Well-posedness of the growth-coagulation equation with singular kernels

TL;DR

This paper proves the existence, uniqueness, and continuous dependence of solutions for the growth-coagulation equation with singular kernels, extending previous results to more general kernel behaviors.

Contribution

It establishes well-posedness for the growth-coagulation equation with singular kernels, using advanced analytical techniques and extending prior work.

Findings

Existence of weak solutions for singular kernels.

Solutions depend continuously on initial data.

Uniqueness of solutions established.

Abstract

The well-posedness of the growth-coagulation equation is established for coagulation kernels having singularity near the origin and growing atmost linearly at infinity. The existence of weak solutions is shown by means of the method of the characteristics and a weak $L_1$-compactness argument. For the existence result, we also show our gratitude to Banach fixed point theorem and a refined version of the Arzel\'{a}-Ascoli theorem. In addition, the continuous dependence of solutions upon the initial data is shown with the help of the DiPerna-Lions theory, Gronwall's inequality and moment estimates. Moreover, the uniqueness of solution follows from the continuous dependence. The results presented in this article extend the contributions made in earlier literature.