Self-Similar Solutions to Coagulation Equations with Time-Dependent Tails: The Case of Homogeneity One

Abstract

We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski’s coagulation equation, for a class of kernels ${K(x,y)}$ which are homogeneous of degree one and satisfy ${K(x,1) \to k_0 > 0}$ as ${x\to 0}$. In particular, we establish the existence of a critical ${\rho_* > 0}$ with the property that for all ${\rho\in(0,\rho_*)}$ there is a positive and differentiable self-similar solution with finite mass M and decay ${A(t)x^{-(2+\rho)}}$ as ${x\to\infty}$, with ${A(t)=e^{M(1+\rho)t}}$. Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter ${\rho}$.