Spectral Gap for the Growth-Fragmentation Equation via Harris's Theorem

We study the long-time behaviour of growth-fragmentation equation, a nonlocal linear evolution equation describing wide range phenomena in structured population dynamics. show existence spectral gap under conditions that generalise those literature by using method based on Harris's theorem, result coming from equilibration Markov processes. The difficulty posed non-conservativeness is overcome performing an $h$-transform, after solving dual Perron eigenvalue problem. direct eigenvector then consequence our methods, which prove exponential contraction equation. Moreover rate convergence explicitly quantifiable terms eigenfunction and coefficients