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Finite propagation and saturation in reaction-diffusion-advection equations governed by p-Laplacian operator

Cristina Marcelli

TL;DR

The paper analyzes traveling wave solutions for the mono-stable reaction-diffusion-advection equation governed by the $p$-Laplacian: $f(u)u_x + g(u)u_\tau = [d(u)|u_x|^{p-2}u_x]_x + \rho(u)$ on $x\in\mathbb{R}$, $ au\ge0$, under continuity and positivity assumptions. It proves existence and non-existence of traveling waves and classifies them into front-type, sharp-type, and those exhibiting finite propagation or finite saturation; the minimal speed $c^*$ governs existence. A central methodological contribution is the reduction to a first-order singular problem for $z(u)$ with $\dot z = cg(u)-f(u) - h(u)/z^{1/(p-1)}$, yielding a precise threshold and sharp integral criteria, e.g., $\int_0^{1/2} \frac{1}{\rho(u)}\,du$ and asymptotics of $d,\rho$. The results extend and refine prior work (notably for $p=2$) and provide a robust framework for understanding finite propagation and saturation in degenerate/advection-dominated diffusion, with explicit connections to the behavior near equilibria $u=0$ and $u=1$ through a singular first-order formulation.

Abstract

The paper concerns front propagation for the following mono-stable reaction-diffusion-advection equation \[f(u)u_x + g(u)u_τ= [d(u)|u_x|^{p-2} u_x]_x+ ρ(u), \quad (x,τ)\in \R\times [0,+\infty).\] Besides existence and non-existence results for traveling wave solutions, the main focus is their classification: we provide criteria to establish if they attain one or both the equilibria at a finite time and in this case, if they are continuable as $C^1$-solutions or if they are sharp solutions.

Finite propagation and saturation in reaction-diffusion-advection equations governed by p-Laplacian operator

TL;DR

The paper analyzes traveling wave solutions for the mono-stable reaction-diffusion-advection equation governed by the -Laplacian: on , , under continuity and positivity assumptions. It proves existence and non-existence of traveling waves and classifies them into front-type, sharp-type, and those exhibiting finite propagation or finite saturation; the minimal speed governs existence. A central methodological contribution is the reduction to a first-order singular problem for with , yielding a precise threshold and sharp integral criteria, e.g., and asymptotics of . The results extend and refine prior work (notably for ) and provide a robust framework for understanding finite propagation and saturation in degenerate/advection-dominated diffusion, with explicit connections to the behavior near equilibria and through a singular first-order formulation.

Abstract

The paper concerns front propagation for the following mono-stable reaction-diffusion-advection equation \[f(u)u_x + g(u)u_τ= [d(u)|u_x|^{p-2} u_x]_x+ ρ(u), \quad (x,τ)\in \R\times [0,+\infty).\] Besides existence and non-existence results for traveling wave solutions, the main focus is their classification: we provide criteria to establish if they attain one or both the equilibria at a finite time and in this case, if they are continuable as -solutions or if they are sharp solutions.
Paper Structure (7 sections, 20 theorems, 122 equations)

This paper contains 7 sections, 20 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. T.w.s. and their properties
  3. An equivalent first order problem
  4. Existence and uniqueness result
  5. Local analysis at the equilibrium 0
  6. Local analysis at the equilibrium 1
  7. Appendix

Key Result

Proposition 1

Let $u$ be a solution to equation in eq:ode_lapl, for some $c\in \mathbb R$, satisfying c:as. Assume that $d$ is bounded in $(0,1)$. Then, if $\alpha=-\infty$ we have $\lim_{t\to -\infty}d(u(t))|u'(t)|^{p-2}u'(t)=0$. Similarly, if $\beta=+\infty$ we have $\lim_{t\to +\infty}d(u(t))|u'(t)|^{p-2}u'(t)

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 5
  • Proposition 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Theorem 11
  • ...and 10 more