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Traveling waves in nonclassical diffusion equations

William Barker, Le Xuan Dong, Vu Trong Luong, Nguyen Duong Toan

TL;DR

This work studies monotone traveling waves for a nonclassical diffusion equation that includes a mixed space-time derivative $\alpha\frac{\partial^3 u}{\partial x^2 \partial t}$ and a delayed reaction term $f(u_t)$. The authors reformulate the traveling-wave problem as a profile equation $\mathcal{L}\phi + H(\phi)=0$ and solve it via a fixed-point approach using $F=-\mathcal{L}^{-1}H$, leveraging a Green's function $G$ for $\mathcal{L}$ and a monotone iteration in a convex set $\Gamma$. Under structural assumptions (C1)–(C3) that ensure Lipschitz control and monotonicity, and a spectral gap ensuring invertibility of $\mathcal{L}$, Schauder's fixed point theorem yields a monotone traveling-wave profile $\phi$ with appropriate asymptotics between $0$ and $K$. In an explicit delayed logistic example, the paper constructs quasi upper- and lower-solutions for small delays and demonstrates the existence of a monotone wavefront connecting $0$ and $1$, illustrating the method’s constructive nature. Overall, the results extend traveling-wave theory to nonclassical diffusion with delays and provide a robust framework for wave existence in higher-order, delay-augmented diffusion models.

Abstract

We study the existence of monotone traveling wave solutions in a class of nonclassical diffusion equations that include both standard diffusion and a higher-order mixed space-time dispersive term. The reaction term is nonlinear and subject to general structural conditions. By employing the method of upper and lower solutions, using less smooth super and subsolutions, we construct a monotone iterative scheme within a convex set and prove its convergence using Schauder's fixed point theorem. Explicit constructions of super and subsolutions are provided.

Traveling waves in nonclassical diffusion equations

TL;DR

This work studies monotone traveling waves for a nonclassical diffusion equation that includes a mixed space-time derivative and a delayed reaction term . The authors reformulate the traveling-wave problem as a profile equation and solve it via a fixed-point approach using , leveraging a Green's function for and a monotone iteration in a convex set . Under structural assumptions (C1)–(C3) that ensure Lipschitz control and monotonicity, and a spectral gap ensuring invertibility of , Schauder's fixed point theorem yields a monotone traveling-wave profile with appropriate asymptotics between and . In an explicit delayed logistic example, the paper constructs quasi upper- and lower-solutions for small delays and demonstrates the existence of a monotone wavefront connecting and , illustrating the method’s constructive nature. Overall, the results extend traveling-wave theory to nonclassical diffusion with delays and provide a robust framework for wave existence in higher-order, delay-augmented diffusion models.

Abstract

We study the existence of monotone traveling wave solutions in a class of nonclassical diffusion equations that include both standard diffusion and a higher-order mixed space-time dispersive term. The reaction term is nonlinear and subject to general structural conditions. By employing the method of upper and lower solutions, using less smooth super and subsolutions, we construct a monotone iterative scheme within a convex set and prove its convergence using Schauder's fixed point theorem. Explicit constructions of super and subsolutions are provided.
Paper Structure (5 sections, 22 theorems, 144 equations)

This paper contains 5 sections, 22 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Main Results
  4. Monotone Iteration method
  5. Applications

Key Result

Theorem 2.5

Let $A \subset \mathbb{C}$ be a simply connected domain, $f, g$ two analytic complex valued functions in $A$. Let $T$ be the (at most denumerable) set of zeros of $f$, $T'$ the set of zeros of $f+g$ in $A$, $\gamma$ a circuit in $A - T$, defined on an interval $I$. Then, if $|g(z)| < |f(z)|$ in $\ga where $j(a;\gamma)$ is the index of $\gamma$ with respect to $a$, and $\omega(a,f)$ is the multipli

Theorems & Definitions (45)

  • Definition 2.1: $L^p$ Spaces
  • Definition 2.2: Sobolev Space $W^{1,p}$
  • Definition 2.3: Sobolev Space $W^{k,\infty}$
  • Definition 2.4: Sobolev Space $W^{3,\infty}$
  • Theorem 2.5
  • Lemma 3.1
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 35 more