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Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices

Isaac Ohavi

TL;DR

The article establishes existence and uniqueness of classical solutions for quasi-linear parabolic PDEs on a bounded ramified network with a nonlinear Neumann-type condition at the vertex. It employs a time-discretization strategy that reduces the problem to solving a sequence of elliptic problems on each edge, leveraging Schauder-type estimates and a parabolic maximum principle to obtain a priori bounds and convergence. The main contributions include a rigorous solvability result in Hölder spaces, a comparison principle ensuring uniqueness, and an extension to unbounded junctions via truncation and passage to the limit. This framework provides well-posedness for diffusion-type dynamics on networks with nonlinear vertex coupling and has potential applications in graph-based diffusion processes and networked systems.

Abstract

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices

TL;DR

The article establishes existence and uniqueness of classical solutions for quasi-linear parabolic PDEs on a bounded ramified network with a nonlinear Neumann-type condition at the vertex. It employs a time-discretization strategy that reduces the problem to solving a sequence of elliptic problems on each edge, leveraging Schauder-type estimates and a parabolic maximum principle to obtain a priori bounds and convergence. The main contributions include a rigorous solvability result in Hölder spaces, a comparison principle ensuring uniqueness, and an extension to unbounded junctions via truncation and passage to the limit. This framework provides well-posedness for diffusion-type dynamics on networks with nonlinear vertex coupling and has potential applications in graph-based diffusion processes and networked systems.

Abstract

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Paper Structure

This paper contains 11 sections, 11 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. main results
  3. Notations and preliminary results
  4. Assumptions and main results
  5. The elliptic problem
  6. The parabolic problem
  7. Estimates on the discretized scheme
  8. Proof of Theorem \ref{['th : exis para']}.
  9. On the existence for unbounded junction
  10. Functionnal spaces
  11. The Elliptic problem

Key Result

Lemma 2.1

Suppose that $u \in \mathcal{C}^{0,1}([0,T]\times[0,R])$ satisfies an Hölder condition in $t$ in $[0,T]\times [0,R]$, with exponent $\alpha\in (0,1]$, constant $\nu_1$, and has derivative $\partial_xu$, which for any $t\in[0,T]$ are Hölder continuous in the variable $x$, with exponent $\gamma\in(0,1

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • Definition 3.2
  • Theorem 3.3
  • Lemma 4.1
  • ...and 12 more