Stability condition of the steady oscillations in aggregation models with shattering process and self-fragmentation
We consider a system of clusters of various sizes or masses, subject to aggregation and fragmentation by collision with monomers or by self-disintegration. The aggregation rate for the cluster of size or mass $k$ is given by a kernel proportional to $k^{a}$, whereas the collision and disintegration kernels are given by $λk^{b}$ and $μk^{a}$, respectively, with $0\le a,b\le 1$ and positive factors $λ$ and $μ$. We study the emergence of oscillations in the phase diagram $(μ,λ)$ for two models: $(a,b)=(1,0)$ and $(1,1)$. It is shown that the monomer population satisfies a class of integral equations possessing oscillatory solutions in a finite domain in the plane $(μ,λ)$. We evaluate analytically this domain and give an estimate of the oscillation frequency. In particular, these oscillations are found to occur generally for small but nonzero values of the parameter $μ$, far smaller than $λ$.