Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution

We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $p(s)$. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $C^{\mathrm{\infty }}$ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $C^{\mathrm{\infty }}$ regularity of solutions for $t>0$. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $p(s)$.