On a mathematical model of age-cycle length structured cell population with non-compact boundary conditions. II (Q2809505)

This study concerns a mathematical model describing an age-cycle length structured cell population.NEWLINENEWLINEThe new challenge is to prove that the bacterial population, described by a full model possessing the asynchronous growth property, is closely related to the strict inequalityNEWLINENEWLINE\[NEWLINE\omega_0 (\mathbb{U}_{\alpha, \beta})> \omega_{\mathrm{ess}}(\mathbb{U}_{\alpha, \beta}), \eqno (1)NEWLINE\]NEWLINE where \(\omega_0 (\mathbb{U}_{\alpha, \beta})\) and \(\omega_{\mathrm{ess}}(\mathbb{U}_{\alpha, \beta})\) denote, respectively, the type and the essential type of the full semigroup \(\mathbb{U}_{\alpha, \beta}=(\mathbb{U}_{\alpha, \beta}(t))_{t \geq 0}\). It is well known that strict inequalities, such as (1), are too hard to prove and therefore, it arises the need to establish adequate strategies. One strategy is to replace (1) by two inequalities such asNEWLINENEWLINE\[NEWLINE \omega_0 (\mathbb{U}_{\alpha, \beta})>\delta\text{ and }\delta \geq \omega_{\mathrm{ess}}(\mathbb{U}_{\alpha, \beta}), \eqno (2)NEWLINE\]NEWLINE for some \(\delta\). Computations show that the most suitable \(\delta\) isNEWLINENEWLINE\[NEWLINE\delta := - \underline{\mu},\quad \text{ where }\underline{\mu} :=\mathrm{ess inf}_{(a,l\in \Omega)}|\mu (a,l)| NEWLINE\]NEWLINE and inequality (2) can be considered instead of (1).NEWLINENEWLINEIn this work, the author is concerned only with the first inequality of (2) (i.e., \(\omega_0 (\mathbb{U}_{\alpha, \beta})> \underline{\mu}\)). Then, he considers relevant hypotheses on the kernel of correlation \(k:=k(\cdot, \cdot)\) and on both transition rates \(\mu\) and \(\eta\). The author estimates the type \(\omega_0 (\mathbb{T}_{\alpha, \beta})\) of the unperturbed semigroup \(\mathbb{T}_{\alpha, \beta}=(\mathbb{T}_{\alpha, \beta}(t))_{t \geq 0}\) (i.e., \(\mu=\eta=0\)) as the unique solution of a certain characteristic equation. Also, it is proved that both semigroups \(\mathbb{T}_{\alpha, \beta}=(\mathbb{T}_{\alpha, \beta}(t))_{t \geq 0}\) and \(\mathbb{U}_{\alpha, \beta}=(\mathbb{U}_{\alpha, \beta}(t))_{t \geq 0}\) are ordered, from which the result of this work (i.e., \(\omega_0(\mathbb{U}_{\alpha, \beta})> \mu\)) can be readily inferred.NEWLINENEWLINEFor Parts I and III, see [ibid. 38, No. 11, 2081--2104 (2015; Zbl 1329.92099); ibid. 39, No. 1, 73--91 (2016; Zbl 1332.92049)].