On the principle of linearized stability for quasilinear evolution equations in time‐weighted spaces

Bogdan‐Vasile Matioc, Lina Sophie Schmitz and Christoph Walker

Mathematische Nachrichten, 2025, vol. 298, issue 12, 3939-3959

Abstract: Quasilinear (and semilinear) parabolic problems of the form v′=A(v)v+f(v)$v^{\prime }=A(v)v+f(v)$ with strict inclusion dom(f)⊊dom(A)$\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the function v↦f(v)$v\mapsto f(v)$ and the quasilinear part v↦A(v)$v\mapsto A(v)$ 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 dom(f)$\mathrm{dom}(f)$ and dom(A)$\mathrm{dom}(A)$ 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.

Date: 2025
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