Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

Abstract

The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted L 1 spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasicompactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.2010 Mathematics Subject Classification. 35B40 (primary), and 35P05, 35Q92, 35R09, 47D06 (secondary).

Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts

“…However in this case the equation can be hypocoercive in the sense (see [36]) that }upt,¨q} ď Ce´ν t }u in } holds for some positive constants C, ν and any initial distribution satisfying xu in , φy " 0. This result is proved in [31,10] for a class of weighted L 1 norms in the case of a constant growth rate g. Roughly speaking this situation of a non-coercive but hypocoercive equation appears when the dissipation of entropy can vanish for a nontrivial set of functions, but this set is unstable for the dynamics of the equation. In our case the equation is not hypocoercive because the set of functions with null entropy dissipation is invariant under the flow, as expressed by the following lemma.…”

“…Since the Perron eigenvalue λ " 1 is strictly positive, it is convenient to consider a rescaled version of our problem It is proved in [20] (see also [10]) that the problem (9) is well-posed in E 1 and admits an associated C 0 -semigroup pT t q tě0 which is positive, meaning that for any u in P E 1 there exists a unique (mild) solution v P CpR`, E 1 q to (9) which is given by vptq " T t u in , and vptq ě 0, t ě 0, for u in ě 0. From Lemma 3 we have that all subspaces E p with p P r1, 8s are invariant under pT t q tě0 .…”

Growth–fragmentation–coagulation equations with unbounded coagulation kernels

“…The reverse process of mass gain can also occur due to the precipitation of matter from the environment. Continuous coagulation and fragmentation processes, combined with a mass transport term that leads to either mass loss or mass gain, have also been studied using functional analytic and, in particular, semigroup methods; for example, see [17–19] and [1, Section 5.2], or [20–22] where, however, the focus is on the long-term behaviour of the linear growth–fragmentation processes. The discrete versions of such models have been comprehensively analysed in [16,23].…”

“…As with (1.5), we re-write (1.9) as (1.6) but with the kernel given by the linear growth–fragmentation semigroup. The main tool is the moment improving property of this semigroup, proven in [20], that makes it a little like an analytic semigroup and allows for an approach similar to that used in [12,14] for pure fragmentation–coagulation problems, where the fragmentation semigroup is indeed analytic. In other words, the growth–fragmentation semigroup retains the moment regularization property of the fragmentation semigroup, but it is not regularizing with respect to the differentiation operator, and hence it is not analytic.…”

The strong Malthusian behavior of growth-fragmentation processes

“…When (1.5) holds, we call λ the Malthus exponent and ν the asymptotic profile. There exists a vast literature on this topic, and we content ourselves here to cite a few contributions [BG20,CDP18,DDGW18,Esc20] amongst the most recent ones, in which many further references can be found. Spectral analysis of the infinitesimal generator A often plays a key role for establishing (1.5).…”