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

TL;DR

This work extends bistable traveling-wave analysis to reaction–diffusion–convection equations with discontinuous density-dependent coefficients and a $p$-Laplacian ($p>1$) under weak regularity. By reducing the second-order ODE to equivalent first-order problems and deploying subinterval analysis on $[0,s_*]$ and $[s_*,1]$, the authors define threshold speeds $c_F$ and $c_B$ and establish that a unique traveling-wave speed $c^*$ exists iff $c_F < c_B$. The approach generalizes prior smooth, $p=2$ results (MMM04) to weak regularity settings and utilizes a carefully crafted notion of monotone solutions to accommodate discontinuities in $d$ and $h$. The results provide explicit conditions under which bistable traveling waves exist or fail to exist, highlighting the roles of diffusion, convection, and the sign-changing reaction term near the unstable equilibrium $s_*$. This advances understanding of wave propagation in media with discontinuous properties and degenerate or singular diffusion, with potential applications to biology, chemistry, and combustion phenomena.

Abstract

Continuing our previous study \cite{DJKZ} on the monostable reaction-diffusion-convection equation, we analyze the bistable case under weak regularity assumptions. Our approach applies monostable results on the subintervals where the reaction term $g$ has constant sign, thereby establishing both existence and nonexistence of bistable traveling wave solutions. We extend the results of \cite{MMM04}, obtained for $p=2$ under higher regularity assumptions ($d \in C^1[0,1]$, $g,h \in C[0,1]$), to the $p$-Laplacian with $p>1$ in our weak regularity setting.

Theorems & Definitions (47)

• Definition 2.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Proposition 3.1
• Lemma 3.1