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Flat Standing Sphere Blow-up Solutions for the Nonlinear Heat Equation

Senhao Duan

Abstract

In this paper, we prove the existence of a singular standing sphere blow-up solution for the nonlinear heat equation with radial symmetry. This solution develops a finite-time singularity on a fixed-radius sphere and exhibits a flat blow-up profile. Our construction refines the method developed by Merle and Zaag \cite{MZJEMS24} which reduces the infinite-dimensional dynamics to a finite-dimensional problem in radial case. The solution satisfies explicit asymptotics near the singular ring and remains regular elsewhere.

Flat Standing Sphere Blow-up Solutions for the Nonlinear Heat Equation

Abstract

In this paper, we prove the existence of a singular standing sphere blow-up solution for the nonlinear heat equation with radial symmetry. This solution develops a finite-time singularity on a fixed-radius sphere and exhibits a flat blow-up profile. Our construction refines the method developed by Merle and Zaag \cite{MZJEMS24} which reduces the infinite-dimensional dynamics to a finite-dimensional problem in radial case. The solution satisfies explicit asymptotics near the singular ring and remains regular elsewhere.
Paper Structure (71 sections, 40 theorems, 297 equations)

This paper contains 71 sections, 40 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Strategy for bounding the "radial singularity"
  4. Expectations for the profile
  5. Profile for $y$ in compact sets
  6. Profile in larger region
  7. Strategy of the proof
  8. Proof of the apriori bound \ref{['eqabU']}
  9. Determination of a modified version of the profile
  10. Dynamics of $q$
  11. • Spectral properties of the linear operator $\mathcal{L}$
  12. • The potential $V$:
  13. • The nonlinear term $B$:
  14. • The remainder term $R$:
  15. • The cut-off term $N$, $H$ and $G$:
  16. ...and 56 more sections

Key Result

Theorem 1.1

(Existence of a singular standing solution for equation eq: NLH introduction with prescribed profile). For any $r_0 > 0$, there exists $T > 0$ such that equation eq: NLH introduction admits a radially symmetric solution $U(x,t)$ defined on $\mathbb{R}^d \times [0,T)$, satisfying:

Theorems & Definitions (81)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 3.1
  • proof
  • Definition 3.2: Definition of the shrinking set $\mathcal{S}(t)$
  • Proposition 4.1
  • proof
  • Lemma 4.2: Estimates for term $R$
  • Proposition 4.3: A delay regularizing estimate for equation \ref{["q's equation"]}
  • proof
  • ...and 71 more