A Uniqueness Result for Self-Similar Profiles to Smoluchowski’s Coagulation Equation Revisited

Abstract

In this note we indicate how to correct the proof of a uniqueness result in [6] for self-similar solutions to Smoluchowski’s coagulation equation for kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in the sense that

$$\begin{aligned} -\varepsilon \le K(x,y)-2 \le \varepsilon \Big ( \Big (\frac{x}{y}\Big )^{\alpha } + \Big (\frac{y}{x}\Big )^{\alpha }\Big ) \end{aligned}$$

for $\alpha \in [0,\frac{1}{2})$. Under the additional assumption, in comparison to [6], that K has an analytic extension to $\mathbb {C}{\setminus } (-\infty ,0]$ and that the precise asymptotic behaviour of K at the origin is prescribed, we prove that self-similar solutions with given mass are unique if $\varepsilon$ is sufficiently small. The complete details of the proof are available in [4]. In addition, we give here the proof of a uniqueness result for a related but simpler problem that appears in the description of self-similar solutions for $x \rightarrow \infty$.