Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non Local Mixed Systems with Neumann Boundary Conditions

Rinaldo M. Colombo, Elena Rossi, Abraham Sylla

TL;DR

The article develops an $L^1$ well-posedness and stability theory for a nonlinear, nonlocal mixed hyperbolic–parabolic system on a bounded domain with Neumann boundary conditions for the parabolic part. The hyperbolic and parabolic components are analyzed separately, yielding existence, uniqueness, positivity, and continuous dependence results, then glued together for the full coupled system via a fixed-point scheme that alternates solving each subproblem. A Neumann Green function representation provides a robust parabolic solution formula, and a careful $L^1$ framework ensures global-in-time solvability under broad, biologically motivated assumptions. These results extend existing $L^2$–based treatments by delivering a rigorous $L^1$ theory applicable to predator–prey and control applications where mass conservation and no-flux boundaries are essential. The combination of approximation arguments, Green function techniques, and contraction on short time intervals enables a global existence proof and quantitative stability estimates with respect to initial data and forcing terms.

Abstract

We prove well posedness and stability in $\mathbf{L}^1$ for a class of mixed hyperbolic-parabolic non linear and non local equations in a bounded domain with no flow along the boundary. While the treatment of boundary conditions for the hyperbolic equation is standard, the extension to $\mathbf{L}^1$ of classical results about parabolic equations with Neumann conditions is here achieved.

Non Local Mixed Systems with Neumann Boundary Conditions

TL;DR

The article develops an well-posedness and stability theory for a nonlinear, nonlocal mixed hyperbolic–parabolic system on a bounded domain with Neumann boundary conditions for the parabolic part. The hyperbolic and parabolic components are analyzed separately, yielding existence, uniqueness, positivity, and continuous dependence results, then glued together for the full coupled system via a fixed-point scheme that alternates solving each subproblem. A Neumann Green function representation provides a robust parabolic solution formula, and a careful framework ensures global-in-time solvability under broad, biologically motivated assumptions. These results extend existing –based treatments by delivering a rigorous theory applicable to predator–prey and control applications where mass conservation and no-flux boundaries are essential. The combination of approximation arguments, Green function techniques, and contraction on short time intervals enables a global existence proof and quantitative stability estimates with respect to initial data and forcing terms.

Abstract

We prove well posedness and stability in for a class of mixed hyperbolic-parabolic non linear and non local equations in a bounded domain with no flow along the boundary. While the treatment of boundary conditions for the hyperbolic equation is standard, the extension to of classical results about parabolic equations with Neumann conditions is here achieved.

Paper Structure

This paper contains 23 sections, 9 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Hyperbolic Problem
  4. The Parabolic Problem
  5. Analytical Proofs
  6. Hyperbolic Problem
  7. Parabolic Problem
  8. Step 1: Uniqueness.
  9. Step 2: Approximation.
  10. Step 3: Existence in $\mathbf{L}^\infty \left([0,T];\mathbf{L}^1 (\Omega;{\mathbb{R}})\right)$.
  11. Step 4: $\mathbf{L}^1(\Omega;{\mathbb{R}})$ Continuity in Time.
  12. Step 5: Representation Formula.
  13. Step 4: \ref{['item:10']} and \ref{['item:32']} hold.
  14. Mixed Problem
  15. Step 1: $c_i$ satisfies \ref{['it:H1']} in Theorem \ref{['thm:hypGen']}.
  16. ...and 8 more sections

Key Result

Theorem 2.2

Fix $T > 0$. Assume that it:omega, it:v, it:alpha, it:a, it:betaFinal and it:b hold. Let $(u_o,w_o) \in (\mathbf{L}^1\cap \mathbf{BV}) (\Omega;{\mathbb{R}}) \times \mathbf{L}^1 (\Omega;{\mathbb{R}})$. Then:

Theorems & Definitions (13)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • ...and 3 more