This article is devoted to study the large time behavior of the solution to a conservative aggregation-fragmentation equation, a class of equations that arises in many applications and that has been widely studied both in the linear case 8141819 and with nonlinearities 6910111320.

Our particular motivation is an extension of the elapsed time neural population model, a partial differential equation structured by “age” studied in 151617, and which gives a new approach to the understanding of synchronization/desynchronization of neural assemblies with respect to the strength of their interconnections. Here, we add a fragmentation term in the model in order to incorporate the fact that the dynamics of the neurons are also related to their past activity, notably that neurons display adaptation and fatigue. That is a progressive decrease of their propensity to firing in response to a step maintained current. This is one of the most common neuronal properties that can introduce correlation in firing times. In this work, we examine whether and how the inclusion of this property can affect the dynamics of neural assemblies. As a consequence, the mathematical study of this equation is more complex. Based on the ideas in 12, we give a new result of exponential decay of the solution to its stationary state in the case where the network is weakly connected.

We consider the following equation:

{ ∂ n ( s , t ) ∂ t + ∂ n ( s , t ) ∂ s + p ( s , N ( t ) ) n ( s , t ) = ∫ K ( s , u ) p ( u , N ( t ) ) n ( u , t ) d u , s ≥ 0 , t ≥ 0 , n ( 0 , t ) = 0 , N ( t ) : = ∫ 0 + ∞ p ( s , N ( t ) ) n ( s , t ) d s , n ( s , 0 ) = n 0 ( s ) ≥ 0 , ∫ 0 ∞ n 0 ( s ) d s = 1 .

• n ( s , t ) denotes the probability density of neurons at time t such that the time elapsed since the last discharge is s. It is a fundamental property which follows from our assumptions that, for all times t ≥ 0 ,

∫ 0 + ∞ n ( s , t ) d s = ∫ 0 + ∞ n 0 ( s ) d s = 1 , n ( s , t ) ≥ 0 .

• N ( t ) represents the flux of neurons which discharge at time t and is identified to the global amplitude of stimulation of the network.

• p ( s , N ) models the firing rate of neurons submitted to a stimulation of amplitude N and such that the time elapsed since the last discharge is s. The coupling between the neurons is taking into account via the function p which varies according to the global activity N(t). Hence, in this model, the strength of interconnections between the neurons is taking into account via the variations of p with respect to the variable N.

• The kernel K ( s , u ) ∈ M ( [ 0 , ∞ ) × [ 0 , ∞ ) ) , the set of nonnegative measures in [ 0 , ∞ ) × [ 0 , ∞ ) , gives the distribution of neurons which take the state s when a discharge occurs after an elapsed time u since their last discharge.

The structured nature of Eq. (1) is related to the choice of the description of the dynamic of the neurons, which is made via the time elapsed since their last discharge. The term “fragmentation” stems from the fact that, at each time, the density of neurons which discharge is fragmented, via K, with respect to the new state of neurons after their discharge; each fragment is given by the flux of neurons which discharge and come back in a same state s.

The main question we address here is to prove exponential convergence as t → ∞ for the nonlinear problem (1) to a steady state solution A ( s ) , that is,

{ ∂ A ( s ) ∂ s + p ( s , A ∗ ) A ( s ) = ∫ K ( s , u ) p ( u , A ∗ ) A ( u ) d u , s ≥ 0 , A ( 0 ) = 0 , ∫ 0 ∞ A ( s ) d s = 1 , A ∗ = ∫ p ( s , A ∗ ) A ( s ) d s .

The existence of a stationary solution is proved in Sect. 5 and we attack to convergence through an adaptation of the strategy in 16. For the linear problem, we construct some kind of spectral gap which opens the door to also treat ‘small’ (in a weak sense) nonlinearities.

The paper is organized as follows. In Sect. 2, we state our main results after giving assumptions on the coefficients; we separate the linear and nonlinear cases because we can prove much stronger results in the linear case. In Sect. 3, we study the solution of the linear version of Eq. (1) more precisely we prove its large time convergence to the stationary state with exponential decay; this is the proof of Theorem 2.1. Section 4 is devoted to the nonlinear case and to the proofs of Theorems 2.2 and 2.3. We prove the existence of stationary states, i.e., solutions to (3) in Sect. 5. In the last Sect. 6, we present numerical results in the case where the nonlinearity is strong enough to obtain periodic solutions to understand the effect of the fragmentation term in regard to the appearance of spontaneous activity in the network. Several general or technical results are postponed to Appendices in order to focus more the proofs on the main arguments.

We need technical assumptions on the coefficients p and K in (1) and before we write them in full generality, we begin with a particular example. For the kernel K, a Dirac mass at 0, K ( s , u ) : = δ s = 0 , the equation is equivalently written as a age structured equation and this situation is covered in 1617. In this case, the interpretation is clear: After they discharge, all neurons take the same state s = 0 , irrespective of the time elapsed since their last discharge.

A more general example of kernel K is to take

K ( s , u ) : = δ ( s − ψ ( u ) ) ,

where ψ is a given increasing function. In this situation, the post discharge state s of the neurons only depends on their discharge state u (a cumulative time elapsed since their last discharge). Still more general is when K ( s , ⋅ ) is a function: this includes variability in the neuron population or randomness in their behavior.

Also, a convenient example of discharge rate p ( s , ⋅ ) is a regularized Heavyside function

p ( s , N ) = H δ ( s − σ ( N ) ) , 0 < σ − ≤ σ ( ⋅ ) ≤ σ + < ∞ ,

with σ a given smooth function. It is a caricature for modeling three desirable properties of the neurons:

• immediately after a discharge, the neuron enters a refractory period, i.e., after a discharge, a neuron cannot discharge again during a certain time interval; this is the assumption that p ( s , N ) = 0 for s small,

• after the end of its refractory period, the neuron rapidly recovers a significative sensitive state,

• for an excitatory system, a larger stimulation on the neuron induces smaller refractory period that ∂ ∂ N p ( s , N ) > 0 and this is written here as σ ′ < 0 even though this assumption is not used in the present analysis.

Those examples of functions p and K are covered by more general assumptions, and links between these quantities, which we explain now. In Appendix A, we give explicitly the conditions on the two functions σ and ψ which are induced by our assumptions below.

We prove the time decay with exponential rate as stated in Theorem 2.1, for the linear equation. In this situation, the function p does not depend on N and assumption (5) can be written as

p ( s , N ) ≡ p ( s ) , p ( s ) = p M ∀ s > σ .

The strategy of the proof of Theorem 2.1 is to observe that exponential decay for | n ( s , t ) − A ( s ) | follows from exponential decay for the functions M ( s , t ) and its first time derivative, which is much easier to prove than for m itself; the counterpart is that it involves a weighted norm, as expressed in Theorem 2.1, at variance with the Poincaré method in 145. Indeed, there are two main advantages of considering the solutions M(s,t), J ( s , t ) , instead of m ( s , t ) : = n ( s , t ) − A ( s ) ; (i) they satisfy a closed equation, (ii) the dual problem to the corresponding stationary equation has a negative first eigenvalue. This directly implies exponential decay of both ∫ 0 + ∞ P ( s ) | M ( s , t ) | d s and ∫ 0 + ∞ P ( s ) | J ( s , t ) | d s .

We split the proof of Theorem 2.1 in two steps:

• In the first part, we check that the proof of Theorem 2.1 is a direct consequence of the exponential decay in L 1 of P ( s ) | M ( s , t ) | and P ( s ) | J ( s , t ) | .

• The second part is devoted to prove exponential decay in L1 of P(s)|M(s,t)| and P ( s ) | J ( s , t ) | .

3.1 Reduction to Exponential Decay on M(s,t) and ∂ ∂ t M ( s , t )

We derive the Theorem 2.1 from the following proposition.

Proposition 3.1 Assume that there are constants λ < 0 , B > 0 and a function P such that

1 B ≤ P ( x ) ≤ B P ( y ) for  0 ≤ x ≤ y ,

and

{ ∫ 0 + ∞ P ( s ) | M ( s , t ) | d s ≤ C e λ t ∫ 0 + ∞ P ( s ) | M ( s , 0 ) | d s , ∫ 0 + ∞ P ( s ) | J ( s , t ) | d s ≤ C e λ t ∫ 0 + ∞ P ( s ) | J ( s , 0 ) | d s .

Then there exists a constant C and a ν>0 such that

∫ 0 + ∞ | n ( s , t ) − A ( s ) | d s ≤ C e − ν t ( ∫ 0 + ∞ P ( s ) | M ( s , 0 ) | d s + ∫ 0 + ∞ P ( s ) | J ( s , 0 ) | d s + ∫ 0 σ | n ( s , 0 ) − A ( s ) | d s ) .

Proof Using Eq. (29) for M (see the first step of the proof of Proposition 3.2), we obtain

m ( s , t ) = ∂ ∂ s M ( s , t ) = − J ( s , t ) − p ( s ) M ( s , t ) − ∫ u = s + ∞ p ′ ( u ) M ( u , t ) f ( s , u ) d u + ∫ Φ ( s , u ) p ( u ) M ( u , t ) d u .

We first use this relation to handle the values s > σ ; then p ′ ( u ) = 0 in (26) and it is reduced to

| m ( s , t ) | ≤ | J ( s , t ) | + | p ( s ) M ( s , t ) | + ∫ Φ ( s , u ) p ( u ) | M ( u , t ) | d u .

Multiplying this inequality by P, integrating between σ and +∞ with respect to the variable s, we obtain using estimate (11) and that Φ ( s , u ) = 0 for s > u , that

∫ σ + ∞ P ( s ) | m ( s , t ) | d s ≤ 2 ∫ σ + ∞ P ( s ) ( | M ( s , t ) | + | J ( s , t ) | ) d s + ∫ σ + ∞ ( ∫ σ + ∞ Φ ( s , u ) P ( s ) d s ) p ( u ) | M ( u , t ) | d u .

Using assumption (12) on Φ, we obtain that for all u ∈ ( σ , + ∞ ) , there exists r u ∈ ( σ , u ) such that

∫ σ + ∞ P ( s ) I s < u Φ ( s , u ) d s ≤ θ P ( r u ) ≤ θ B P ( u ) ,

the second inequality being a consequence of assumption (25) on P. We deduce that there exists a constant C such that

∫ σ ∞ P ( s ) | m ( s , t ) | d s ≤ C ∫ 0 + ∞ P ( s ) ( | M ( s , t ) | + | J ( s , t ) | ) d s .

Finally, with the exponential decay assumptions on M and J in Proposition 3.1, we conclude that

∫ σ ∞ P ( s ) | m ( s , t ) | d s ≤ C e − ν t ∫ 0 + ∞ P ( s ) ( | M ( s , 0 ) | + | J ( s , 0 ) | ) d s .

Next, we control the small values of s, namely ∫ 0 σ | m ( s , t ) | d s . We cannot control this quantity directly and proceed to estimate ∫ 0 σ e − μ s | m ( s , t ) | d s for μ given in condition (16).

We set v ( s , t ) : = e − μ s m ( s , t ) . Then, for s ∈ [ 0 , σ ] , v satisfies the equation

∂ ∂ t v + ∂ ∂ s v + ( μ + p ) v = ∫ 0 + ∞ p ( u ) K ( s , u ) e − μ s m ( u , t ) d u

and thus

∂ ∂ t | v | + ∂ ∂ s | v | + ( μ + p ) | v | ≤ ∫ 0 + ∞ p ( u ) K ( s , u ) e − μ s | m ( u , t ) | d u .

Integrating the above equation, using that m ( 0 , t ) = 0 and (8), we obtain

d d t ∫ 0 σ | v ( t , s ) | d s + μ ∫ 0 σ | v ( t , s ) | d s ≤ ∫ 0 σ ∫ 0 + ∞ p ( u ) K ( s , u ) e − μ s | m ( u , t ) | d u d s − ∫ 0 σ p ( u ) | v ( u , t ) | d u ≤ ∫ 0 σ p ( u ) ( ∫ 0 u K ( s , u ) e μ ( u − s ) d s − 1 ) | v ( u , t ) | d u + p M ∫ σ + ∞ | m ( t , u ) | d u .

Therefore, using assumption (16) and estimate (28), there is a ν>0 such that

d d t ∫ 0 σ | v ( t , s ) | d s ≤ − ν ∫ 0 σ | v ( t , s ) | d s + C e λ t ∫ 0 ∞ P ( s ) ( | M ( s , 0 ) | + | J ( s , 0 ) | ) d s .

This concludes the proof of Proposition 3.1. □

We now establish the assumptions in Proposition 3.1 on exponential decay for M and J. This is stated in the following proposition.

Proposition 3.2 There exist a constant λ < 0 and a function P satisfying properties (25) such that the following estimates hold

∫ 0 + ∞ P ( s ) | M ( s , t ) | d s ≤ C e λ t ∫ 0 + ∞ P ( s ) | M ( s , 0 ) | d s , ∫ 0 + ∞ P ( s ) | J ( s , t ) | d s ≤ C e λ t ∫ 0 + ∞ P ( s ) | J ( s , 0 ) | d s ,

assuming the initial bounds (20) are satisfied.

Proof We divide the proof in two steps. We first derive a closed form for the equation on M(s,t); this is our main observation, which allows to extend the argument in 12 to nonconstant coefficients. Then, thanks to dual problem that we study in Appendix C, we conclude our proof.

Step 1. The equation on M(s,t). Integrating Eq. (1) once subtracted to the same equation for A, we find successively

∂ M ( s , t ) ∂ t + ∂ M ( s , t ) ∂ s + ∫ 0 s p ( x ) ∂ M ( x , t ) ∂ x d x = ∫ x = 0 s ∫ K ( x , u ) p ( u ) ∂ M ( u , t ) ∂ u d u d x , ∂ M ( s , t ) ∂ t + ∂ M ( s , t ) ∂ s + p ( s ) M ( s , t ) = ∫ 0 s ∂ p ( x ) ∂ x M ( x , t ) d x − ∫ x = 0 s ∫ ∂ ∂ u ( K ( x , u ) p ( u ) ) M ( u , t ) d u d x , ∂ M ( s , t ) ∂ t + ∂ M ( s , t ) ∂ s + p ( s ) M ( s , t ) = ∫ u = 0 s ∂ p ( u ) ∂ u M ( u , t ) d u − ∫ u = 0 ∞ ∂ p ( u ) ∂ u M ( u , t ) ∫ x = 0 s K ( x , u ) d x d u − ∫ x = 0 s ∫ ∂ K ( x , u ) ∂ u p ( u ) M ( u , t ) d u d x , ∂ M ( s , t ) ∂ t + ∂ M ( s , t ) ∂ s + p ( s ) M ( s , t ) = ∫ u = 0 s ∂ p ( u ) ∂ u ( 1 − f ( s , u ) ) M ( u , t ) d u − ∫ u = s ∞ ∂ p ( u ) ∂ u f ( s , u ) M ( u , t ) d u + ∫ Φ ( s , u ) p ( u ) M ( u , t ) d u .

As f ( s , u ) = 1 for s ≥ u , this equality reduces to

∂ M ( s , t ) ∂ t + ∂ M ( s , t ) ∂ s + p ( s ) M ( s , t ) = − ∫ u = s ∞ ∂ p ( u ) ∂ u f ( s , u ) M ( u , t ) d u + ∫ p ( u ) Φ ( s , u ) M ( u , t ) d u .

We can now insert the absolute values and find

∂ | M ( s , t ) | ∂ t + ∂ | M ( s , t ) | ∂ s + p ( s ) | M ( s , t ) | ≤ ∫ u = s ∞ | p ′ ( u ) | f ( s , u ) | M ( u , t ) | d u + ∫ p ( u ) Φ ( s , u ) | M ( u , t ) | d u .

There are to routes to go further. To cover the case of interest where p can vanish during the refractory period, we use a duality argument. However, we can use directly formula (30) under different assumptions on p; this is performed in Appendix B.

Step 2. End of the proof of Proposition 3.2. By construction, we have M ( 0 , t ) = M ( ∞ , t ) = 0 . Therefore, we may multiply inequality (30) by P, using Lemma C.1 and assumption (20), we may integrate by parts and we find that there exists λ < 0 such that

d d t ∫ | M ( s , t ) | P ( s ) d s ≤ λ ∫ | M ( s , t ) | P ( s ) d s ,

which proves the decay of | M | in Proposition 3.2 thanks to the Gronwall lemma.

Because time only enters in Eq. (29) through M(s,t), we may differentiate in time and find that J still satisfies (29), therefore, the inequality (30) also holds for | J | and, since 0 = J ( 0 , t ) = J ( ∞ , t ) , we conclude as before that

d d t ∫ | J ( s , t ) | P ( s ) d s ≤ λ ∫ | J ( s , t ) | P ( s ) d s

which concludes the proof of Proposition 3.2. □

The proofs of Theorems 2.2 and 2.3 follow the strategy used to prove Theorem 2.1. The main difficulty is that the control on M is a weak control on m while the nonlinear term, which involves N(t), is in a strong dependency. To solve this difficulty, we had to assume that the function p is regular enough; this allows us, via an integration by parts to increase the regularity of the nonlinear term N(t) at the expense of Lipschitz regularity on p.

Step 1. Proof of (21) and (22). Let n be the solution of Eq. (1) and A be the stationary state in (3). Then m : = n − A is solution of the equation

{ ∂ m ( s , t ) ∂ t + ∂ m ( s , t ) ∂ s + p ( s , A ∗ ) m ( s , t ) = ∫ K ( s , u ) p ( u , A ∗ ) m ( u , t ) d u + [ p ( s , A ∗ ) − p ( s , N ( t ) ) ] n ( s , t ) − ∫ K ( s , u ) [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] n ( u , t ) d u , s ≥ 0 , t ≥ 0 , m ( 0 , t ) = 0 , m ( s , 0 ) = n 0 ( s ) − A ( s ) , ∫ 0 ∞ m 0 ( s ) d s = 0 , A ∗ : = ∫ p ( s , A ∗ ) A ( s ) d s .

Following the calculation in the linear case, the function M ( s , t ) : = ∫ 0 s m ( u , t ) d u is solution of the equation:

{ ∂ M ( s , t ) ∂ t + ∂ M ( s , t ) ∂ s + p ( s , A ∗ ) M ( s , t ) = − ∫ u = s ∞ ∂ p ( u , A ∗ ) ∂ u f ( s , u ) M ( u , t ) d u + ∫ Φ ( s , u ) p ( u , A ∗ ) M ( u , t ) d u + ∫ 0 s [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] n ( u , t ) d u − ∫ f ( s , u ) [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] n ( u , t ) d u .

We introduce the remainder term

R ( s , t ) : = ∫ 0 s [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] n ( u , t ) d u − ∫ f ( s , u ) [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] n ( u , t ) d u

which can be written using (11) as

R ( s , t ) = − ∫ s + ∞ f ( s , u ) [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] n ( u , t ) d u .

Using assumptions (7) and (5), we find that

| R ( s , t ) | ≤ { 0 for  s ≥ σ + : = max σ ( ⋅ ) , 2 η | A ∗ − N ( t ) | else.

As before, we may enter the absolute values and find

∂ | M ( s , t ) | ∂ t + ∂ | M ( s , t ) | ∂ s + p ( s , A ∗ ) | M ( s , t ) | ≤ ∫ u = s ∞ [ | ∂ p ( u , A ∗ ) ∂ u | f ( s , u ) + Φ ( s , u ) p ( u , A ∗ ) ] | M ( u , t ) | d u + | R ( s , t ) | .

Multiplying by P the equation obtained for M, where P is as in Lemma C.1, we conclude that there is λ<0 (and close to 0) such that

d d t ∫ P ( s ) | M ( s , t ) | d s ≤ λ ∫ P ( s ) | M ( s , t ) | d s + 2 η ∥ P ∥ L ∞ ( 0 , σ + ) | A ∗ − N ( t ) | .

To proceed further, using the notation (17) and M ( 0 , t ) = 0 , we write

A ∗ − N ( t ) = ∫ 0 ∞ [ p ( s , A ∗ ) A ( s ) − p ( s , N ( t ) ) n ( s , t ) ] d s = − ∫ 0 ∞ p ( s , A ∗ ) m ( s , t ) d s + ∫ 0 ∞ [ p ( s , A ∗ ) − p ( s , N ( t ) ) ] n ( s , t ) d s = ∫ 0 ∞ ∂ ∂ s p ( s , A ∗ ) M ( s , t ) d s + ∫ 0 ∞ [ p ( s , A ∗ ) − p ( s , N ( t ) ) ] n ( s , t ) d s

from which we conclude

| A ∗ − N ( t ) | ≤ 1 inf s ≥ 0 P ∥ ∂ s p ∥ L ∞ ( [ 0 , ∞ ) × [ 0 , ∞ ) ) ∫ P ( s ) | M ( s , t ) | d s + η | N ( t ) − A ∗ |

and finally

| A ∗ − N ( t ) | ≤ 1 ( 1 − η ) inf s ≥ 0 P ∥ ∂ s p ∥ L ∞ ( [ 0 , ∞ ) × [ 0 , ∞ ) ) ∫ P ( s ) | M ( s , t ) | d s .

Since η > 0 is supposed small enough, we may insert this estimate in (34) and we deduce that there exists ν>0 such that

d d t ∫ P ( s ) | M ( s , t ) | d s ≤ − ν ∫ P ( s ) | M ( s , t ) | d s

which proves the inequality (21) using the Gronwall lemma.

Inserting this exponential decay on ∫ P ( s ) | M ( s , t ) | d s in (35) proves (22).

Step 2. Proof of (23). In order to better explain our strategy, we begin with a global Lipschitz estimate on N and then prove the exponential decay.

From the definition of N, we have

N ′ ( t ) = ∫ 0 + ∞ N ′ ( t ) ∂ N p ( s , N ( t ) ) n ( s , t ) d s + ∫ 0 + ∞ p ( s , N ( t ) ) ∂ t n ( s , t ) d s .

Using (2), and (7), we obtain

( 1 − η ) | N ′ ( t ) | ≤ | ∫ 0 + ∞ p ( s , N ( t ) ) ∂ t n ( s , t ) d s | .

We first prove a Lipschitz estimate. We write

∫ 0 + ∞ p ( s , N ( t ) ) ∂ t n ( s , t ) d s = − ∫ 0 + ∞ p ( s , N ( t ) ) ∂ s n ( s , t ) d s − ∫ 0 + ∞ p 2 ( s , N ( t ) ) n ( s , t ) d s + ∫ 0 + ∞ p ( s , N ( t ) ) ∫ 0 + ∞ K ( s , u ) p ( u , N ( t ) ) n ( u , t ) d u d s .

Integration by parts implies that

∫ 0 + ∞ p ( s , N ( t ) ) ∂ s n ( s , t ) d s = − ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) n ( s , t ) d s .

The Lipschitz bound on p and (2) gives the estimate | N ′ | ≤ C .

We now prove estimate (23) based on inequality (36). We recall the notations m = n − A = ∂ s M ( s , t ) and conclude

∫ 0 + ∞ p ( s , N ( t ) ) ∂ t n ( s , t ) d s = ∫ 0 + ∞ p ( s , N ( t ) ) ∂ t m ( s , t ) d s = − ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∂ t M ( s , t ) d s .

Because M satisfies the closed equation (32), we deduce that

− ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∂ t M ( s , t ) d s = ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∂ s M ( s , t ) d s + ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) p ( s , A ∗ ) M ( s , t ) d s − ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∫ u = s ∞ ∂ p ( u , A ∗ ) ∂ u f ( s , u ) M ( u , t ) d u d s + ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∫ Φ ( s , u ) p ( u , A ∗ ) M ( u , t ) d u d s + ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) R ( s , t ) d s .

To continue, we control each term by quantities as M(s,t) for which we have already proved exponential decay

| ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∂ s M ( s , t ) d s | = | ∫ 0 + ∞ ∂ s 2 p ( s , N ( t ) ) M ( s , t ) d s | ≤ C ∫ 0 + ∞ | M ( s , t ) | d s , | ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) p ( s , A ∗ ) M ( s , t ) d s | ≤ C ∫ 0 + ∞ | M ( s , t ) | d s .

Using assumption (5)

| ∫ 0 + ∞ ∂ s p ( s , N ( t ) ) ∫ u = s ∞ ∂ p ( u , A ∗ ) ∂ u f ( s , u ) M ( u , t ) d u d s | ≤ C ∫ 0 + ∞ | M ( s , t ) | d s

because these integrals vanish for s > σ + and u > σ + . Next, assumption (12) gives

| ∫ 0 + ∞ ∫ ∂ s p ( s , N ( t ) ) Φ ( s , u ) p ( u , A ∗ ) M ( u , t ) d u d s | ≤ C ∫ 0 + ∞ | M ( s , t ) | d s .

Finally, estimate (33) and Theorem 2.2 imply that there exist C and ν>0 such that

∫ 0 + ∞ | R ( s , t ) | d s ≤ C e − ν t .

All these terms have exponential decay, and thus, back to inequality (36), this concludes the proof of (23).

The proof of Theorem 2.2 is now complete.  □

Step 3. Proof Theorem 2.3. Thanks to (32), the function ∂ t M ( s , t ) : = J ( s , t ) satisfies the equation

∂ J ( s , t ) ∂ t + ∂ J ( s , t ) ∂ s + p ( s , A ∗ ) J ( s , t ) = − ∫ u = s ∞ ∂ p ( u , A ∗ ) ∂ u f ( s , u ) J ( u , t ) d u + ∫ Φ ( s , u ) p ( u , A ∗ ) J ( u , t ) d u + R ˜ ( s , t )

with

{ R ˜ ( s , t ) ≡ 0 for  s ≥ σ + , R ˜ ( s , t ) = − ∫ s σ + f ( s , u ) N ′ ( t ) ∂ N p ( u , N ( t ) ) n ( u , t ) d u R ˜ ( s , t ) = − ∫ s σ + f ( s , u ) [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] ∂ t n ( u , t ) d u for  s ≤ σ + .

Inserting absolute values in Eq. (37) and multiplying by P built in Lemma C.1, we obtain that there exists λ > 0 such that

d d t ∫ P ( x ) | J ( x , t ) | d x ≤ − λ ∫ P ( x ) | J ( x , t ) | d x + ∫ 0 + ∞ P ( x ) | R ˜ ( x , t ) | d x .

We claim that there exist C>0, ν>0 such that

∫ 0 + ∞ P ( x ) | R ˜ ( x , t ) | d x = ∫ 0 σ + P ( x ) | R ˜ ( x , t ) | d x ≤ C e − ν t .

To prove it, we proceed in estimating each term of R ˜ . For s ≤ σ + , we control the first term as

∫ s σ + P ( s ) f ( s , u ) | N ′ ( t ) ∂ N p ( u , N ( t ) ) | n ( u , t ) d u ≤ | N ′ ( t ) | ∥ ∂ N p ∥ L ∞ ( [ 0 , ∞ ) × [ 0 , ∞ ) ) ∥ P ∥ L ∞ [ 0 , σ + ] ≤ C e − ν t

thanks to the exponential decay on N ′ in (23).

The second contribution to R˜, namely

∫ s = 0 σ + | ∫ s σ + f ( s , u ) [ p ( u , A ∗ ) − p ( u , N ( t ) ) ] ∂ t n ( u , t ) d u | d s