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arXiv · 2412.13940

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

Bogdan-Vasile Matioc·Lina Sophie Schmitz·Christoph Walker
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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.

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Bogdan-Vasile Matioc, Lina Sophie Schmitz, Christoph Walker. 2024-12-18. On the principle of linearized stability for quasilinear evolution equations in time-weighted spaces. https://arxiv.org/abs/2412.13940

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