Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T16:56:41.127Z Has data issue: false hasContentIssue false
  • English
  • Français

ON EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PANTOGRAPH TYPE EQUATION

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  03 March 2021

M. MOHSIN Open the ORCID record for M. MOHSIN [Opens in a new window]  and
A. A. ZAIDI Open the ORCID record for A. A. ZAIDI [Opens in a new window]
M. MOHSIN
Affiliation:
Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan; e-mail: muhammad.mohsin@lums.edu.pk
A. A. ZAIDI*
Affiliation:
Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan; e-mail: muhammad.mohsin@lums.edu.pk
*
e-mail: ali.zaidi@lums.edu.pk
Get access Rights & Permissions [Opens in a new window]

Abstract

We show existence and uniqueness of solutions to an initial boundary value problem that entails a pantograph type functional partial differential equation with two advanced nonlocal terms. The problem models cell growth and division into two daughter cells of different sizes. There is a paucity of information about the solution to the problem for an arbitrary initial cell distribution.

Keywords

hyperbolic partial differential equationsfunctional partial differential equationsCauchy problems

MSC classification

Primary: 35R10: Partial functional-differential equations
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 4 , October 2020 , pp. 489 - 512
Check for updates
Copyright
© Australian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Diekmann, O., Heijmans, H. J. A. M. and Thieme, H. R., “On the stability of the cell size distribution”, J. Math. Biol. 19 (1984) 227–248; doi:10.1007/BF00277748.CrossRefGoogle Scholar
Efendiev, M., van-Brunt, B., Zaidi, A. A. and Shah, T. H., “Asymmetric cell division with stochastic growth rate”, Math. Methods Appl. Sci. 41 (2018) 8059–8069; doi:10.1002/mma.5269.CrossRefGoogle Scholar
Efendiev, M. A., van Brunt, B., Wake, G. C. and Zaidi, A. A., “A functional partial differential equation arising in a cell growth model with dispersion”, Math. Methods Appl. Sci. 41 (2018) 1541–1553; doi:10.1002/mma.4684.CrossRefGoogle Scholar
Gönczy, P., “Mechanisms of asymmetric cell division: flies and worms pave the way”, Nat. Rev. Mol. Cell Biol. 9 (2008) 355–366; doi:10.1038/nrm2388.CrossRefGoogle ScholarPubMed
Hall, A. J. and Wake, G. C., “A functional differential equation arising in modelling of cell growth”, J. Aust. Math. Soc. Ser. B 30 (1989) 424–435; doi:10.1017/S0334270000006366.CrossRefGoogle Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation determining steady size distributions for populations of cells growing exponentially”, J. Aust. Math. Soc. Ser. B 31 (1990) 344–353; doi:10.1017/S0334270000006779.CrossRefGoogle Scholar
Morrison, S. J. and Kimble, J., “Asymmetric and symmetric stem-cell divisions in development and cancer”, Nature 441 (2006) 1068–1074; doi:10.1038/nature04956.CrossRefGoogle ScholarPubMed
Neumüller, R. A. and Knoblich, J. A., “Dividing cellular asymmetry: asymmetric cell division and its implications for stem cells and cancer”, Genes Dev. 23 (2009) 2675–2699; doi:10.1101/gad.1850809.CrossRefGoogle ScholarPubMed
Perthame, B. and Ryzhik, L., “Exponential decay for the fragmentation or cell-division equation”, J. Differ. Equ. 210 (2005) 155–177; doi:10.1016/j.jde.2004.10.018.CrossRefGoogle Scholar
Thorpe, P. H., Bruno, J. and Rothstein, R., “Modeling stem cell asymmetry in yeast”, Cold Spring Harb. Symp. Quant. Biol. 73 (2008) 81–88; doi:10.1101/sqb.2008.73.010.CrossRefGoogle ScholarPubMed
van Brunt B, B., Almalki, A., Lynch, T. and Zaidi, A., “On a cell division equation with a linear growth rate”, ANZIAM J. 59 (2018) 293–312. doi:10.1017/S1446181117000591.Google Scholar
Wake, G. C., Cooper, S., Kim, H.K. and van Brunt, B., “Functional differential equations for cell-growth models with dispersion”, Commun. Appl. Anal. 4 (2000) 561–574.Google Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “A model for asymmetrical cell division”, Math. Biosci. Eng. 12 (2015) 491–501; doi:10.3934/mbe.2015.12.491.CrossRefGoogle Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “Solutions to an advanced functional partial differential equation of the pantograph type”, Proc. R. Soc. A 471 (2015) 20140947; doi:10.1098/rspa.2014.0947.CrossRefGoogle Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “Probability density function solutions to a Bessel type pantograph equation”, Appl. Anal. 95 (2016) 2565–2577; doi:10.1080/00036811.2015.1102890.CrossRefGoogle Scholar