Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

Details Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

2011-11-28

arxiv.org/abs/1111.6379v1

Mathematics

Analysis of PDEs

Scientific paper

The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. 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 $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely, for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$.

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