On the non existence of non negative solutions to a critical Growth-Fragmentation Equation

Escobedo, Miguel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 177-220.

Nous considérons une équation de croissance fragmentation dont les taux de croissance et de fragmentation s’équilibrent et dont le noyau de dislocation est constant. Suivant la valeur de cette constante l’équation vérifie ou non la condition $\left(M_{+}\right)$ introduite par J. Bertoin et A. Watson dans [8]. Nous démontrons que, comme ces auteurs l’avaient suggéré, lorsque la condition n’est pas vérifiée l’équation ne possède pas de solution globale non négative dont les moments satisfont certaines estimations naturelles. Nous montrons également que l’équation peut aussi ne pas avoir de solution locale vérifiant de telles estimations. Lorsqu’une telle solution existe, locale ou globale, une formule explicite est obtenue.

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $\left(M_{+}\right)$ of J. Bertoin and A. Watson [8] is not necessarily satisfied. It is proved that, as it was first suggested by these authors, when that happens, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exists. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.

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