Paper · Jun 5, 2021

Global dynamics of a parabolic type equation arising from the curvature flow

Xueli Bai, Fang Li, Xiaoliu Wang

math.APWider flowsarXiv:2106.02911
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Abstract

This paper studies a type of degenerate parabolic problem with nonlocal term \beginequation* \begincases u_t=u^p(u_xx+u-\baru) & 0<t<T_{\max},\ 0<x<a, u_x(0,t)=u_x(a,t)=0 & 0<t<T_{\max}, u(x,0)=u_0(x) & 0<x<a, \endcases \endequation* where , . In this paper, the classification of the finite-time blowup/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. Moreover, we combine the applications of Lojasiewicz-Simon inequality and energy estimates to derive that any bounded solution with positive initial data converges to some steady state as .

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Submitted
Jun 5, 2021
Subjects
math.AP
DOI
10.1088/1361-6544/ac9a95
Other curvature flowsSingularitiesLong-time existence & convergence

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper General curvature flow without singularities (Ling Xiao, 2016)7Xiao 2016 Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes (Bendong Lou, Xiaoliu Wang and Lixia Yuan, 2019)4Lou 2019 The Prescribed Q-Curvature Flow for Arbitrary Even Dimension in a Critical Case (Yuchen Bi and Jiayu Li, 2021)5Bi 2021 Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions (Xinfu Chen, Bendong Lou, Xiaoliu Wang et al., 2023)2Chen 2023 Two nonlocal inverse curvature flows of convex closed plane curves (Zezhen Sun, 2023)6Sun 2023 Long time regularity of the p-Gauss curvature flow with flat side (G. Huang, X. -J. Wang and Y. Zhou, 2024)3Huang 2024 On the Yamabe flow in a bounded domain (Fei Fang and Zhong Tan, 2026)1Fang 2026 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Yamabe flow in a bounded domain

    Fei Fang and Zhong TanWider flows

    Topics Long-time existence & convergence, Singularities · Shared terms “steady states”, “time blowup”, “initial data”

  2. Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

    Xinfu Chen, Bendong Lou, Xiaoliu Wang et al.Wider flows

    Topics Long-time existence & convergence · Same author Xiaoliu Wang · Shared terms “degenerate”

  3. Long time regularity of the -Gauss curvature flow with flat side

    G. Huang, X. -J. Wang and Y. ZhouWider flows

    Topics Other curvature flows · Same author Xiaoliu Wang

  4. Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes

    Bendong Lou, Xiaoliu Wang and Lixia YuanWider flows

    Topics Other curvature flows · Same author Xiaoliu Wang

  5. The Prescribed -Curvature Flow for Arbitrary Even Dimension in a Critical Case

    Yuchen Bi and Jiayu LiWider flows

    Topics Other curvature flows, Singularities · Shared terms “explicit expression”, “energy functional”

  6. Two nonlocal inverse curvature flows of convex closed plane curves

    Zezhen SunWider flows

    Topics Long-time existence & convergence, Other curvature flows, Singularities · Shared terms “nonlocal”

  7. General curvature flow without singularities

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows, Singularities

  8. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Long-time existence & convergence, Other curvature flows, Singularities · Shared terms “finite time”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.