Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel

This article is concerned with the question of uniqueness self-similar profiles for Smoluchowski’s coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel K K can be written as K equals 2 plus epsilon upper W"> = 2 + ε W encoding="application/x-tex">K=2+\varepsilon W . The perturbation assumed to have homogeneity zero and might also singular both Under further regularity assumptions on encoding="application/x-tex">W , will show that sufficiently small alttext="epsilon"> encoding="application/x-tex">\varepsilon there exists, up normalisation tail behaviour infinity, most one profile. Establishing generally considered difficult problem still essentially open. Concerning fat-tailed this actually gives first statement non-solvable kernel.