# American Institute of Mathematical Sciences

December  2021, 41(12): 6023-6046. doi: 10.3934/dcds.2021105

## Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity

 1 Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Reggio Emilia, 42122, Italy 2 Department of Mathematics and Computer Science, University of Ferrara, Ferrara, 44121, Italy

* Corresponding author: Andrea Corli

Received  November 2020 Revised  April 2021 Published  December 2021 Early access  June 2021

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or even more than once; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviors at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviors are new and unusual.

Citation: Diego Berti, Andrea Corli, Luisa Malaguti. Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 6023-6046. doi: 10.3934/dcds.2021105
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##### References:
Typical plots of the functions $g$ and $D$.
The thresholds $c^*_{p,r}$, $c^*_{n,l}$ used in (2.7) and $c^*_{n,r}$, $c^*_{p,l}$, used below in (2.15).
Some possible wavefronts joining $1$ with $0$ in case $\rm (D_{pn})$: a classical wavefront $\varphi^1$ (for $c>c_{pn}^*$ and either $\dot D(\alpha)<0$ or $c>h(\alpha)$); a wavefront $\varphi^2$ which is sharp at $1$, with finite right derivative at $\xi_1^2$ and $( \varphi^2)'(\xi_\alpha^2) = -\infty$ (for $c = c_{n,l}^*>h(1)$, $D(1) = 0$, $\dot D(1)>0$, $\dot D(\alpha) = 0$ and $c\le h(\alpha)$); a wavefront $\varphi^3$ which is sharp at $0$ with $( \varphi^3)'(\xi_0^3) = -\infty$ (for $c = c_{p,r}^*> h(0)$, $D(0) = 0 = \dot D(0)$).
$D$ satisfies $(\rm D_{pn})$: Left: the plots of $D$, $g$ in $[0,\alpha]$ (solid lines) and in $[\alpha,1]$ (dashed lines); right, a corresponding profile.
Construction of the profile $\varphi$ in the case $\xi_0,\xi_1\in \mathbb R$.
$D$ satisfies $(\rm D_{np})$. Left: the plots of $D$, $g$ in $[0,\beta]$ (dashed lines) and $[\beta,1]$ (solid lines); right, a corresponding profile.
Some possible wavefronts joining $1$ with $0$ in case $\rm (D_{np})$; Profiles are labelled according to the cases (1)–(4) of Remark 6; $\varphi^5$ occurs in both cases (3) and (4). For simplicity we only represented strictly monotone profiles.
Typical plots of the functions $D$.
The pasting of the profiles.
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