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ON A CELL DIVISION EQUATION WITH A LINEAR GROWTH RATE

Part of:General first-order equations and systems

Published online by Cambridge University Press:  26 February 2018

B. VAN BRUNT,
A. ALMALKI,
T. LYNCH and
A. ZAIDIOpen the ORCID record for A. ZAIDI[Opens in a new window]
B. VAN BRUNT
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand email b.vanbrunt@massey.ac.nz, alone-h1@hotmail.com, t.a.lynch@massey.ac.nz
A. ALMALKI
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand email b.vanbrunt@massey.ac.nz, alone-h1@hotmail.com, t.a.lynch@massey.ac.nz
T. LYNCH
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand email b.vanbrunt@massey.ac.nz, alone-h1@hotmail.com, t.a.lynch@massey.ac.nz
A. ZAIDI*
Affiliation:
Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan email ali.zaidi@lums.edu.pk
*
✉ ali.zaidi@lums.edu.pk
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Abstract

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We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

Keywords

functional partial differential equationsnonlocal partial differential equations

MSC classification

Primary:35F16: Initial-boundary value problems for linear first-order equations

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018, pp. 293 - 312
Copyright
© 2018 Australian Mathematical Society 

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