Spectral gap for the growth-fragmentation equation via Harris's theorem
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AuthorCañizo Rincón, José Alfredo
Society for Industrial and Applied Mathematics
Growth-fragmentation equationsHarris's theoremSpectral gapLong-time behavior of solutionsStructured population dynamics
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]
SponsorshipBERC; 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 Fund
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.