Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels

Abstract

The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree ${\gamma \in (-\infty,1)}$ and satisfy K(x, y) ≤ C (x ^γ + y ^γ). More precisely, for any ${\rho \in (\gamma,1)}$ we establish the existence of a continuous weak nonnegative self-similar profile with decay ${x^{-(1{+}\rho)}}$ as x → ∞.

For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem.