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Infinitely many saturated travelling waves for a degenerate Fisher-KPP equation not in divergence form

Matthieu Alfaro, Maxime Herda, Andrea Natale

TL;DR

This work analyzes travelling waves for a degenerate Fisher-KPP equation not in divergence form, arising from a distributed-contacts epidemic model. The authors transform the travelling-wave problem into a singular first-order ODE for an auxiliary function $h$ and perform a rigorous phase-plane analysis, proving that for every speed $c\ge 2$ there exists a unique non-saturated wave and infinitely many saturated waves, with distinct tail behaviors. The tails exhibit novel features: right tails may decay with different rates from the non-saturated counterpart, and saturated fronts can reach the unstable state with sharp, logarithmic-front asymptotics. These findings contrast strongly with classical Fisher-KPP or $I^m\Delta I$-type models and have implications for the well-posedness and long-time behavior of the corresponding Cauchy problem.

Abstract

We consider an epidemic model with distributed-contacts. When the contact kernel concentrates, one formally reaches a very degenerate Fisher-KPP equation with a diffusion term that is not in divergence form. We make an exhaustive study of its travelling waves. For every admissible speed, there exist not only a unique non-saturated (smooth) wave but also infinitely many saturated (sharp) ones. Furthermore their tails may differ from what is usually expected. These results are thus in sharp contrast with their counterparts on related models.

Infinitely many saturated travelling waves for a degenerate Fisher-KPP equation not in divergence form

TL;DR

This work analyzes travelling waves for a degenerate Fisher-KPP equation not in divergence form, arising from a distributed-contacts epidemic model. The authors transform the travelling-wave problem into a singular first-order ODE for an auxiliary function and perform a rigorous phase-plane analysis, proving that for every speed there exists a unique non-saturated wave and infinitely many saturated waves, with distinct tail behaviors. The tails exhibit novel features: right tails may decay with different rates from the non-saturated counterpart, and saturated fronts can reach the unstable state with sharp, logarithmic-front asymptotics. These findings contrast strongly with classical Fisher-KPP or -type models and have implications for the well-posedness and long-time behavior of the corresponding Cauchy problem.

Abstract

We consider an epidemic model with distributed-contacts. When the contact kernel concentrates, one formally reaches a very degenerate Fisher-KPP equation with a diffusion term that is not in divergence form. We make an exhaustive study of its travelling waves. For every admissible speed, there exist not only a unique non-saturated (smooth) wave but also infinitely many saturated (sharp) ones. Furthermore their tails may differ from what is usually expected. These results are thus in sharp contrast with their counterparts on related models.

Paper Structure

This paper contains 7 sections, 13 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Travelling wave solutions
  4. Reaching a first order ODE
  5. A Cauchy problem
  6. Boundary behaviors of the $h$'s
  7. Back to the wave profile $u$

Key Result

Theorem 2.4

There is no travelling wave solution to edp with $c<2$. Next, let $c\geq 2$ be fixed. Then there exist a non-saturated and infinitely many saturated waves normalized by $u(0)=\frac{1}{2}$. More precisely, defining there are constants $C_i>0$ such that for any $0<\varepsilon\ll 1$ the following hold.

Figures (2)

  • Figure 1: Some travelling waves of Theorem \ref{['th:tw']} (for a given speed $c=2.1$). The profile (1), (2) and (3) are saturated waves. Specifically, (1) and (2) are profiles of type (c) (convex), whereas (3) is a profile of type (a) (convex-concave-convex). Profile (4) corresponds to the unique non-saturated wave.
  • Figure 2: The $(r,h)$-phase plane in linear (left) and log-scale (right) for $c=2.1$. The black, blue and red lines, labelled $(1)$, $(2)$ and $(3)$, correspond to three different saturated wave solutions and are computed numerically. The green line, labelled (4), corresponds to the unique non-saturated wave. In particular, with respect to Lemma \ref{['lem:other-large']}, $(1)$ and $(2)$ are two solutions of type $(c)$, whereas $(3)$ is a solution of type $(a)$. With respect to Lemma \ref{['lem:behavior-zero']}, $(1)$ is a solution of type $(c)$, whereas $(2)$ and $(3)$ are solutions of type $(a)$. These waves are represented in physical space in Figure \ref{['fig:waves']}.

Theorems & Definitions (30)

  • Definition 2.1: Weak solution
  • Definition 2.2: Travelling wave profile
  • Definition 2.3: Travelling wave solution to \ref{['edp']}
  • Theorem 2.4: Travelling waves
  • Lemma 3.1: A priori facts
  • proof
  • Remark 3.2
  • Lemma 3.3: Equivalent formulations
  • proof
  • Lemma 3.4: Small speeds
  • ...and 20 more