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Fragmentation and growth-fragmentation equations is a family of problems withvaried and wide applications. This paper is devoted to description of the longtime time asymptotics of two critical cases of these equations, when thedivision rate is constant and the growth rate is linear or zero. The study ofthese cases may be reduced to the study of the following fragmentationequation:$$\frac{\partial}{\partial t} u(t,x) + u(t,x)=\int\limits\_x^\inftyk\_0(\frac{x}{y}) u(t,y) dy.$$Using the Mellin transform of the equation, wedetermine the long time behavior of the solutions. Our results show inparticular the strong dependence of this asymptotic behavior with respect tothe initial data.

Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations