }\frac {a_{j+k}Q_{j+k}}{a_{j-1}Q_{j-1}}\right ]}_{\,:\!=\, \gamma _k}z^k \end{equation*}

For all $k\geq 0$ , we prove that $\gamma _k \lt +\infty$ . Let $k\geq 0$ , there exists $j_0 \gg 1$ such that for $j\geq j_0$ we have $Q_{j+k} \leq Q_j$ . Thus, using (H4), the sequence of ratio is bounded by $K \geq 0$ and the hypothesis of the second case, we have

\begin{equation*} \sum _{j\geq j_0} \frac {a_{j+k}Q_{j+k}}{a_{j-1}Q_{j-1}} \leq \sum _{j\geq j_0} \frac {a_{j+k}Q_{j}}{a_{j-1}Q_{j-1}} = \sum _{j\geq j_0} \frac {a_{j+k}}{b_j} \leq \sum _{j\geq j_0} \frac {a_{j+k}}{j^\nu a_j} \leq K^{k+1} \sum _{j\geq j_0} \frac {1}{j^\nu } \lt +\infty . \end{equation*}

This is where the hypothesis of the second case, mainly $\nu \gt 1$ , is crucial. Otherwise, $\gamma _k = +\infty$ and the computation does not allow us to conclude. Computing the ratio of $\gamma _k$ , we have

\begin{align*} \frac {\gamma _{k+1}}{\gamma _k} = \frac {\sum \limits _{j=2}^{+\infty }\dfrac {a_{j+k+1}Q_{j+k+1}}{a_{j-1}Q_{j-1}}}{\sum \limits _{j=2}^{+\infty }\dfrac {a_{j+k}Q_{j+k}}{a_{j-1}Q_{j-1}}} = \frac {\sum \limits _{j=2}^{+\infty }\dfrac {a_{j+k}Q_{j+k}}{a_{j-1}Q_{j-1}}\dfrac {a_{j+k+1}}{b_{j+k+1}}}{\sum \limits _{j=2}^{+\infty }\dfrac {a_{j+k}Q_{j+k}}{a_{j-1}Q_{j-1}}}. \end{align*}

However, for $k\gg 1$ using the hypothesis of the second case, we have

\begin{equation*} \frac {a_{j+k+1}}{b_{j+k+1}} \lt \frac {a_{j+k+1}}{(j+k+1)^\nu a_{j+k+1}} \lt \frac {1}{(k+1)^\nu }. \end{equation*}

Therefore,

\begin{equation*} \frac {\gamma _{k+1}}{\gamma _k} \leq \frac {1}{(k+1)^\nu } \xrightarrow {k\to +\infty } 0. \end{equation*}

Using D’Alembert’s ratio test, the sum $\sum a_iC_i$ converges for all $z\geq 0$ . To obtain a steady state, it remains to satisfy the first equation and if there is any condition on $\lambda$ , $z$ or $J$ . The equation

\begin{equation*} a_1 z^2 + z |J| \sum _{k=0}^{+\infty } \gamma _k z^k + \sum _{k=1}^{\infty } a_kQ_kz^{k+1} = \lambda \end{equation*}

admits, for all $J\lt 0$ , a unique $z$ such that the above equation is satisfied, since the left-hand side of the equation is a strictly increasing function of $z$ from 0 to $+\infty$ . We, therefore, obtain an infinity of steady states parameterised by $J$ .

5.3. Local exponential stability: Sub-critical case

Let $\Theta _i = Q_iz^i$ for all $i\geq 1$ . We introduce the ansatz $C_i = \Theta _i(1+h_i)$ where $h_i$ denotes the fluctuation around $\Theta _i$ . We introduce the Hilbert space $\mathcal{H} \,:\!=\, \ell ^2(\Theta _i)$ and a subspace $\mathcal{H}_{\mathcal{D}} = \ell ^2(\Theta _i(1+\sigma _i^2))$ namely

\begin{equation*} \mathcal{H} = \left \{ h = (h_i)_{i\geq 1} \;:\; \|h\|_{\mathcal{H}} \,:\!=\, \left (\sum _{i=1}^{+\infty } \Theta _ih_i^2\right )^{1/2} \lt \infty \right \}\!, \end{equation*}

and

\begin{equation*} \mathcal{H}_{\mathcal{D}} = \left \{ h = (h_i)_{i\geq 1} \;:\; \|h\|_{\mathcal{H}_{\mathcal{D}}} \,:\!=\, \left (\sum _{i=1}^{+\infty } \Theta _i(1+\sigma _i^2)h_i^2\right )^{1/2} \lt \infty \right \}\!, \end{equation*}

where the sequence $(\sigma _i)_{i\geq 1}$ is defined later in (5.19). The inner scalar product of $\mathcal{H}$ is denoted by $\langle \cdot ,\cdot \rangle _{\mathcal{H}}$ . We define two linear operators $S$ and $P$ on $\mathcal{H}$ by the following weak form

(5.13) \begin{equation} \sum _{i=1}^{+\infty } S_i(h)\Theta _i\varphi _i = \sum _{i=1}^{+\infty } [\varphi _{i+1}-\varphi _i-\varphi _1]a_i\Theta _1\Theta _i(h_i+h_1-h_{i+1}), \end{equation}

and

(5.14) \begin{equation} \sum _{i=1}^{+\infty } P_i(h)\Theta _i\varphi _i = - \varphi _1\left (a_1\Theta _1^2h_2+\sum _{i=1}^{+\infty }a_i\Theta _1\Theta _ih_{i+1}\right )\!, \end{equation}

for any compactly supported sequences $(h_i)_{i\geq 1}$ and $(\varphi _i)_{i\geq 1}$ . We denote by $L$ the sum of the previous operators, namely

(5.15) \begin{equation} L = S + P. \end{equation}

We also define a bilinear operator $\Gamma$ by the following weak form

\begin{equation*} \sum _{i=1}^{+\infty } \Gamma _i(f,g)\Theta _i\varphi _i = \frac {1}{2}\sum _{i=1}^{+\infty }a_i\Theta _1\Theta _i(f_1g_i+f_ig_1)(\varphi _{i+1}-\varphi _i-\varphi _1), \end{equation*}

for any compactly supported sequences $(f_i)_{i\geq 1}$ , $(g_i)_{i\geq 1}$ , and $(\varphi _i)_{i\geq 1}$ . We note that $\Gamma (0,0)=0$ .

The fluctuation $h$ satisfies

(5.16) \begin{equation} \frac {d}{dt}h_i(t) = L_i(h(t)) + \Gamma _i(h(t),h(t)). \end{equation}

The aforementioned decomposition of the linear operator $L$ is due to the fact that the operator $S$ is the linear operator studied by Cañizo and Lods [Reference Cañizo and Lods15] for the Becker–Döring equations. Consequently, it is established that $S$ is a symmetric operator in $\mathcal{H}$ and dissipative. Cañizo and Lods proved that the operator $S$ admits a spectral gap in $\mathcal{H}$ , and they have also proved a lower bound on this spectral gap. In order to prove that the steady state is then locally exponentially stable, it is necessary to establish estimates on the quadratic term $\Gamma$ . However, finding estimates on the quadratic term $\Gamma$ in a weighted $\ell ^2$ space is quite complicated. The approach proposed by Cañizo and Lods [Reference Cañizo and Lods15] is to extend the operator $S$ in some weighted $\ell ^1$ space, with the aim of showing that the operator S still has a spectral gap in this space. Then, we control the quadratic term in this weighted $\ell ^1$ space, which is more treatable.

Our goal is to prove the same spectral gap holds whenever we control the perturbation operator $P$ .

Remark 5.4. We may also write the operator $S$ in the strong form, for all compactly supported sequence $h = (h_i)_{i\geq 1}$ ,

(5.17) \begin{equation} S_1(h) = -\frac {1}{\Theta _1}\left (W_1(h)+\sum _{k=1}^{+\infty }W_k(h)\right )\!, \,\,\, S_i(h) = \frac {1}{\Theta _i}\left (W_{i-1}(h)-W_i(h)\right )\!, \,i\geq 2, \end{equation}

where

(5.18) \begin{equation} W_k(h) = a_k\Theta _1\Theta _k(h_k+h_1-h_{k+1}). \end{equation}

As well as the operator $L$ , for all compactly supported sequence $h = (h_i)_{i\geq 1}$ ,

\begin{equation*} L_i(h) = -\sigma _ih_i + \sum _{j=1}^{+\infty }\xi _{i,j}h_j, \,\text{ for } i\geq 1, \end{equation*}

where $\sigma _i$ are defined by

(5.19) \begin{equation} \sigma _1 = 3a_1z + \sum _{i=1}^{+\infty } a_i\Theta _i, \,\, \sigma _i = a_iz+b_i \,\text{ for }i\geq 2, \end{equation}

and $\xi _{i,j} = 0$ for $j\not \in \left \{ 1,i-1,i+1\right \}$ . The other $\xi _{i,j}$ are not equal to zero but are not relevant in the following.

The operator $L$ is in general not continuous on $\mathcal{H}$ ; however, we give a dense subspace $\mathcal{H}_{\mathcal{D}}$ of $\mathcal{H}$ in which it is bounded:

Lemme 5.5. Assume hypotheses (H1)–(H5). Then, there exists a constant $K\gt 0$ , depending only on $z$ and the coefficients $(a_i)_{i\geq 1}$ and $(b_i)_{i\geq 1}$ , such that for any compactly supported sequence $h = (h_i)_{i\geq 1}$ , we have

\begin{equation*} \|L(h)\|_{\mathcal{H}} \leq K \|h\|_{\mathcal{H}_{\mathcal{D}}}. \end{equation*}

Proof. Using triangular inequality, it is sufficient to prove this bound separately for $S$ then $P$ . We already know from Cañizo and Lods [Reference Cañizo and Lods15, Lemma 2.4] that this is true for the operator $S$ . We prove it for the operator $P$ . We show equivalently that

\begin{equation*} \left | \langle P(h), \phi \rangle _{\mathcal{H}} \right | \leq K \|h\|_{\mathcal{H}_{\mathcal{D}}}\|\phi \|_{\mathcal{H}}, \text{ for all } \phi \in \mathcal{H}. \end{equation*}

We have from the weak form of $P$ (5.14), and using Cauchy–Schwarz’s inequality,

\begin{align*} \left | \langle P(h), \phi \rangle _{\mathcal{H}} \right | &\leq \sqrt {\Theta _1}\|\phi \|_{\mathcal{H}}\left (a_1\Theta _1|h_2| + \sum _{i=1}^{+\infty }a_i\Theta _i|h_{i+1}| \right ) \\ &\leq K\|\phi \|_{\mathcal{H}}\left (\sum _{i=1}^{+\infty }\Theta _i\right )^{\frac {1}{2}}\left (\sum _{i=1}^{+\infty }a_i^2\Theta _i|h_{i+1}|^2\right )^{\frac {1}{2}}. \end{align*}

However, using (H3) and the definition of $(\sigma _i)_{i\geq 1}$ (5.19), we have

\begin{align*} \left (\sum _{i=1}^{+\infty }a_i^2\Theta _i|h_{i+1}|^2\right )^{\frac {1}{2}} &= \sum _{i=1}^{+\infty } \frac {\Theta _{i+1}}{\Theta _i}b_{i+1}^2\Theta _{i+1}|h_{i+1}|^2 \\ &\leq K\sum _{i=2}^{+\infty }\sigma _i^2\Theta _i|h_i|^2 \leq K\|h\|_{\mathcal{H}_{\mathcal{D}}}. \end{align*}

Combining the previous inequalities, we obtain the desired estimation on $P$ and therefore on $L$ .

Proof. As mentioned earlier, the operator $S$ has been studied by Cañizo and Lods [Reference Cañizo and Lods15]. They proved that the eigenvalues are non-positive and that $S$ admits a spectral gap $\mu _0\gt 0$ when $z \in (0,z_s)$ . Moreover, they found [Reference Cañizo and Lods15, Theorem 2.15] a lower bound on the spectral gap; namely,

\begin{equation*} \frac {1}{4D}\leq \mu _0, \end{equation*}

where

(5.20) \begin{equation} D \,:\!=\, \sup _{k\geq 1} \left (\sum _{j=k+1}^{+\infty } Q_jz^j\right )\left (\sum _{j=1}^k\frac {1}{a_jQ_jz^j}\right ) \in (0,+\infty ). \end{equation}

The fact that $D$ is finite comes from our hypotheses on the coefficients (H1)–(H5). Moreover, for all $h\in \mathcal{H}$ , using the definition of $P$ and (H5), we have

\begin{align*} \|P(h)\|_{\mathcal{H}}^2 \leq \frac {4b^2}{z}\left (\sum _{k=1}^{+\infty }k^\beta \Theta _kh_k\right )^2. \end{align*}

Then, using Cauchy–Schwarz’s inequality, we obtain

\begin{align*} \|P(h)\|_{\mathcal{H}}^2 \leq \frac {4b^2}{z}\left (\sum _{k=1}^{+\infty }k^{2\beta }Q_kz^k\right )\|h\|_{\mathcal{H}}^2, \end{align*}

this means that

(5.21) \begin{equation} \left \lvert \hspace {-1 pt}\left \lvert \hspace {-1 pt}\left \lvert P\right \lvert \hspace {-1 pt}\right \lvert \hspace {-1 pt}\right \lvert \leq \frac {2b}{\sqrt {z}}\sqrt {\sum _{k=1}^{+\infty }k^{2\beta }Q_kz^k} \lt +\infty . \end{equation}

We now compare the bound on the norm of $P$ and the one on the spectral gap $\mu _0$ . A sufficient condition for the norm $\left \lvert \hspace {-1 pt}\left \lvert \hspace {-1 pt}\left \lvert P\right \lvert \hspace {-1 pt}\right \lvert \hspace {-1 pt}\right \lvert$ to be smaller than the spectral gap $\mu _0$ is

\begin{equation*} \frac {1}{4D} \gt \frac {2b}{\sqrt {z}}\sqrt {\sum _{k=1}^{+\infty }k^{2\beta }Q_kz^k}, \end{equation*}

which is exactly

(2.13) \begin{equation} \frac {8b}{\sqrt {z}}\sqrt {\sum _{i=1}^{+\infty }i^{2\beta }Q_iz^i}\sup _{k\geq 1} \left (\sum _{j=k+1}^{+\infty } Q_jz^j\right )\left (\sum _{j=1}^k\frac {1}{a_jQ_jz^j}\right )\lt 1. \quad \end{equation}

Using [Reference Engel and Nagel28, Theorem III.1.3] on bounded perturbation operator, we have that under the condition (2.13) the linearised operator $L$ admits a spectral gap $\mu _1\gt 0$ .

To prove that the steady-state $(Q_iz^i)_{i\geq 1}$ of the full system (1.1) is locally exponentially stable, we now need to find some estimates on the quadratic term $\Gamma (h,h)$ . As previously announced, we extend the operator $L$ in a weighted $\ell ^1$ space.

Let $\eta \in (0,\frac {1}{2}\log (\frac {z_s}{z}))$ . We introduce two subspaces of $\ell ^1$ , $\mathcal{Z} = \ell ^1(e^{\eta i}\Theta _i)$ and $\mathcal{Z}_{\mathcal{D}} = \ell ^1((1+\sigma _i)e^{\eta i}\Theta _i)$ namely

\begin{equation*} \mathcal{Z} = \left \{ h = (h_i)_{i\geq 1} \;:\; \|h\|_{\mathcal{Z}} \,:\!=\, \sum _{i=1}^{+\infty } e^{\eta i}\Theta _i|h_i|\lt +\infty \right \}\!, \end{equation*}

and

\begin{equation*} \mathcal{Z}_{\mathcal{D}} = \left \{ h = (h_i)_{i\geq 1} \;:\; \|h\|_{\mathcal{Z}_{\mathcal{D}}} \,:\!=\, \sum _{i=1}^{+\infty } (1+\sigma _i)e^{\eta i}\Theta _i|h_i| \lt +\infty \right \}\!. \end{equation*}

The choice of $\eta$ ensures that the space $\mathcal{Z}$ is larger than the previous Hilbert space $\mathcal{H} = \ell ^2(\Theta )$ , since

\begin{equation*} \|h\|_{\mathcal{Z}} = \sum _{k=1}^{+\infty }\exp\!(\eta i)\Theta _i|h_i| \leq \left (\sum _{i=1}^{+\infty } h_i^2\Theta _i \right )^{\frac {1}{2}}\left (\sum _{i=1}^{+\infty } \exp\!(2\eta i)\Theta _i\right )^{\frac {1}{2}} = \left (\sum _{i=1}^{+\infty } \exp\!(2\eta i)\Theta _i\right )^{\frac {1}{2}} \|h\|_{\mathcal{H}}, \end{equation*}

and the last factor in the previous inequality is finite for $\eta \lt \frac {1}{2}\log (\frac {z_s}{z})$ . Indeed, we have

\begin{equation*} \sum _{i=1}^{+\infty }Q_iz^i\exp\!(2\eta i) = \sum _{i=1}^{+\infty } Q_ir^i \text{ where } r = \exp\!(2\eta +\log (z)), \end{equation*}

and recalling that the radius of convergence of the power series $\sum iQ_iz^i$ is $z_s$ , then the above sum is finite whenever $r\lt z_s$ , this is exactly when $\eta \lt \frac {1}{2}\log ({z_s}/{z})$ .

As before, the operator $L$ is in general not continuous on $\mathcal{Z}$ ; however, we give a dense subspace $\mathcal{Z}_{\mathcal{D}}$ of $\mathcal{Z}$ in which it is bounded:

Lemme 5.7. Assume hypotheses (H1)–(H5). Then, there exists a constant $K\gt 0$ , depending only on $(a_i)_{i\geq 1}$ , $(b_i)_{i\geq 1}$ , $z$ and $\eta$ , such that for all compactly supported sequence $h=(h_i)_{i\geq 1}$ , we have

\begin{equation*} \|L(h)\|_{\mathcal{Z}} \leq K\|h\|_{\mathcal{Z}_{\mathcal{D}}}. \end{equation*}

Proof. Using triangular inequality, it is sufficient to prove this bound separately for $S$ then $P$ . We already know from Cañizo and Lods [Reference Cañizo and Lods15, Lemma 3.3] that this is true for the operator $S$ . We prove it for the operator $P$ . Recalling the definition of $P$ and using (H5), we have

\begin{align*} \|P(h)\|_{\mathcal{Z}} &\leq \exp\!(\eta )z\left (\frac {1}{z}\left (b_2\Theta _2|h_2| + \sum _{k=2}^{+\infty }b_k\Theta _k|h_k|\right )\right ) \\ &\leq 2b\exp\!(\eta )\sum _{k=1}^{+\infty }k^\eta \Theta _k|h_k|. \end{align*}

Then, using that $k^\beta \leq K(\eta ,\beta ) \exp\!(\eta k)$ for all $k\geq 1$ and the positivity of $\sigma _k$ (5.19) for all $k\geq 1$ , we have

\begin{align*} \|P(h)\|_{\mathcal{Z}} &\leq K(\eta ,\beta ) \sum _{k=1}^{+\infty }\exp\!(\eta k) \Theta _k|h_k| \\ &\leq K(\eta ,\beta ) \sum _{k=1}^{+\infty } (1+\sigma _k)\exp\!(\eta k) \Theta _k|h_k| = C\|h\|_{\mathcal{Z}_{\mathcal{D}}}. \end{align*}

Combining this estimation with the one on $S$ , we obtain the desired estimation.

Following Cañizo and Lods [Reference Cañizo and Lods15], we extend the operator $L$ to an unbounded operator on $\mathcal{Z}$ that has a positive spectral gap using techniques based upon a suitable decomposition of the linearised operator into a dissipative part and a “regularising” part. To extend the operator, we use [Reference Cañizo and Lods15, Theorem 3.1], which is a slight improvement over some of the consequences of the work of Gualdani, Mischler and Mouhot [Reference Gualdani, Mischler and Mouhot30].

Theorem 5.8 (Extension of the spectral gap). Assume hypotheses (H1)–(H5). Let $\eta \in (0,\frac {1}{2}\log (\frac {z_s}{z}))$ . If ( 2.13 ) is satisfied, then the extension of $L$ on $\mathcal{Z}_{\mathcal{D}}$ that we note $\Lambda$ generates a strongly continuous semigroup $(\!\exp\!(t\Lambda ))_{t\geq 0}$ on $\mathcal{Z}_{\mathcal{D}}$ and there exists $\mu _\ast \in (0,\mu _1]$ , a constant $K\gt 0$ such that

(5.22) \begin{equation} \|\exp\!(t\Lambda )x\|_{\mathcal{Z}} \leq K\exp\!(\!-\mu _\ast t)\|x\|_{\mathcal{Z}}, \,\forall t\geq 0 \text{ and } x\in \mathcal{Z}. \end{equation}

Proof. The proof follows the one of [Reference Cañizo and Lods15, Theorem 3.5]; to that end, we apply [Reference Cañizo and Lods15, Theorem 3.1]. The main modification is to add the operator $P$ in the regularising part of the decomposition of $\Lambda$ . We mean that an operator $\mathcal{A}$ is regularising if for all $h\in \mathcal{Z}$ , $\mathcal{A}h \in \mathcal{H}$ . We just have to show that this is true for our operator $P$ . Let $h\in \mathcal{Z}$ , we have

\begin{align*} \|P(h)\|_{\mathcal{H}}^2 &\leq \frac {4b^2}{z}\left (\sum _{k=1}^{+\infty }b_kQ_kz^kh_k\right )^2 \\ &\leq \frac {4b^2}{z}K(\beta ,\eta )\|h\|_{\mathcal{Z}}^2. \end{align*}

Then, the proof is straightforward following the proof of [Reference Cañizo and Lods15, Theorem 3.5]. We obtain that our operator has a spectral gap $\mu _\ast \in (0,\mu _1)$ .

5.3.1. Proof of Theorem2.11

As stated before, we can obtain an estimation on the quadratic term in a weighted $\ell ^1$ space. Indeed, Cañizo and Lods [Reference Cañizo and Lods15, Proposition 3.2] proved an estimation on the quadratic term $\Gamma$ , it reads as follows:

Proposition 5.9. Let $\eta \in (0,\frac {1}{2}\log (\frac {z_s}{z}))$ . There is a constant $K\gt 0$ , depending only on $(a_i)_{i\geq 1}$ , $(b_i)_{i\geq 1}$ , and $z$ , such that

(5.23) \begin{equation} \|\Gamma (h,h)\|_{\mathcal{Z}} \leq K\|h\|_{\mathcal{Z}}\|h\|_{\mathcal{Z}_{\mathcal{D}}},\,\forall h\in \mathcal{Z}_{\mathcal{D}}. \end{equation}

Let $\eta \in (0,\min \{\nu ,\frac {1}{2}\log \frac {z_s}{z}\})$ , where $\nu$ comes from (2.7). The fluctuation $h$ satisfies (5.16) in $\mathcal{Z}$ , namely

\begin{equation*} \frac {d}{dt}h = \Lambda h + \Gamma (h,h).\end{equation*}

Therefore, if $(V_t)_{t\geq 0}$ denotes the semigroup generates by $\Lambda$ , we have

\begin{equation*} h(t) = V_th(0) + \int _0^tV_{t-s}\Gamma (h(s),h(s))ds. \end{equation*}

Using Proposition5.9 and Theorem5.8, we have for some constant $C\gt 0$ ,

\begin{align*} \|h(t)\|_{\mathcal{Z}} &\leq \|V_th(0)\|_{\mathcal{Z}} + \int _0^t\|V_{t-s}\Gamma (h(s),h(s))\|_{\mathcal{Z}} ds \\ &\leq K\|h(0)\|_{\mathcal{Z}}\exp\!(\!-\mu _\ast t) + K\exp\!(\!-\mu _\ast t)\int _0^t \exp\!(\mu _\ast s)\|\Gamma (h(s),h(s))\|_{\mathcal{Z}} ds \\ &\leq K\|h(0)\|_{\mathcal{Z}}\exp\!(\!-\mu _\ast t) + K\exp\!(\!-\mu _\ast t)\int _0^t \exp\!(\mu _\ast s)\|h(s)\|_{\mathcal{Z}}\|h(s)\|_{\mathcal{Z}_{\mathcal{D}}} ds, \end{align*}

For any $\upsilon \in (0,\nu -\eta )$ , using Cauchy–Schwarz’s inequality, we have

\begin{align*} \|h(t)\|_{\mathcal{Z}_{\mathcal{D}}} &= \sum _{i=1}^{+\infty } |h_i(t)|i\Theta _i\exp\!(\eta i) \\ &\leq \left (\sum _{i=1}^{+\infty } |h_i(t)|i\Theta _i\exp\!(\eta i-\upsilon i)\right )^{\frac {1}{2}}\left (\sum _{i=1}^{+\infty }|h_i(t)|i\Theta _i\exp\!(\eta i+\upsilon i)\right )^{\frac {1}{2}}. \end{align*}

However, $i\exp\!(\!-\upsilon i) \leq K$ , and $i\exp\!(\eta i +\upsilon i) \leq K\exp\!(\nu i)$ , for all $i\geq 1$ . Therefore, we deduce from (2.8) that

\begin{equation*}\|h(t)\|_{{\mathcal{Z}}_{\mathcal{D}}} \leq K \|h(t)\|_{\mathcal{Z}}^{\frac {1}{2}} \left (\sum _{i=1}^{+\infty }|h_i(t)|\Theta _i\exp\!(\nu i)\right )^{\frac {1}{2}} \leq K \Xi _1^\frac {1}{2}\|h(t)\|_{\mathcal{Z}}^{\frac {1}{2}}, \,\forall t\geq 0.\end{equation*}

All-in-one, we have

\begin{equation*} \|h(t)\|_{\mathcal{Z}} \leq K\|h(0)\|_X\exp\!(\!-\mu _\ast t) + K\exp\!(\!-\mu _\ast t)\int _0^t \exp\!(\mu _\ast s)\|h(s)\|_{\mathcal{Z}}^\frac {3}{2} ds. \end{equation*}

Let $u(t) = \|h(t)\|_{\mathcal{Z}}\exp\!(\mu _\ast t)$ , we have

\begin{align*} u(t) &\leq K\|h(0)\|_{\mathcal{Z}} + K\int _0^t u(s)\|h(s)\|_{\mathcal{Z}}^{\frac {1}{2}} ds \\ &\leq K\varepsilon + K\int _0^t \exp \left (-\frac {\mu _\ast }{2}s\right )u(s)^{\frac {3}{2}} ds. \end{align*}

Using Grönwall’s lemma, we have for some $K\gt 0$ and $\varepsilon$ small enough,

\begin{equation*} u(t) \leq K. \end{equation*}

This means that

\begin{equation*} \|h(t)\|_{\mathcal{Z}} \leq K\exp\!(\!-\mu _\ast t), \,\forall t\geq 0, \end{equation*}

which is exactly what we want. Indeed, recalling the ansatz $C_i = \Theta _i(1+h_i)$ , we have

\begin{equation*} \|h(t)\|_{\mathcal{Z}} = \sum _{i=1}^{+\infty } e^{\eta i}Q_iz^i|h_i(t)| = \sum _{i=1}^{+\infty }e^{\eta i}|C_i(t)-Q_iz^i|, \end{equation*}

and therefore, the previous inequality is exactly (2.15).

6.1. Numerical scheme

In this section, we describe a numerical scheme used for the conservative truncation (2.16). At the end, we will make a few comments about the non-conservative truncation.

Let $n\geq 2$ be an integer. Recalling the PDE (2.17) and (2.18), it remains to add the boundary conditions. As we are considering the conservative truncation, we have on the right side of our domain the following boundary condition:

(6.1) \begin{equation} J(t,n) = 0. \end{equation}

Now, on the left side of our domain, which is the production of monomers, we have the following ODE:

(6.2) \begin{equation} \frac {d}{dt}C(t,1) = \lambda - \int _1^n a(x)C(t,1)C(t,x)dx - a(1)C(t,1)^2. \end{equation}

6.1.1. Verification of the choice of PDE

We verify that with the right discretisation and the right mesh, we retrieve our system of equations (2.16). To do this, we use a uniform mesh with a size step $\Delta x = 1$ , giving

\begin{equation*} C_i(t) \approx C(t,i), \,i=1,\ldots ,n. \end{equation*}

We discretise (2.17) using centre differences

\begin{align*} \frac {\partial J(t,x)}{\partial x}\bigg |_{x=x_i} &\approx \frac {J(t,x_{i+1/2})-J(t,x_{i-1/2})}{\Delta x},\\ \frac {\partial }{\partial x}\left (\frac {C(t,x)}{Q(x)C(t,1)^x}\right )\bigg |_{x=x_{i+1/2}} &\approx \frac {1}{\Delta x}\left ( \frac {C_{i+1}(t)}{Q_{i+1}C_1^{i+1}(t)} - \frac {C_i(t)}{Q_iC_1^i(t)}\right )\!. \end{align*}

Therefore, since $\Delta x = 1$ and from (2.19), we obtain

\begin{equation*} \frac {\partial C(t,x)}{\partial t}\bigg |_{x=i} = -\frac {\partial J(t,x)}{\partial x}\bigg |_{x=i} \approx \frac {J_{i-1}(t)-J_i(t)}{\Delta x} = \frac {d}{dt}C_i(t). \end{equation*}

We just need to check the case $i=1$ , which comes from the left boundary condition (6.2). Using the rectangle rule to approximate the integral, we have

\begin{equation*} \frac {d}{dt}C(t,1) = \lambda - \int _1^n a(x)C(t,1)C(t,x) dx - a(1)C(t,1)^2 \approx \lambda - \sum _{j=1}^{n-1} a_jC_1(t)C_j(t) - a_1C_1(t)^2 = \frac {d}{dt}C_1(t). \end{equation*}

We have indeed found back our ODE (2.16) with a given scheme and a given mesh.

6.1.2. Reaction-diffusion (RD) scheme

We now follow the idea in [Reference Goudon and Monasse29] on a non-uniform mesh. Let $K\gt 0$ and $\mathbb L$ the set of all the nodes in our non-uniform mesh:

\begin{equation*} \mathbb{L} = \left \{ n_1,\ldots ,n_K \,:\, n_j\lt n_{j+1},\, n_1=1,\,n_K=n \right \}\!. \end{equation*}

We then define the following size step:

\begin{equation*} \Delta x_{j+1/2} = n_{j+1} - n_j \text{ and } \Delta x_j = n_{j+1/2}-n_{j-1/2} \text{ where } n_{j+1/2} = \frac {n_{j+1}+n_j}{2} \text{ with }n_j,n_{j+1}\in \mathbb{L}. \end{equation*}

Using finite volume scheme, we discretise (2.17) and (2.18) as follows:

(6.3) \begin{equation} C_{n_j}^{k+1}-C_{n_j}^k = \frac {\Delta t}{\Delta x_j}\left (J_{n_{j-1/2}}^{k+1}-J_{n_{j+1/2}}^{k+1}\right )\!, \end{equation}

with

(6.4) \begin{equation} J_{n_{j-1/2}}^{k+1} = -a_{n_{j-1/2}}Q_{n_{j-1/2}}\big(C_1^k\big)^{n_{j-1/2}+1/2}\frac {1}{\Delta x_{j-1/2}}\left (\frac {C_{n_j}^{k+1}}{Q_{n_j}\big(C_1^k\big)^{n_j}}-\frac {C_{n_{j-1}}^{k+1}}{Q_{n_{j-1}}\big(C_1^k\big)^{n_{j-1}}}\right )\!. \end{equation}

At this point, it remains to choose what is $a_{n_{j-1/2}}$ and $Q_{n_{j-1/2}}$ . To remain consistent with the discrete equations (i.e. when we take the step size equal to 1), we impose $a_{n_{j-1/2}} = a_{n_{j-1}}$ et $Q_{n_{j-1/2}} = Q_{n_{j-1}}$ . We have tried to impose something else, and the numerical results were not satisfying. Let $M_{n_j}^k = Q_{n_j}\big(C_1^k\big)^{n_j}$ , we have

\begin{equation*} Q_{n_{j-1}}\big(C_1^k\big)^{n_{j-1}} = M_{n_{j-1}} = \sqrt {M_{n_{j-1}}M_{n_j}}\sqrt {\frac {M_{n_{j-1}}}{M_{n_j}}} = \sqrt {M_{n_{j-1}}M_{n_j}}\sqrt {\prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}}\left (\sqrt {C_1^k}\right )^{-(n_j-n_{j-1})}. \end{equation*}

Using the above equality, we can rewrite (6.3), we obtain for all $2\leq j\leq K-1$ ,

\begin{align*} C_{n_j}^{k+1}-C_{n_j}^k &= \frac {\Delta t}{\Delta x_j}\sqrt {M_j^k}\left (\frac {a_{n_j}\sqrt {C_1^k}}{\Delta x_{j+1/2}}\sqrt {\prod _{k=n_{j}+1}^{n_{j+1}}\frac {b_k}{a_{k-1}}}\frac {C_{n_{j+1}}^{k+1}}{\sqrt {M_{n_j}^k}} + \frac {a_{n_{j-1}}\sqrt {C_1^k}}{\Delta x_{j-1/2}} \sqrt {\prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}} \frac {C_{n_{j-1}}^k}{\sqrt {M_{n_{j-1}}^k}} \right . \\ & \quad \left . - \left ( \frac {a_{n_j}}{\Delta x_{j+1/2}}\left (\sqrt {C_1^k}\right )^{n_{j+1}-n_j+1} + \frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \left [ \prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}\right ]\left (\sqrt {C_1^k}\right )^{1-(n_j-n_{j-1})} \right ) \frac {C_{n_j}^{k+1}}{\sqrt {M_{n_j}^k}} \right )\!. \end{align*}

Let $h_{n_j}^\ast \,:\!=\, \dfrac {C_{n_j}^{k+1}}{\sqrt {M_{n_j}^k}}$ and $h_{n_j} = \dfrac {C_{n_j}^k}{\sqrt {M_{n_j}^k}}$ , we then have for all $2\leq j \leq K-1$ ,

\begin{align*} h_{n_j}^{\ast } &= h_{n_j}^k + \frac {\Delta t}{\Delta x_j}\left (\frac {a_{n_j}}{\Delta x_{j+1/2}}\sqrt {\prod _{k=n_{j}+1}^{n_{j+1}}\frac {b_k}{a_{k-1}}}\sqrt {C_1^k}h_{n_{j+1}}^\ast + \frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \sqrt {\prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}} \sqrt {C_1^k} h_{n_{j-1}}^\ast \right . \\ &\quad \left . - \left ( \frac {a_{n_j}}{\Delta x_{j+1/2}}\left (\sqrt {C_1^k}\right )^{n_{j+1}-n_j+1} + \frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \left [ \prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}\right ]\left (\sqrt {C_1^k}\right )^{1-(n_j-n_{j-1})}\right )h_{n_j}^\ast \right )\!. \end{align*}

As mentioned in the beginning of this section, we present this scheme on the conservative truncation (2.16); thus, we impose the zero flux condition on the right boundary, meaning $J_{n_{K+1/2}} = 0$ . This condition may be rewritten as follows:

(6.5) \begin{equation} \frac {C_{n_{K+1}}^{k+1}}{M_{n_{K+1}}^k} = \frac {C_{n_K}^{k+1}}{M_{n_K}^k},\,\, \forall k\gt 0. \end{equation}

Therefore, the scheme becomes

(6.6) \begin{align} C_1^{k+1} &= C_1^k + \Delta t \left ( \lambda - C_1^{k+1}\left (\sum _{j=1}^{K-1} a_{n_j}\Delta x_{j+1/2}C_{n_j}^k+a_1C_1^k\right )\right ) \nonumber\\ h_1^\ast &= \frac {1}{\sqrt {C_1^k}} \frac {C_1^k + \lambda \Delta t}{1+\Delta t\left (\sum \limits _{j=1}^{K-1} a_{n_j}\Delta x_{j+1/2}C_{n_j}^k+a_1C_1^k\right )} \nonumber\\ h_{n_j}^{\ast } &= h_{n_j}^k + \frac {\Delta t}{\Delta x_j}\left (\frac {a_{n_j}}{\Delta x_{j+1/2}}\sqrt {\prod _{k=n_{j}+1}^{n_{j+1}}\frac {b_k}{a_{k-1}}}\sqrt {C_1^k}h_{n_{j+1}}^\ast + \frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \sqrt {\prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}} \sqrt {C_1^k} h_{n_{j-1}}^\ast \right . \nonumber\\ & \quad \left . - \left ( \frac {a_{n_j}}{\Delta x_{j+1/2}}\left (\sqrt {C_1^k}\right )^{n_{j+1}-n_j+1} + \frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \left [ \prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}\right ]\left (\sqrt {C_1^k}\right )^{1-(n_j-n_{j-1})}\right )h_{n_j}^\ast \right ) \nonumber\\ h_{n_K}^\ast &= h_{n_K} + \frac {\Delta t}{\Delta x_K}\left ( \frac {a_{n_{K-1}}}{\Delta x_{K-1/2}}\sqrt {\prod _{k=n_{K-1}+1}^{n_K} \frac {b_k}{a_{k-1}}}\sqrt {C_1^k}h_{n_{K-1}} \right . \nonumber\\ & \qquad\qquad\qquad \left . - \frac {a_{n_{K-1}}}{\Delta x_{K-1/2}}\left [\prod _{k=n_{K-1}+1}^{n_K} \frac {b_k}{a_{k-1}} \right ] \left (\sqrt {C_1^k}\right )^{1-(n_K-n_{K-1})}h_{n_K} \right )\!. \end{align}

Numerically to implement (6.6), we solve the following linear system:

(6.7) \begin{equation} \left (I_{K-1} - \Delta t\mathsf{W}^k\right )H^\ast = H + \beta ^k, \end{equation}

where

\begin{equation*} H^\ast = \begin{pmatrix} h_2^\ast \\ \vdots \\ \vdots \\ \vdots \\ h_{n_K}^\ast \end{pmatrix}, \,\, H = \begin{pmatrix} h_2 \\ \vdots \\ \vdots \\ \vdots \\ h_{n_K} \end{pmatrix}, \,\, \beta ^k = \frac {\Delta t}{\Delta x_2 \Delta x_{2-1/2}}a_{n_1}\sqrt {C_1^k}\sqrt {\prod _{k=n_1+1}^{n_2}\frac {b_k}{a_{k-1}}} \begin{pmatrix} h_1^\ast \\ 0 \\ \vdots \\ \vdots \\ \vdots \\ 0 \end{pmatrix}, \end{equation*}

and the tri-diagonal matrix

with for $2\leq j\leq K-1$ ,

\begin{equation*} \omega _{n_j}^k = \frac {1}{\Delta x_j}\left (\frac {a_{n_{j}}}{\Delta x_{j+1/2}}\left (\sqrt {C_1^k}\right )^{n_{j+1}-n_j+1} + \frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \left [ \prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}\right ]\left (\sqrt {C_1^k}\right )^{1-(n_j-n_{j-1})} \right )\!, \end{equation*}

\begin{equation*} \alpha _{n_{j+1/2}} = \frac {1}{\Delta x_j}\frac {a_{n_j}}{\Delta x_{j+1/2}}\sqrt {\prod _{k=n_{j}+1}^{n_{j+1}}\frac {b_k}{a_{k-1}}}\sqrt {C_1^k}, \,\, \gamma _{n_{j-1/2}} = \frac {1}{\Delta x_{j}}\frac {a_{n_{j-1}}}{\Delta x_{j-1/2}} \sqrt {\prod _{k=n_{j-1}+1}^{n_j}\frac {b_k}{a_{k-1}}} \sqrt {C_1^k}, \end{equation*}

and

\begin{equation*} \nu _{n_K}^k = \frac {1}{\Delta x_K}\frac {a_{n_{K-1}}}{\Delta x_{K-1/2}}\left [\prod _{k=n_{K-1}+1}^{n_K} \frac {b_k}{a_{k-1}} \right ] \left (\sqrt {C_1^k}\right )^{1-(n_K-n_{K-1})}. \end{equation*}

Remark 6.1. As opposed to uniform mesh, the matrix $\mathsf{W}^k$ is no longer symmetric, which can translate in a loss of computation time. Therefore, to symmetrise this matrix while keeping the tri-diagonal form, we can multiply the matrix by

\begin{equation*} \mathsf{\Theta } = \text{diag}\left (\frac {\Delta x_j}{\Delta x_1}\right )_{2\leq j\leq K}. \end{equation*}

The scheme on $h^\ast$ (6.6) is unconditionally stable in time and size. Indeed, the only restriction could be on the time step because of Euler’s explicit–implicit scheme for $C_1$ , but this scheme is also unconditionally stable. We have

\begin{equation*} C_1^{k+1}\left (1+\Delta t\left (a_1C_1^k+\sum _{j=1}^{K-1}a_{n_j}\Delta x_{j+1/2}C_{n_j}^k\right )\right ) = C_1^k + \Delta t \lambda \end{equation*}

thus

\begin{equation*} C_1^{k+1} \leq C_1^k + \Delta t \lambda , \end{equation*}

which leads to, for $0\leq t^k \leq T$ ,

\begin{equation*} C_1^{k} \leq C_1^0 + \lambda t^k \leq C_1^0 + \lambda T. \end{equation*}

Remark 6.2. We could just do an Euler explicit scheme for ( 6.2 ), but then we would have two conditions on the time step $\Delta t$ : stability and to preserve positivity. This would lead to having an adaptive time step $\Delta t^k$ .

We now have a well-balanced scheme that is unconditionally stable in time and size. However, there is a difference with the uniform mesh with $\Delta x = 1$ or with the ODEs, which is the parameter $z$ of the steady-state. Indeed, equation (6.6) for $j\geq 2$ implies that the steady state is of the form $Q_{n_j}\eta ^{n_j}$ , and the parameter $\eta$ is determined with the equation on $C_1$ . The difference comes from the following two equations:

(6.8) \begin{equation} \lambda - \sum _{i=1}^{N-1} a_iQ_iz^{i+1} - a_1z^2 = 0 \text{ and } \lambda - \sum _{j=1}^{K-1} a_{n_j}\Delta x_{n_{j+1/2}}Q_{n_j}\tilde z^{n_j+1} - a_{n_1}\tilde z^2 = 0, \end{equation}

which do not have the same solutions. In fact, the difference of the two solutions is of order $Q_{n_l}\tilde z^{n_l}$ where $l$ is defined as follows:

\begin{equation*} l = \min \left \{ j \in \left \{1,\ldots ,K-1\right \} \,:\, \Delta x_{j+1/2} \gt 1 \right \}\!. \end{equation*}

A difference is expected since we reduce the linear system; therefore, we lost information that translates into a slight difference in the steady state. We note that we can compute the steady state on a uniform mesh with $\Delta x = 1$ and, therefore, choose the right mesh to control the error. Morally, if we choose a mesh with $\Delta x = 1$ for small size and whatever we want for bigger size, then we approach the right steady state.

Remark 6.3 (Non-conservative truncation). Changing the right boundary condition ( 6.5 ) to one taking account things that can leave the system, we obtain the same scheme except for the bottom right term in the iteration matrix. However, the scheme is no longer well-balanced since we still use the same form of steady state. One modification could be to use the steady state of the non-conservative truncation and doing again the same scheme.

6.2. Numerical results

Before showing numerical results, we have to mention the fact that the conservative truncation (2.16) has the same steady state as the infinite system, meaning $(Q_iz^i)_{i\geq 1}$ ; however, the $z$ differs due to the truncation. In fact, the conservative truncation (2.16) has a unique steady state, which is this one, no matter $\lambda$ . There is no sub-critical or super-critical cases in any truncation; indeed, if $z_s$ is finite, then for a fixed $n$ and $\lambda$ there always exists $z(n)\gt z_s$ such that $(Q_iz(n)^i)_{1\leq i\leq n}$ is the only steady state of (2.16).

Following the above discussion on $z$ for different meshes, we denote $z_{RK4} = z_{RDU}$ the parameter $z$ obtained with the full sum, meaning with $\Delta x = 1$ , and we denote $z_{RDNU}$ the parameter $z$ obtained with a non-uniform mesh. Consequently, we denote $C^{eq}_{RK4} = C^{eq}_{RDU}$ and $C^{eq}_{RDNU}$ the corresponding steady state.

In every numerical simulation used to obtain graphs below, we start with a system containing nothing:

\begin{equation*} C_i(0) = 0, \,\text{for } i = 1,\ldots ,n. \end{equation*}

For the size discretisation, we use a non-uniform mesh with $\Delta x = 1$ close to the boundaries and then nodes are spaced evenly on a log scale until the step size reaches the maximum size that we impose.

We start with the convergence towards the steady state of our numerical solution. Figure 1 shows results for a system of size $n = 30 000$ with kinetics coefficients and production rate as follows:

(6.9) \begin{equation} a_i = i^{1/2}, \,b_i = i^{2/3} \text{ and } \lambda = 10. \end{equation}

These parameters give $z_s = +\infty$ , and therefore, $\lambda _s = +\infty$ and z $\approx$ 1.119. Using our coarse-grain mesh, we have $|\texttt {z}_{RDU}-\texttt {z}_{RDNU}| \leq \texttt {1.8e-8}$ .

Figure 1a shows that the numerical solution of our scheme converges towards the steady-state $(Q_iz_{RDNU}^i)_{1\leq i\leq n}$ at the given precision, and the green curve shows what we expected, which is that the two equilibria differ at the expected order of magnitude. Focusing on the red and blue curves, our scheme preserves the exponential decay towards the steady state. Figure 1b shows that the relative error between the numerical solution obtained from our scheme and the one obtained with RK4 scheme (Runge–Kutta method) converges towards the relative error between the different equilibria, which is expected from Figure 1b and (6.8). However, unlike the scheme of Duncan and Soheili [Reference Duncan and Soheili26, Figure 3], the relative error does not behave similarly. Indeed, their relative error is close to 0 for small time but remains around $10^{-2}$ when the solution reaches the steady state, whereas our relative error is bigger for small time but decreases towards the expected error (the sub-sampling error from (6.8)) as the solution reaches the steady state.

Figure 1.

Comparison of the convergence towards the steady state. System size $K = 649$ with maximum step size $\Delta x_{max} = 50$ . The relative error is only taken from the nodes on the non-uniform mesh, no interpolation is used.

We now compare computation time, since for the super-critical case, we may need to compute very large size for a very long time. We, therefore, want a scheme that is able to reduce the computation time. In our scheme, we take $\Delta x_{max}$ such that it represents at most 0.05% of the truncation size $n$ . Figure 2 shows that, for large enough truncation size $n$ , it is efficient to use our scheme.

Figure 2a shows results with kinetics coefficients and production rate from (6.9), and Figure 2b shows results with kinetics coefficients and production rate as follows:

(6.10) \begin{equation} a_i = i^{1/2}, \,b_i = 0.1+0.75i^{1/2} \text{ and } \lambda = 1. \end{equation}

These parameters give $z_s = 0.75$ , $\lambda _s$ $\approx$ 5.34375 and z $\approx$ 0.516.

We mention that with other coefficients (e.g. (6.11)), the convergence is much slower towards the steady-state and therefore the gain can be much better. Taking the maximum size $n=30000$ and allowing only $\Delta x_{max}$ to be roughly $0.05\%$ of $n$ , we reduce the computational time by $65\%$ . Taking $\Delta x_{max}$ to be roughly $0.5\%$ of $n$ , the reduction becomes $80\%$ . The gain of computational time also comes from the time size, which is $10$ times bigger for our scheme.

(6.11) \begin{equation} a_i = i^{1/2}, \,b_i = 0.05+0.1i^{2/3} \text{ and } \lambda = 10. \end{equation}

Figure 2.

Comparison of computation time.

We now compare the dynamics of our numerical solution. We know that the convergence towards the steady state is the same and that for large truncation it is more efficient time wise. However, is the behaviour, quantitatively and qualitatively speaking, of our solution the same or not? Figure 3 shows that it globally conserves the dynamical behaviour of the solution with the same order of magnitude. We only show the first coordinate since the dynamical behaviour is led by the latter. We note that in those simulations, neither $\Delta t$ nor $\Delta x_j$ is the same as in the RK4 scheme, which already creates an error.

Figure 3 shows results with kinetics coefficients and production rate from (6.9).

One more reason we want to develop a different scheme and not settle for classical ODE schemes is for the super-critical case. Indeed, as stated before, either truncation have a unique steady state, but they are not necessarily the one (or ones) of the infinite system, and moreover, they do not necessarily converge as $n$ goes to infinity to the one (or ones) of the infinite system. In practice, we observe that the numerical solution of the RK4 scheme (Runge–Kutta method) always converges towards the steady state of the truncation even in the theoretical super-critical case.

Figure 3.

Comparison of the dynamical behaviour of $C_1(t)$ .

We might expect a different behaviour of our scheme, leading to hints of what is happening in the super-critical case. There is no steady state, but as in the Becker–Döring equations, it still could converge towards $(Q_iz_s^i)_{i\geq 1}$ in some sense, even if it is not a steady state, unlike in the Becker–Döring equations. We then believe that our scheme may be able to catch a very slow convergence towards this sequence that a classical numerical scheme on the truncation system would definitely not catch.