On the principle of linearized stability for quasilinear evolution equations in time-weighted spaces
Bogdan-Vasile Matioc, Lina Sophie Schmitz, Christoph Walker
TL;DR
The paper develops a principle of linearized stability for quasilinear and semilinear parabolic evolution equations in time-weighted, phase-space interpolations between $\mathrm{dom}(f)$ and $\mathrm{dom}(A)$. By combining time-weighted well-posedness with an evolution-operator framework, it proves exponential stability of equilibria whenever the linearization has negative spectrum, and provides instability criteria when spectrum crosses the imaginary axis. It introduces and analyzes a critical index $\alpha_{\rm crit}$ governing the admissible phase spaces $E_\alpha$, including cases with scaling invariance, and extends prior results to intermediate spaces via interpolation theory. The results are illustrated through three applications: quasilinear problems with quadratic semilinearity, a parabolic-parabolic chemotaxis system, and a scaling-invariant quasilinear equation, demonstrating stability and instability phenomena in critical and noncritical spaces with explicit decay estimates.
Abstract
Quasilinear (and semilinear) parabolic problems of the form $v'=A(v)v+f(v)$ with strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the function $v\mapsto f(v)$ and the quasilinear part $v\mapsto A(v)$ are considered in the framework of time-weighted function spaces. This allows one to establish the principle of linearized stability in intermediate spaces lying between $\mathrm{dom}(f)$ and $\mathrm{dom}(A)$ and yields a greater flexibility with respect to the phase space for the evolution. In applications to differential equations such intermediate spaces may correspond to critical spaces exhibiting a scaling invariance. Several examples are provided to demonstrate the applicability of the results.