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Extinction and propagation phenomena for semilinear parabolic equations on metric trees

Fabio Punzo, Alberto Tesei

TL;DR

This work analyzes semilinear parabolic equations on regular metric trees, establishing an extinction/propagation dichotomy governed by the spectrum of the Neumann Laplacian, $E_0=\min\sigma(-\\Delta)$, and the linearization $f'(0)$. By reducing to symmetric one-dimensional problems and employing comparison principles with carefully constructed sub- and super-solutions, it proves that for $f'(0)>E_0$ solutions converge to the stable state $1$, while for homogeneous trees and $f'(0)<E_0$, small data lead to extinction. When propagation occurs, the paper derives explicit speed bounds: an upper bound $\hat{c}=M\rho_1\frac{b_1}{b_1-1}$ for small data and a lower bound $\check{c}=2\sqrt{f'(0)-E_0}$ under suitable geometric constraints, with the true speed sandwiched as $\check{c}<c_0<\hat{c}$. The analysis hinges on spectral theory for regular trees, symmetry reductions, and a robust PDE framework on metric graphs, extending classical Euclidean and hyperbolic results to branching networks with explicit dependence on edge lengths and branching numbers.

Abstract

We study the Cauchy-Neumann problem on a regular metric tree T for the semilinear heat equation with forcing term of KPP type. Propagation and extinction of solutions, as well as asymptotical speed of propagation are investigated.

Extinction and propagation phenomena for semilinear parabolic equations on metric trees

TL;DR

This work analyzes semilinear parabolic equations on regular metric trees, establishing an extinction/propagation dichotomy governed by the spectrum of the Neumann Laplacian, , and the linearization . By reducing to symmetric one-dimensional problems and employing comparison principles with carefully constructed sub- and super-solutions, it proves that for solutions converge to the stable state , while for homogeneous trees and , small data lead to extinction. When propagation occurs, the paper derives explicit speed bounds: an upper bound for small data and a lower bound under suitable geometric constraints, with the true speed sandwiched as . The analysis hinges on spectral theory for regular trees, symmetry reductions, and a robust PDE framework on metric graphs, extending classical Euclidean and hyperbolic results to branching networks with explicit dependence on edge lengths and branching numbers.

Abstract

We study the Cauchy-Neumann problem on a regular metric tree T for the semilinear heat equation with forcing term of KPP type. Propagation and extinction of solutions, as well as asymptotical speed of propagation are investigated.
Paper Structure (24 sections, 26 theorems, 284 equations)

This paper contains 24 sections, 26 theorems, 284 equations.

Table of Contents

  1. introduction
  2. Motivations
  3. Statement of the problem and assumptions
  4. The Euclidean case
  5. The hyperbolic case
  6. Outline of results
  7. Main results
  8. Extinction versus propagation
  9. Asymptotic speed of propagation
  10. Spectrum of the Neumann Laplacian on regular trees
  11. Semilinear parabolic equations on regular trees
  12. Concept of solution and well-posedness
  13. Monotonicity results
  14. Symmetric super- and subsolutions
  15. Extinction versus propagation: Proofs
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $(H_0)$-$(H_2)$ and $(H_4)$ be satisfied. Suppose that and let $u_0\not \equiv 0$ in $\mathcal{T}$. Then for the corresponding solution of problem CN there holds

Theorems & Definitions (59)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 49 more