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Soliton resolution for the energy-critical nonlinear heat equation in the radial case

Shrey Aryan

Abstract

We establish the Soliton Resolution Conjecture for the radial critical non-linear heat equation in dimension $D\geq 3.$ Thus, every finite energy solution resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.

Soliton resolution for the energy-critical nonlinear heat equation in the radial case

Abstract

We establish the Soliton Resolution Conjecture for the radial critical non-linear heat equation in dimension Thus, every finite energy solution resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.
Paper Structure (16 sections, 24 theorems, 224 equations)

This paper contains 16 sections, 24 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Setting of the Problem
  3. Statement of the Main result
  4. Summary of the Proof
  5. Notation
  6. Acknowledgments
  7. Preliminaries
  8. Local Cauchy Theory
  9. Multi-Bubble Configuration
  10. Localized Energy Inequalities and Energy Trapping
  11. Sequential bubbling for finite time blow-up solutions
  12. Sequential bubbling for global solutions
  13. Decomposition of the solution and collision intervals
  14. Collision intervals
  15. Decomposition of the solution
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $D \geq 3$ and let ${u}(t)$ be a finite energy solution to eqn:NLH with initial data ${u}(0)={u}_0 \in \mathcal{E}$, defined on its maximal forward interval of existence $\left[0, T_{+}\right)$. Suppose that, Then either $\mathrm{(i)}$$T_{+}=\infty$, there exist a time $T_0>0$, an integer $N \geq 0$, continuous functions $\lambda_1(t), \ldots, \lambda_N(t) \in C^0\left(\left[T_0, \infty\right

Theorems & Definitions (53)

  • Theorem 1.1: Soliton Resolution
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5: Multi-bubble configuration
  • Lemma 1.6: Localized sequential bubbling
  • Lemma 1.7: Compactness Lemma
  • Lemma 2.1: Local well-posedness
  • Lemma 2.2
  • Definition 2.3
  • ...and 43 more