Stationary solutions to Smoluchowski's coagulation equation with source

TL;DR

This paper investigates the existence of stationary solutions to Smoluchowski's coagulation equation with sources, establishing conditions based on the source's integrability and kernel properties, and exploring their connection.

Contribution

It provides new criteria for existence and non-existence of stationary solutions, linking source integrability with solution properties under specific kernel conditions.

Findings

Existence of stationary solutions depends on source integrability and kernel monotonicity.

Non-existence results are established for certain source and kernel configurations.

Connections between source properties and stationary solutions are characterized.

Abstract

Existence and non-existence of integrable stationary solutions to Smoluchowski's coagulation equation with source are investigated when the source term is integrable with an arbitrary support in (0, $\infty$). Besides algebraic upper and lower bounds, a monotonicity condition is required for the coagulation kernel. Connections between integrability properties of the source and the corresponding stationary solutions are also studied.