We study a size-structured tree growth model from [4–6], described by the nonlinear renewal equation $\phi(t) = \mathfrak{F} \phi_t, \ \phi_t \in L^1_\rho(\mathbb{R}_{-}),$ with reproduction, death, and growth rates $\beta$, $\mu$, and $g$. We prove that, under mild conditions on these rates, the equation generates a semiflow in $L^1_\rho(\mathbb{R}_{-})$ that is permanent and possesses a compact global attractor $\mathcal{A}$. If $\beta$ is monotone, $\mathcal{A}$ reduces to a single asymptotically stable equilibrium attracting all compact sets with positive initial data. Adapting an approach from [21], originally developed for simpler renewal equations, we investigate stability and persistence in this more complex setting via the one-dimensional recurrence $b_{n+1} = \mathfrak{F} b_n,$ thereby complementing the functional-analytic framework of [13].

Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.