Spectral gap for the growth-fragmentation equation via Harris's theorem
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Cañizo Rincón, José AlfredoEditorial
Society for Industrial and Applied Mathematics
Materia
Growth-fragmentation equations Harris's theorem Spectral gap Long-time behavior of solutions Structured population dynamics
Date
2021-09-16Referencia bibliográfica
Cañizo, J. A., Gabriel, P., & Yoldas, H. (2021). Spectral gap for the growth-fragmentation equation via Harris's Theorem. SIAM Journal on Mathematical Analysis, 53(5), 5185-5214. DOI. [10.1137/20M1338654]
Sponsorship
BERC; French Ministry of Research; Hausdorff Research Institute for Mathematics ANR-20-CE40-0015; Spanish government; European Commission 639638 EC; European Research Council; Ministerio de Economía y Competitividad ¡SEV-2017-0718 MINECO; European Regional Development FundAbstract
We study the long-time behavior of the growth-fragmentation equation, a nonlocal
linear evolution equation describing a wide range of phenomena in structured population dynamics.
We show the existence of a spectral gap under conditions that generalize those in the literature
by using a method based on Harris's theorem, a result coming from the study of equilibration of
Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by
performing an h-transform, after solving the dual Perron eigenvalue problem. The existence of the
direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction
of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the
dual eigenfunction and the coefficients of the equation.