Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

Jacek Banasiak

doi:10.3934/dcdss.2020161

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2020, Volume 13, Issue 12: 3319-3334. Doi: 10.3934/dcdss.2020161

This issue Previous Article Fractional Ostrowski-Sugeno Fuzzy univariate inequalities Next Article Existence of minimizers for some quasilinear elliptic problems

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

Jacek Banasiak^1,2,

1.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
2.
Institute of Mathematics, Łódź University of Technology, Łódź, Poland

The paper is dedicated to Giséle Ruiz Goldstein on the occasion of her birthday

Received: January 31, 2019

Revised: March 31, 2019

Early access: December 2019

Published: August 2020

The research has been partially supported by the National Science Centre of Poland Grant 2017/25/B/ST1/00051 and the National Research Foundation of South Africa Grant 82770

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In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

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Mathematics Subject Classification: Primary: 35R09, 47J35; Secondary: 35K58, 47D06, 82C22, 82C05.

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