Exponential decay for the growth-fragmentation/cell-division equation

By Philippe Laurencçot, Benoit Perthame. 2009

64 citations recorded by the source corpus.

We consider the linear growth-fragmentation equation arising in the modelling of cell division or polymerisation processes. For constant coefficients, we prove that the dynamics converges to the steady state with an exponential rate. The control on the initial data uses an elaborate $L^1$-norm that seems to be necessary. It also reflects the main idea of the proof which is to use an anti-derivative of the solution. The main technical difficulty is related to the entropy dissipation rate which is too weak to produce a Poincar\'e inequality.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/b91edabcf5ee80710abce4a5a40433bff236b4d2

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Eigenelements of a General Aggregation-Fragmentation Model (2009, unknown): “More precisely, it is proved for a constant fragmentation rate β that u ( x, t ) e − λt converges exponentially fast to ρ U ( x ) where ρ = u 0 ( y ) dy (see [22, 32]).”
Statistical estimation of a growth-fragmentation model observed on a genealogical tree (2012, methodology): “By use of the general relative entropy principle, Michel, Mischler and Perthame showed that the approximation n ( t, x ) e − λ 0 t ≈ N ( x ) is valid [29], with λ 0 > 0, and where ( λ 0 , N ) is the dominant eigenpair related to the corresponding eigenvalue problem, see [28, 32, 12, 25, 8, 2].”