#### S. Sutrima, M. Mardiyana, R. Setiyowati, R. Respatiwulan, A new approach on the well-posedness of nonautonomous Cauchy problems: An application in population growth, Vol. 2021 (2021), Article ID 29, pp. 1-15

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DOI: 10.23952/jnfa.2021.29

Received May 15, 2021; Accepted September 1, 2021; Published October 1, 2021

Abstract. This paper addresses the well-posedness of nonautonomous Cauchy problems by using strongly continuous quasi-semigroups ($C_{0}$-quasi-semigroups) on Banach spaces. Necessary and sufficient conditions for the nonautonomous Cauchy problem $\dot{u}(t)=A(t)u(t)$ to be well-posed are that $A(t)$ is the infinitesimal generator of a $C_{0}$-quasi-semigroup. This paper also verifies the sufficiency for an operator to be the infinitesimal generator of a $C_{0}$-quasi semigroup. A simple application in predicting that the persistence or extinction of a population growth model with diffusion and intrinsic growth rate are time-dependent is considered.