Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations
Provides theoretical insights into the long-time behavior of fragmentation equations, which are relevant to various applications, but the results are incremental.
The paper analyzes the long-time asymptotics of critical cases in fragmentation and growth-fragmentation equations, showing that the asymptotic behavior strongly depends on initial data.
Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation:$$\frac{\partial}{\partial t} u(t,x) + u(t,x)=\int\limits\_x^\infty k\_0(\frac{x}{y}) u(t,y) dy.$$Using the Mellin transform of the equation, we determine the long time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.