Steady distribution of the incremental model for bacteria proliferation. (English) Zbl 1428.35623
Summary: We study the mathematical properties of a model of cell division structured by two variables – the size and the size increment – in the case of a linear growth rate and a self-similar fragmentation kernel. We first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue \(1\) from a solution of a certain one dimensional fixed point problem. Then we prove the existence and uniqueness of this fixed point in the appropriate \(\mathrm{L}^1\) weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35P05 | General topics in linear spectral theory for PDEs |
| 45K05 | Integro-partial differential equations |
| 45P05 | Integral operators |
| 92D25 | Population dynamics (general) |
| 35A22 | Transform methods (e.g., integral transforms) applied to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
structured populations; cell division; transport equation; eigenproblem; long-time asmptotics; integral equationReferences:
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