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Self-similar blow up for energy supercritical semilinear wave equation

Jihoi Kim

Abstract

We analyse the energy supercritical semilinear wave equation $$Φ_{tt}-ΔΦ-|Φ|^{p-1}Φ=0$$ in $\mathbb R^d$ space. We first prove in a suitable regime of parameters the existence of a countable family of self similar profiles which bifurcate from the soliton solution. We then prove the non radial finite codimensional stability of these profiles by adapting the functional setting of arXiv:1912.11005.

Self-similar blow up for energy supercritical semilinear wave equation

Abstract

We analyse the energy supercritical semilinear wave equation in space. We first prove in a suitable regime of parameters the existence of a countable family of self similar profiles which bifurcate from the soliton solution. We then prove the non radial finite codimensional stability of these profiles by adapting the functional setting of arXiv:1912.11005.
Paper Structure (22 sections, 27 theorems, 462 equations)

This paper contains 22 sections, 27 theorems, 462 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Existence of self-similar profiles
  4. non-linear stability
  5. Notations
  6. Construction of exterior solutions
  7. Fundamental solutions and exterior resolvent
  8. Exterior solutions
  9. Construction of interior solutions
  10. The matching
  11. Dissipativity of linearized operator
  12. Subcoercivity
  13. Dissipativity
  14. Growth bounds for the dissipative operators
  15. Finite codimensional stability
  16. ...and 7 more sections

Key Result

Theorem 1

Assume vneoneneoivne. There exists $N\in\mathbb N$ such that for all $n\ge N$, there exists a smooth radially symmetric self-similar solution to equation eq: Nonlinear Wave such that for $\Lambda u_n$ vanishes exactly $n$ times on $(0,\infty)$. Moreover: (i) Behaviour at infinity: as $n\rightarrow \infty$ the solutions $u_n$ converge to the explicit singular solution to eq: Self-similar Profile

Theorems & Definitions (54)

  • Theorem 1: Existence and asymptotes of excited self-similar solutions
  • Theorem 2: Non-linear stability
  • Lemma 3.1: Fundamental solutions of $\mathcal{L}_\infty$
  • proof
  • Proposition 3.2: Solutions to singular ODEs, V
  • Corollary 3.3
  • proof
  • Proposition 3.4: Exterior resolvent
  • proof
  • Lemma 3.5: Non-linear bounds
  • ...and 44 more