Stability condition of the steady oscillations in aggregation models with shattering process and self-fragmentation

Abstract

Abstract 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 λ k b and μ k a , respectively, with 0 ⩽ a , b ⩽ 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.