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Sharp non-explicit blow-up profile for perturbed nonlinear heat equations with gradient terms

Maissâ Boughrara

TL;DR

The paper tackles blow-up for the perturbed nonlinear heat equation with gradient perturbations and establishes single-point blow-up at a blow-up point $a$ with a final profile $u_*$ on $\mathbb{R}^N\setminus\{a\}$. By introducing similarity variables and a shrinking set $V_A$, it linearizes around an explicit profile, analyzes the linear operator $\mathcal{L}$ spectrally, and derives sharp $L^2_\rho$ and $L^\infty_{y,s}$ controls including a no-blow-up threshold to bound the gradient perturbation effects. A central contribution is the sharp non-explicit blow-up profile, obtained by comparing with the unperturbed solution $\hat u$, and a precise estimate of difference between two blow-up solutions: the main result shows that $u_1$ and $u_2$ are related by a space-time similarity transformation, with the difference decaying in the selfsimilar variables. The results culminate in a complete description of the final profile $u_*$ away from the blow-up point and a rigorous characterization of the blow-up dynamics in all dimensions, including a refined 1D analysis. Overall, the work advances the understanding of how gradient perturbations affect blow-up profiles and provides a rigorous framework for sharp asymptotics in perturbed semilinear heat equations.

Abstract

We consider a class of blow-up solutions for perturbed nonlinear heat equations involving gradient terms. We first prove the single point blow-up property for this equation and determine its final blow-up profile. We also give a sharper description of its blow-up behaviour, where we take as a profile some suitably chosen solution of the unperturbed semilinear heat equation. The proof relies on selfsimilar variables with involved arguments to control the gradient term.

Sharp non-explicit blow-up profile for perturbed nonlinear heat equations with gradient terms

TL;DR

The paper tackles blow-up for the perturbed nonlinear heat equation with gradient perturbations and establishes single-point blow-up at a blow-up point with a final profile on . By introducing similarity variables and a shrinking set , it linearizes around an explicit profile, analyzes the linear operator spectrally, and derives sharp and controls including a no-blow-up threshold to bound the gradient perturbation effects. A central contribution is the sharp non-explicit blow-up profile, obtained by comparing with the unperturbed solution , and a precise estimate of difference between two blow-up solutions: the main result shows that and are related by a space-time similarity transformation, with the difference decaying in the selfsimilar variables. The results culminate in a complete description of the final profile away from the blow-up point and a rigorous characterization of the blow-up dynamics in all dimensions, including a refined 1D analysis. Overall, the work advances the understanding of how gradient perturbations affect blow-up profiles and provides a rigorous framework for sharp asymptotics in perturbed semilinear heat equations.

Abstract

We consider a class of blow-up solutions for perturbed nonlinear heat equations involving gradient terms. We first prove the single point blow-up property for this equation and determine its final blow-up profile. We also give a sharper description of its blow-up behaviour, where we take as a profile some suitably chosen solution of the unperturbed semilinear heat equation. The proof relies on selfsimilar variables with involved arguments to control the gradient term.
Paper Structure (21 sections, 30 theorems, 268 equations, 2 figures)

This paper contains 21 sections, 30 theorems, 268 equations, 2 figures.

Table of Contents

  1. Introduction and statement of the results
  2. Setting the problem
  3. A shrinking set $V_A$ to zero
  4. Linearisation in the neighbourhood of the explicite profile
  5. Decomposition of the solution and shrinking set $V_A$
  6. Difference between two blow-up solutions
  7. Strategy of the proof of Theorem \ref{['Thm: ch1']}'
  8. $L^\infty_{y,s}$ estimates and parabolic regularity
  9. $L^2_\rho$ estimate on $g$
  10. On the existence of a dominating component
  11. Further refinement related to the dominating component
  12. Conclusion of the proof of Proposition \ref{['prp:g in L^2_rho']}
  13. $L^\infty(\mathbb{R}^N)$ estimate in the blow-up region and near the blow-up point
  14. No blow-up under some threshold
  15. No blow-up under a threshold
  16. ...and 6 more sections

Key Result

Theorem 1

Assume $N\geq 1$ and let be $(a,T)\in \mathbb{R}^N\times \mathbb{R}^+$, $h$ verifies condition h and $u\in S_{a,T,h}$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (68)

  • Definition 1.1
  • Remark
  • Theorem 1: Single-point blow-up and final profile
  • Theorem 2: Sharp non-explicit blow-up profile
  • Remark
  • Remark
  • Definition 1.2
  • Remark
  • Proposition 2.1: A set shrinking to zero
  • proof
  • ...and 58 more