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Linearized Stability of Non-Isolated Equilibria of Quasilinear Parabolic Problems in Interpolation Spaces

Bogdan-Vasile Matioc, Christoph Walker

Abstract

The stability of non-isolated equilibria to quasilinear parabolic problems of the form $u' = A(u)u + f(u)$ is established in interpolation spaces (and thus extending previous results relying on maximal regularity). The approach allows full flexibility in choosing the interpolation methods and requires only low regularity assumptions on the semilinear part $f$. Applications to concrete problems are presented, including the capillarity-driven Hele--Shaw problem and the fractional mean curvature flow.

Linearized Stability of Non-Isolated Equilibria of Quasilinear Parabolic Problems in Interpolation Spaces

Abstract

The stability of non-isolated equilibria to quasilinear parabolic problems of the form is established in interpolation spaces (and thus extending previous results relying on maximal regularity). The approach allows full flexibility in choosing the interpolation methods and requires only low regularity assumptions on the semilinear part . Applications to concrete problems are presented, including the capillarity-driven Hele--Shaw problem and the fractional mean curvature flow.
Paper Structure (7 sections, 9 theorems, 205 equations)

This paper contains 7 sections, 9 theorems, 205 equations.

Table of Contents

  1. Introduction and Main Results
  2. Proof of Theorem \ref{['MT1']}
  3. Proof of Theorem \ref{['MT2']}
  4. Examples
  5. Example (Fractional Mean Curvature Flow)
  6. Example (Hele-Shaw Problem Driven by Capillarity)
  7. Example (A Quasilinear Problem with Scaling Invariance)

Key Result

Theorem 1.1

Assume assA. Then, for each $u^0\in O_\alpha$, the Cauchy problem CP possesses a unique maximal classical solution for every $0\le \theta\le \alpha$, where $t^+=t^+(u^0)\in(0,\infty]$ is the maximal existence time. Moreover, if the orbit $u([0,t^+(u^0));u^0)$ is relatively compact in $O_\alpha$, then $t^+(u^0)=\infty$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • proof : Proof of Theorem \ref{['MT1']}
  • Remark 2.1
  • Proposition 3.1
  • ...and 11 more