## Some results based on maximal regularity regarding population models with age and spatial structure.(English)Zbl 1387.35373

Summary: We review some results on abstract linear and nonlinear population models with age and spatial structure. The results are mainly based on the assumption of maximal $$L_p$$-regularity of the spatial dispersion term. In particular, this property allows us to characterize completely the generator of the underlying linear semigroup and to give a simple proof of asynchronous exponential growth of the semigroup. Moreover, maximal regularity is also a powerful tool in order to establish the existence of nontrivial positive equilibrium solutions to nonlinear equations by fixed point arguments or bifurcation techniques. We illustrate the results with examples.

### MSC:

 35K59 Quasilinear parabolic equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35B32 Bifurcations in context of PDEs 47D06 One-parameter semigroups and linear evolution equations 92D25 Population dynamics (general) 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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