Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monotonicity results for semilinear parabolic equations on metric graphs

Fabio Punzo, Alberto Tesei

TL;DR

This work extends monotonicity methods for semilinear parabolic equations to the setting of locally finite metric graphs, analyzing $u_t = \Delta u + f(u)$ with either Neumann or Dirichlet Laplacians. It develops $L^2$ and $L^\infty$ sub-/supersolution frameworks, proves a comparison principle, and shows convergence of solutions to minimal and maximal stationary states, providing extremal stationary solutions within admissible bounds. A key contribution is handling infinite graphs via a growth condition $(H_2)$ and finite-graph truncations to obtain global-in-time results, complemented by a treatment of regular metric trees. The results illuminate monotone dynamics on networks and offer concrete implications for radial symmetry reductions on trees, enhancing understanding of long-time behavior in graph-based diffusion models.

Abstract

We prove monotonicity results for semilinear parabolic problems on locally finite connected metric graphs. Applications to regular metric trees are discussed.

Monotonicity results for semilinear parabolic equations on metric graphs

TL;DR

This work extends monotonicity methods for semilinear parabolic equations to the setting of locally finite metric graphs, analyzing with either Neumann or Dirichlet Laplacians. It develops and sub-/supersolution frameworks, proves a comparison principle, and shows convergence of solutions to minimal and maximal stationary states, providing extremal stationary solutions within admissible bounds. A key contribution is handling infinite graphs via a growth condition and finite-graph truncations to obtain global-in-time results, complemented by a treatment of regular metric trees. The results illuminate monotone dynamics on networks and offer concrete implications for radial symmetry reductions on trees, enhancing understanding of long-time behavior in graph-based diffusion models.

Abstract

We prove monotonicity results for semilinear parabolic problems on locally finite connected metric graphs. Applications to regular metric trees are discussed.

Paper Structure

This paper contains 10 sections, 8 theorems, 157 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Concepts of solution
  4. Subsolutions and supersolutions
  5. Main results
  6. Proofs
  7. An example: metric trees
  8. Metric graphs
  9. Laplacian on metric graphs
  10. Metric trees

Key Result

Theorem 3.1

Let $u_0\in L^2(\mathcal{G})$, and let $(H_0)$-$(H_1)$ hold. Then there exists $T=T(u_0)\in\mathbb{R}_+$ such that problem CN has a unique solution in $[0,T]$. In addition, if $u_0\ge0$ in $\mathcal{G}$ and $f:\overline{\mathbb{R}}_+\mapsto\overline{\mathbb{R}}_+$, there holds $u(\cdot,t)\ge0$ in $\

Theorems & Definitions (28)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 18 more