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Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients

Pavel Drábek, Soyeun Jung, Eunkyung Ko, Michaela Zahradníková

TL;DR

This work analyzes traveling waves for a monostable reaction-diffusion-convection equation with density-dependent diffusion of $p$-Laplacian type and possible discontinuities in diffusion and convection coefficients. By introducing a non-smooth traveling wave profile and transforming the second-order profile equation into a non-Lipschitz first-order boundary value problem, the authors establish existence and nonexistence results for waves with speeds above a convection-modified threshold $c^*$. The key contributions include a robust weak-solution framework that handles jumps in $d$ and $h$, a backward-ODE positivity argument enabling uniqueness and comparison, and detailed asymptotics near the equilibria $0$ and $1$ under power-type diffusion and reaction terms. The findings reveal how convection shifts the minimal wave speed and provide precise conditions for wave existence, monotonicity, and asymptotic behavior, significantly extending traveling wave theory to highly irregular density-dependent settings.

Abstract

This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. We also provide brief asymptotic analysis of the profiles, for which we need to assume power-type behavior of the diffusion and reaction terms.

Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients

TL;DR

This work analyzes traveling waves for a monostable reaction-diffusion-convection equation with density-dependent diffusion of -Laplacian type and possible discontinuities in diffusion and convection coefficients. By introducing a non-smooth traveling wave profile and transforming the second-order profile equation into a non-Lipschitz first-order boundary value problem, the authors establish existence and nonexistence results for waves with speeds above a convection-modified threshold . The key contributions include a robust weak-solution framework that handles jumps in and , a backward-ODE positivity argument enabling uniqueness and comparison, and detailed asymptotics near the equilibria and under power-type diffusion and reaction terms. The findings reveal how convection shifts the minimal wave speed and provide precise conditions for wave existence, monotonicity, and asymptotic behavior, significantly extending traveling wave theory to highly irregular density-dependent settings.

Abstract

This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with -Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. We also provide brief asymptotic analysis of the profiles, for which we need to assume power-type behavior of the diffusion and reaction terms.
Paper Structure (11 sections, 18 theorems, 143 equations, 2 figures)

This paper contains 11 sections, 18 theorems, 143 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Definition of solution
  4. Existence and nonexistence of traveling waves
  5. Equivalent first order ode
  6. Monotonicity of a traveling wave profile
  7. Transformation into the first order boundary value problem
  8. Existence, uniqueness, and nonexistence of the first order ode
  9. Asymptotic analysis of the traveling wave
  10. Asymptotics near $1$
  11. Asymptotics near $0$

Key Result

Theorem 2.1

Assume that $g$, $d$ and $h$ satisfy (H1)--(H3), and let where $\frac{1}{p}+\frac{1}{p'}=1$. Then, in the case $h(0)>H(1)$, there exists a number such that the problem eqU, bc has a unique traveling wave $U=U(z)$, $z \in {\mathbb R}$, if and only if $c\geq c^*$. In the case $h(0) \leq H(1)$, the same result holds for Moreover, recalling the notation from MKN, $U$ has the following properties:

Figures (2)

  • Figure 1: Sign conditions characterizing (a) Fisher-KPP-type (monostable); (b) combustion-type; (c) Nagumo-type (bistable) reaction terms
  • Figure 2: The region of the threshold speed $c^*$, as discussed in Theorem \ref{['exthm']}, is shown as a blue line. In the case that $H(1) < h(0)+(p')^{\frac{1}{p'}}p^{\frac{1}{p}}\nu^{\frac{1}{p'}}$ (top figure), the problem \ref{['ode']} has no positive solution for any $c$ between $H(1)$ and $h(0)+(p')^{\frac{1}{p'}}p^{\frac{1}{p}}\nu^{\frac{1}{p'}}$ (red line), so that the region of $c^*$ described in Theorem \ref{['exthm']} is reduced to \ref{['range of c^*']}. Otherwise, the nonexistence result does not effect the region of $c^*$ (bottom figure).

Theorems & Definitions (46)

  • Definition 2.1: cf. DZ22
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.1: Existence
  • Theorem 2.2: Nonexistence
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 36 more