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Cauchy problems with state-dependent time evolution. (English) Zbl 0711.34005

Summary: We consider a class of quasilinear Cauchy problems which frequently arise in the context of structured population dynamics and which have in common that they can be reduced to a semilinear problem by a time-scale argument. We prove existence and uniqueness of solutions, study positivity and regularity properties, and prove the principle of linearized stability. These abstract results are applied to a model describing the dynamics of a size-structured cell population whose individuals are subject to a nonlinar growth law.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
92D25 Population dynamics (general)
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