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

Diego Berti111Department of Sciences and Methods for Engineering, University
of Modena and Reggio Emilia, Italy Andrea Corli222Department of Mathematics
and Computer Science, University of Ferrara, Italy Luisa
Malaguti11footnotemark: 1

###### Abstract

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.

AMS Subject Classification: 35K65; 35C07, 34B40, 35K57

Keywords: Sign-changing diffusivity, traveling-wave solutions, diffusion-
convection reaction equations, sharp profiles.

##  1 Introduction

This paper deals with traveling-wave solutions to degenerate parabolic
equations of forward-backward type. More precisely, we consider the equation

$\rho_{t}+f(\rho)_{x}=\left(D(\rho)\rho_{x}\right)_{x}+g(\rho),\qquad t\geq
0,\,x\in\mathbb{R}.$ (1.1)

We denote with $\rho=\rho(t,x)$ the unknown function; also in view of
applications we understand $\rho$ as a (normalized) density or a concentration
and then assume that $\rho$ is valued in the interval $[0,1]$. On the
convective term $f$ we only assume

  * (f)

$f\in C^{1}[0,1]$, $f(0)=0$.

The condition $f(0)=0$ just fixes a flux representative, since convection is
only defined up to an additive constant. In the following, for brevity, we
denote the derivative of $f$ as $h(\rho)=\dot{f}(\rho)$. We assume that $g$
satisfies

  * (g)

$g\in C^{0}[0,1]$, $g>0$ in $(0,1)$, $g(0)=g(1)=0$.

Condition (g) is natural in this framework and then $g$ is said of monostable
type. The diffusivity $D$ is required to satisfy one of the following
assumptions, for some $\alpha,\beta\in(0,1)$:

  * $(\rm D_{pn})$

$D\in C^{1}[0,1]$, $D>0$ in $(0,\alpha)$ and $D<0$ in $(\alpha,1)$;

  * $(\rm D_{np})$

$D\in C^{1}[0,1]$, $D<0$ in $(0,\beta)$ and $D>0$ in $(\beta,1)$.

Each condition is labelled following the sign of $D$: “pn” in $\rm(D_{pn})$
means that $D$ is first positive and then negative, and the other so on.
Above, $\alpha$ denotes a zero of $D$ such that $\dot{D}(\alpha)\leq 0$ while
$\beta$ denotes a zero of $D$ such that $\dot{D}(\beta)\geq 0$.

$0$$\rho$$f$$1$$\rho$$g$$1$$\rho$$D$$\alpha$$1$$(\rm
D_{pn})$$\rho$$D$$\beta$$1$$(\rm D_{np})$ Figure 1: Typical plots of the
functions $f$, $g$ and $D$.

The above assumptions on $D$ are the main issue of this paper, and make (1.1)
a forward parabolic equation where $D>0$ and a backward parabolic equation
where $D<0$. The above conditions leave also open the possibility that $D$
vanishes at $0$ or $1$.

There are several motivations to study forward-backward parabolic equations as
(1.1): for a short list of different modeling we quote [18, 21, 27] for
biology, [13] for geophysics, [19] for thermodynamics. However, our main
source of inspiration has been the recent modeling of collective movements,
namely of vehicular flows and crowds dynamics. About this topic, we refer to
[15, 16, 28] for general information, to [4, 5, 7] for the modeling using
degenerate parabolic equations, to [9] for the study of star graphs, while for
sign-changing diffusivities we refer to [11, 12, 26] and the references in the
two former papers.

Our interest in this paper is about traveling-wave solutions (TWs for short)
to (1.1); they are solutions to (1.1) of the form $\rho(t,x)=\varphi(x-ct)$.
Here, the function $\varphi=\varphi(\xi)$ is the profile of the TW and the
real number $c$ is its speed. The equation for the profiles is then

$\left(D(\varphi)\varphi^{\prime}\right)^{\prime}+\left(c-h(\varphi)\right)\varphi^{\prime}+g(\varphi)=0,$
(1.2)

where ′ denotes the derivative with respect to $\xi$. More precisely we focus
on wavefronts, i.e., globally-defined, nonconstant and monotone profiles. To
fix ideas we deal with non-increasing profiles and this leads us, because of
(g), to impose the conditions

$\varphi(-\infty)=1,\qquad\varphi(+\infty)=0.$ (1.3)

The study of non-decreasing profiles, in which the conditions in (1.3) are
switched, is not explicitly treated in this paper. Nevertheless, all the
results can be rephrased in that case, once that (roughly speaking) the
direction of speeds is reversed. Clearly, even under (1.3) the solutions to
(1.2) are at most unique up to horizontal shifts. An interesting issue is
whether the equilibria $0$ and $1$ can be reached by a wavefront $\varphi$ at
a finite value $\xi_{0}$. This possibly occurs if $D$ vanishes at those
points, and in such cases $\varphi$ is necessarily constant on either
$(\xi_{0},\infty)$ or $(-\infty,\xi_{0})$, with values $0$ and $1$,
respectively. The profile is called sharp if it is not differentiable at
$\xi_{0}$. In this case $\varphi$ is not a classical solution of (1.2) (see
Definition 2.1). We refer to [17] for more information on traveling waves. The
case where $D$ changes sign has been considered by several authors but only
when $f=0$. About this case, we quote [2, 21] for $D$ satisfying $(\rm
D_{np})$ and $(\rm D_{pn})$, respectively, and monostable $g$; [22] for the
case $(\rm D_{pn})$ and bistable $g$ (i.e., $g$ changes sign once).

The main result of the current paper is that there still exist wavefronts
joining $1$ with $0$, which travel across regions where $D$ changes sign. In
our approach the profiles are constructed by suitably pasting two semi-
wavefronts and possibly a _traveling wave solution_ in a bounded interval (see
the next section for a definition), as in [11]. As a consequence of this
procedure, one realizes that the assumptions on $D$ can be somewhat relaxed,
as in [2]. Indeed, for instance in case $(\rm D_{pn})$, it is sufficient to
require $D\in C^{0}[0,1]$ and $D\in C^{1}[0,\alpha]\cap C^{1}[\alpha,1]$: the
derivatives at $\alpha^{-}$ and $\alpha^{+}$ may be different.

We prove that the wavefronts constructed in this way are unique (up to shifts)
and provide results about the strict monotony of profiles; in particular, we
characterize when they are sharp. At last, we give rather precise bounds on
the critical thresholds by exploiting the estimates obtained in [6]; there, in
turn, we used some related recent results proved in [25]. The main tool
underlying our approach is a well-known reduction (in regions where $D$ has
constant sign) of Equation (1.2) to singular first-order systems [24] and its
study by comparison-type techniques. More precisely, if we denote
$z(\varphi):=D(\varphi)\varphi^{\prime}$, where $\varphi^{\prime}$ is computed
at $\varphi^{-1}(\varphi)$ (notice that the inverse function of $\varphi$
exists because of the monotony of $\varphi$), then we are reduced to consider
the problems, for instance in the case $(\rm D_{pn})$,

$\begin{cases}\dot{z}(\varphi)=h(\varphi)-c-\frac{\left(Dg\right)(\varphi)}{z(\varphi)},\
&\varphi\in(0,\alpha),\\\ z(\varphi)<0,\ &\varphi\in(0,\alpha),\\\
z(0)=0,\end{cases}\hbox{ or
}\begin{cases}\dot{z}(\varphi)=h(\varphi)-c-\frac{\left(Dg\right)(\varphi)}{z(\varphi)},\
&\varphi\in(\alpha,1),\\\ z(\varphi)>0,\ &\varphi\in(\alpha,1),\\\
z(1)=0,\end{cases}$ (1.4)

and similar problems in the other cases. We refer to [6] for a detailed study
of $\eqref{e:zIntro}_{1}$.

Problems (1.4) seems to suggest that the roles played by $D$ and $g$ are
interchangeable; this is not true, in general. Consider for instance the
bistable (or Allen-Cahn) equation, where $D>0$ in $(0,1)$ but $g$ satisfies
$g(0)=g(\alpha)=g(1)=0$, $g<0$ in $(0,\alpha)$ and $g>0$ in $(\alpha,1)$. In
this case it is known [1], at least if $f=0$, that equation (1.1) admits a
unique admissible speed $c$ corresponding to a wavefront from $1$ to $0$. In
the current case $(\rm D_{np})$, on the contrary, where the role of $D$ and
$g$ is commuted with respect to the bistable case, we shall find a whole half
line of admissible speeds. The same result holds for the case $(\rm D_{pn})$.

Here follows an outline of the paper. Section 2 contains the statements of our
main results. In Section 3, we first briefly recall some preliminary results
in order to keep the paper self-contained; then we prove the main result under
$(\rm D_{pn})$. Section 4 deals with the case $(\rm D_{np})$. In Section 5 we
give some explicit examples; they aim at showing the role played by the
convection term $f$, the qualitative difference of the thresholds in the cases
$(\rm D_{pn})$ and $(\rm D_{np})$, the occurrence of non-regular profiles. At
last, Section 6 extends the previous results to the case when $D$ changes sign
twice.

##  2 Main results

As we mentioned in the Introduction, traveling-waves can fail to be of class
$C^{1}$ in the whole of their domain. The following definition makes precise
what we mean by a TWs, see [17].

######  Definition 2.1.

Assume $f,D,g\in C^{0}[0,1]$ and let $I\subset\mathbb{R}$ be an open interval.
Let $\varphi\in C^{0}(I)$ be a function valued in $[0,1]$, which is
differentiable a.e. and such that $D(\varphi)\varphi^{\,\prime}\in L_{\rm
loc}^{1}(I)$; at last, let $c$ be a real constant.

Then the function $\rho(x,t):=\varphi(x-ct)$, for $(x,t)$ with $x-ct\in I$, is
a traveling-wave solution (briefly, a TW) to equation (1.1) with wave speed
$c$ and wave profile $\varphi$ if we have

$\int_{I}\left(D\left(\varphi(\xi)\right)\varphi^{\prime}(\xi)-f\left(\varphi(\xi)\right)+c\varphi(\xi)\right)\psi^{\prime}(\xi)-g\left(\varphi(\xi)\right)\psi(\xi)\,d\xi=0,$
(2.1)

for every $\psi\in C_{0}^{\infty}(I)$.

We now give some definitions about TWs. A TW is: global if $I=\mathbb{R}$;
strict if $I\neq\mathbb{R}$ and $\varphi$ cannot be extended to $\mathbb{R}$;
classical if $\varphi$ is differentiable, $D(\varphi)\varphi^{\prime}$ is
absolutely continuous and (1.2) holds a.e.; sharp at $\ell$ if there exists
$\xi_{\ell}\in I$, with $\varphi(\xi_{\ell})=\ell$, such that $\varphi$ is
classical in $I\setminus\\{\xi_{\ell}\\}$ and not differentiable at
$\xi_{\ell}$. Analogously, a TW is classical at $\ell$ if it is classical in a
neighborhood of $\xi_{\ell}$.

Moreover, a TW is: a wavefront if it is global, with a monotonic, non-constant
profile $\varphi$ which satisfies either (1.3) or the converse condition; a
semi-wavefront to $1$ (or to $0$) if $I=(a,\infty)$ for $a\in\mathbb{R}$, the
profile $\varphi$ is monotonic, non-constant and $\varphi(\xi)\to 1$
(respectively, $\varphi(\xi)\to 0$) as $\xi\to\infty$; a semi-wavefront from
$1$ (or from $0$) if $I=(-\infty,b)$ for $b\in\mathbb{R}$, the profile
$\varphi$ is monotonic, non-constant and $\varphi(\xi)\to 1$ (respectively,
$\varphi(\xi)\to 0$) as $\xi\to-\infty$. About semi-wavefronts, we say that
$\varphi$ connects $\varphi(a^{+})$ ($1$ or $0$) with $1$ or $0$ (resp., with
$\varphi(b^{-})$).

The problem of the loss of regularity of $\varphi$ depends on whether the
parabolic equation degenerates or not; more precisely, by arguing on the very
equation (1.2), it is easy to see that if $f,D$ are of class $C^{1}$ and $g$
of class $C^{0}$, then $\varphi$ is classical in every interval
$I_{\pm}\subseteq I$ where $\pm D\left(\varphi(\xi)\right)>0$ for $\xi\in
I_{\pm}$; moreover, $\varphi\in C^{2}(I_{\pm})$ (see e.g. [17, Lemma 2.20]).

We recall that for a function $q:[0,1]\to\mathbb{R}$, the notation
$D_{+}q(\rho_{0})$ and $D_{-}q(\rho_{0})$, with $\rho_{0}\in[0,1]$, stands for
the right, resp., left lower Dini-derivative of $q$ at $\rho_{0}$;
analogously, $D^{\pm}(q)$ represent the right and left upper Dini-derivatives
of $q$. More explicitly,

$D_{\pm}q(\rho_{0}):=\liminf_{\rho\to\rho_{0}^{\pm}}\frac{q(\rho)-q(\rho_{0})}{\rho-\rho_{0}},\quad
D^{\pm}q(\rho_{0}):=\limsup_{\rho\to\rho_{0}^{\pm}}\frac{q(\rho)-q(\rho_{0})}{\rho-\rho_{0}}.$

In addition to the main assumptions (f), ${\rm(D_{pn})}$-${\rm(D_{np})}$ and
(g) stated in the Introduction, we also need for some results two further
regularity conditions on the product of $Dg$ at the boundary of the interval
$[0,1]$, which are stated using the above notation:

$D^{+}Dg(0)<+\infty\ \mbox{ and }\ D^{-}Dg(1)<+\infty.$ (2.2)

In general, $\eqref{Dg}_{1}$ implies that problem $\eqref{e:zIntro}_{1}$ has a
super-solution for sufficiently large $c$, see [23] in the case $f=0$, and
hence it is solvable for those $c$. Condition $\eqref{Dg}_{1}$ is always
satisfied if $D(0)=0$, while, if $D(0)\neq 0$, it requires that $g$ is
sublinear close to $0$; condition $\eqref{Dg}_{2}$ is commented analogously.
At last, we denote the difference quotient of a function $F=F(\varphi)$ with
respect to a point $\varphi_{0}$ as

$\delta(F,\varphi_{0})(\varphi):=\frac{F(\varphi)-F(\varphi_{0})}{\varphi-\varphi_{0}}.$
(2.3)

We first focus on the case $(\rm D_{pn})$. The construction of a wavefront to
(1.1) takes place by properly joining two semi-wavefronts, each of them with
an intrinsic threshold. The existence of such semi-wavefronts in the region
$(0,\alpha)$ has been done in [6, Theorem 2.2]; the main content of that
result is the following. Under (f), $(\rm D_{pn})$, (g) and $\eqref{Dg}_{1}$
there exists a real number, denoted by $c^{*}_{p,r}$, satisfying

$\max\left\\{\sup_{(0,\alpha]}\delta(f,0),h(0)+2\sqrt{D_{+}Dg(0)}\right\\}\leq\\\
c^{*}_{p,r}\leq\sup_{(0,\alpha]}\delta(f,0)+2\sqrt{\sup_{(0,\alpha]}\delta(Dg,0)},$
(2.4)

such that Equation (1.1) admits strict semi-wavefronts to $0$, connecting
$\alpha$ to $0$, with speed $c$ if and only if $c\geq c^{*}_{p,r}$. It is
worth noting that the left- and the right-hand side of (2.4) describe a non-
empty interval (possibly degenerating to a single point) of real numbers, as a
direct inspection trivially shows. Moreover, in [25, Theorem 3.1] the authors
proved that in case $Dg$ differentiable at $\varphi=0$ (e.g. in case $D(0)=0$)
the second addend of the right-hand side of (2.4) can be further enhanced by
replacing $\delta(Dg,0)(\varphi)$ with its mean value in $(0,\varphi)$, that
is

$c^{*}_{p,r}\leq\sup_{(0,\alpha]}\delta(f,0)+2\sqrt{\sup_{\varphi\in(0,\alpha]}\frac{1}{\varphi}\int_{0}^{\varphi}\frac{D(s)g(s)}{s}\,ds}.$
(2.5)

We warn the reader that those semi-wavefronts are proved to be intrinsically
nonunique, i.e., nonuniqueness holds even understanding profiles differing by
a horizontal shift as a same profile (see Proposition 3.1). The subscripts
$p,r$ in $c^{*}_{p,r}$ mean that we are considering a case where $D$ is
positive in an interval $I_{+}$ and vanishes in the right extremum of $I_{+}$,
while $g$ vanishes at the opposite extremum. Similarly, we shall use the
notation $c^{*}_{n,l}$, $c^{*}_{p,l}$ and $c^{*}_{n,r}$.

An analogous result for semi-wavefronts from $1$, connecting $1$ to $\alpha$,
is first deduced in this paper. Assume (f), $(\rm D_{pn})$, (g) and
$\eqref{Dg}_{2}$. There exists $c^{*}_{n,l}\in\mathbb{R}$ satisfying

$\max\left\\{\sup_{[\alpha,1)}\delta(f,1),h(1)+2\sqrt{D_{-}Dg(1)}\right\\}\leq
c^{*}_{n,l}\leq\sup_{[\alpha,1)}\delta(f,1)+2\sqrt{\sup_{[\alpha,1)}\delta(Dg,1)},$
(2.6)

such that Equation (1.1) admits strict semi-wavefronts from $1$, connecting
$1$ to $\alpha$, with speed $c$, if and only if $c\geq c^{*}_{n,l}$; see
Proposition 3.2. Also in this case profiles are intrinsically nonunique.

We now present our main results on wavefronts. Define, see Figure 2,

$c^{*}_{pn}:=\max\left\\{c^{*}_{p,r},c^{*}_{n,l}\right\\}.$ (2.7)

$\rho$$c_{p,r}^{*}$$D>0$$c_{n,l}^{*}$$D<0$$\rho$$0$$1$$\alpha$$\rho$$c_{n,r}^{*}$$D<0$$c_{p,l}^{*}$$D>0$$\rho$$0$$1$$\beta$
Figure 2: The thresholds $c^{*}_{p,r}$, $c^{*}_{n,l}$ used in (2.7) and
$c^{*}_{n,r}$, $c^{*}_{p,l}$, used below in (2.16).

For what concerns an estimate on $c^{*}_{pn}$, see Remark 3.3. We introduce
the quantity $s_{\pm}(\gamma,c)$, defined formally by

$s_{\pm}(\gamma,c):=\frac{1}{2}\left[h(\gamma)-c\pm\sqrt{\left(h(\gamma)-c\right)^{2}-4\dot{D}(\gamma)g(\gamma)}\right].$
(2.8)

In the next result, we make use of (2.8) with $\gamma=\alpha$. In this
particular case, $s_{-}(\alpha,c)$ is well-defined since
$\dot{D}(\alpha)g(\alpha)\leq 0$.

######  Theorem 2.1.

Assume (f), (g), $\rm(D_{pn})$ and (2.2). Equation (1.1) admits a (unique up
to space shifts) wavefront, with speed $c$ and profile $\varphi$ satisfying
(1.3), if and only if $c\geq c^{*}_{pn}$.

For such $c$, we have $\varphi^{\prime}{}(\xi)<0$ when
$\varphi(\xi)\in(0,1)\setminus\\{\alpha\\}$; there exists a unique
$\xi_{\alpha}\in\mathbb{R}$ such that $\varphi(\xi_{\alpha})=\alpha$ and

$\varphi^{\prime}{}(\xi_{\alpha}^{\pm})=\begin{cases}\frac{g(\alpha)}{s_{-}(\alpha,c)}<0\
&\mbox{ if }\ \dot{D}(\alpha)<0\ \mbox{ or }\ c>h(\alpha),\\\ -\infty\ &\mbox{
if }\ \dot{D}(\alpha)=0\mbox{ and }c\leq h(\alpha).\end{cases}$ (2.9)

At last, it holds that:

  1. (i)

if $D(0)D(1)<0$, then $\varphi$ is strictly decreasing and hence classical in
$\mathbb{R}\setminus\left\\{\xi_{\alpha}\right\\}$;

  2. (ii)

if $D(0)D(1)=0$ and $c>c^{*}_{pn}$, then $\varphi$ is classical in
$\mathbb{R}\setminus\\{\xi_{\alpha}\\}$; $\varphi$ is strictly decreasing if

$c>\max\left\\{h(0)+D^{+}g(0),\ h(1)+D^{-}\\{-g\\}(1)\right\\};$

  3. (iii)

if $D(0)=0$ and $c=c_{pn}^{*}=c^{*}_{p,r}>h(0)$, then $\varphi$ is sharp at
$0$ (reached at some $\xi_{0}>\xi_{\alpha}$) and if $D(1)=0$ and
$c=c_{pn}^{*}=c^{*}_{n,l}>h(1)$, then $\varphi$ is sharp at $1$ (reached at
some $\xi_{1}<\xi_{\alpha}$). In these cases we have

$\displaystyle\varphi^{\prime}{}(\xi_{0}^{-})=\left\\{\begin{array}[]{ll}\frac{h(0)-c^{*}_{p,r}}{\dot{D}(0)}<0&\mbox{
if }\dot{D}(0)>0,\\\\[5.69054pt] -\infty&\mbox{ if
}\dot{D}(0)=0,\end{array}\right.\
\varphi^{\prime}{}(\xi_{1}^{+})=\left\\{\begin{array}[]{ll}\frac{h(1)-c^{*}_{n,l}}{\dot{D}(1)}<0&\mbox{
if }\dot{D}(1)>0,\\\\[5.69054pt] -\infty&\mbox{ if
}\dot{D}(1)=0.\end{array}\right.$

We refer to Figure 3 for a pictorial representation of Theorem 2.1.

$\xi$$\varphi$$1$$\alpha$$\xi_{\alpha}^{1}$$\xi_{\alpha}^{3}$$\xi_{\alpha}^{2}$$\xi_{1}^{2}$$\varphi^{1}$$\varphi^{2}$$\varphi^{2}$$\varphi^{3}$$\xi_{0}^{3}$$\varphi^{3}$
Figure 3: 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
$(\varphi^{2})^{\prime}(\xi_{\alpha}^{2})=-\infty$ (for $c=c_{n,l}^{*}>h(1)$,
$D(1)=0$, $\dot{D}(1)>0$, $\dot{D}(\alpha)=0$ and $c\leq h(\alpha)$); a
wavefront $\varphi^{3}$ which is sharp at $0$ with
$(\varphi^{3})^{\prime}(\xi_{0}^{3})=-\infty$ (for $c=c_{p,r}^{*}>h(0)$,
$D(0)=0=\dot{D}(0)$).

Theorem 2.1 extends [21, Theorem 1] to the case of a non-zero convection term
$f$; if $f=0$ the estimates on $c_{pn}^{*}$ deduced from (2.4) and (2.6)
coincide with those in [21]. Estimates (2.4) and (2.6) imply
$c_{pn}^{*}\geq\max\left\\{h(0),h(1)\right\\}$. However, if $h(\varphi)\geq
h(0)$ for every $\varphi$ in a right neighborhood of $0$, then
$c_{p,r}^{*}>h(0)$, see [6, Remark 6.4], and so $c_{pn}^{*}>h(0)$. An
analogous remark holds for the point $1$. This means that the lower estimate
on $c_{pn}^{*}$ is not sharp, in general. Therefore we also extend the
corresponding result of [21], since they proved that the threshold $c^{*}$,
which plays the role of $c_{pn}^{*}$ here, satisfies the stricter estimate
$c^{*}>0$.

On the other hand, a new result that could not occur in [21] is that we can
have $\varphi^{\prime}(\xi_{\alpha})=-\infty$, see
$\eqref{e:phip(xialpha)}_{2}$, while in the case $f=0$ profiles are always
$C^{1}$ at $\xi_{\alpha}$. Explicit examples of fronts passing from a
positive- to a negative-diffusivity region such that
$\eqref{e:phip(xialpha)}_{2}$ occurs, are given in Example 5.1. From a formal
point of view, if $f=0$ then this could occur if $c\leq 0$, see
$\eqref{e:phip(xialpha)}_{2}$, while in that case $c>0$ holds; more
rigorously, see [21, (29)]. We emphasize that this phenomenon does not depend
on the change of sign of $D$ at $\alpha$, but on the occurrence of the two
conditions in $\eqref{e:phip(xialpha)}_{2}$, see [6, Remark 9.3]. Moreover,
just to get an insight on the problem, assume $\dot{D}(\alpha)=0$; in order
that $\eqref{e:phip(xialpha)}_{2}$ takes place it is necessary that

$\max\bigl{\\{}\sup_{(0,\alpha]}\delta(f,0),\sup_{[\alpha,1)}\delta(f,1)\bigr{\\}}\leq
h(\alpha),$ (2.11)

so that we have room for $c$. Condition (2.11) has a simple geometric
interpretation: the slope of the tangent to the graph of $f$ at $\alpha$ must
be larger that the slope of any chord joining $(0,0)$ with any other point of
the graph of $f$, in the interval in $(0,\alpha]$, and analogously for the
interval $[\alpha,1)$. We easily see that $\sup_{(0,\alpha]}\delta(f,0)\leq
h(\alpha)$ fails if $f$ is concave in $[0,\alpha]$, while it holds if $f$ is
convex; the converse result holds for the other condition. This means that
$\eqref{e:phip(xialpha)}_{2}$ may hold only if $f$ changes its convexity.
Moreover, if $f$ changes convexity only once, for instance at $\alpha$, then
$f$ must be convex in $[0,\alpha]$ and concave in $[\alpha,1]$, and not
conversely. In other words, the profile may become vertical if, at least in
some subintervals, the behavior of $f$ strongly contrast that of $D$.

We now comment on (iii). The lack of results for the cases $c^{*}_{p,r}=h(0)$
or $c^{*}_{n,l}=h(1)$ is due to the fact that in these extremal cases the
regularity depends on further properties of $D$ and $g$. Indeed, under the
mild assumptions of Theorem 2.1 the profile $\varphi$ can be either sharp or
classical when reaches the equilibria $0$ or $1$ (for explicit examples at
$0$, we refer to [6, Remark 10.1]; the discussion at $1$ is analogous).

We focus now on $\rm(D_{np})$. Condition $\rm(D_{np})$ is specular to
$\rm(D_{pn})$, in an obvious sense: if $D$ satisfies $\rm(D_{pn})$ then $-D$
satisfies $\rm(D_{np})$. Despite this fact, the results in this case only
partially mimic those of Theorem 2.1. In particular, if we focus on the
interval $(\beta,1)$, where $D>0$, the contrast with [6, Theorem 2.2] emerges
evident. Apart from a trivial horizontal translation, we are under the
hypotheses considered in [10, Theorem 2.7] (compare $\rm(D_{np})$-(g) in the
interval $(\beta,1)$ with the corresponding assumptions of [10, Theorem 2.7]).
In contrast with [6, Theorem 2.2] cited before, [10, Theorem 2.7] affirms
that, for each $c\in\mathbb{R}$, Equation (1.1) admits (unique up to space
shifts) strict semi-wavefronts from $1$, connecting $1$ to $\beta$.
Nevertheless, the system still admits a threshold speed, in the following
sense. There exists $c^{*}_{p,l}\in\mathbb{R}$ such that the (unique up to
space shifts) non-increasing wave profile $\varphi$, defined maximally in
$(-\infty,\xi_{\beta})$, satisfies [10, Theorem 2.6]

$\left(D(\varphi)\varphi^{\prime}{}\right)(\xi_{\beta}^{-})=\left\\{\begin{array}[]{ll}0&\mbox{
if }\ c\geq c^{*}_{p,l},\\\ \ell<0&\mbox{ if }\
c<c^{*}_{p,l}.\end{array}\right.$ (2.12)

By improving the estimates of $c^{*}_{p,l}$ in [10], as for (2.4)-(2.5), it
results that $c^{*}_{p,l}$ must satisfy

$\max\left\\{\sup_{(\beta,1]}\delta(f,\beta),h(\beta)+2\sqrt{\dot{D}(\beta)g(\beta)}\right\\}\leq
c^{*}_{p,l}\leq\\\
\sup_{(\beta,1]}\delta(f,\beta)+2\sqrt{\sup_{\varphi\in(\beta,1]}\frac{1}{\varphi-\beta}\int_{\beta}^{\varphi}\frac{D(s)g(s)}{s-\beta}\,ds}.$
(2.13)

See Proposition 4.1. Similarly, Equation (1.1) admits a (unique up to shifts)
strict semi-wavefront to $0$, connecting $\beta$ to $0$, for every
$c\in\mathbb{R}$. Moreover, there exists $c^{*}_{n,r}\in\mathbb{R}$ satisfying

$\max\left\\{\sup_{[0,\beta)}\delta(f,\beta),h(\beta)+2\sqrt{\dot{D}(\beta)g(\beta)}\right\\}\leq
c^{*}_{n,r}\leq\\\
\sup_{[0,\beta)}\delta(f,\beta)+2\sqrt{\sup_{\varphi\in[0,\beta)}\frac{1}{\beta-\varphi}\int_{\varphi}^{\beta}\frac{D(s)g(s)}{s-\beta}\,ds},$
(2.14)

such that the (unique up to shifts) non-increasing profile $\varphi$, defined
maximally in $(\xi_{\beta},+\infty)$, satisfies

$\left(D(\varphi)\varphi^{\prime}{}\right)(\xi_{\beta}^{+})=\left\\{\begin{array}[]{ll}0&\mbox{
if }\ c\geq c^{*}_{n,r},\\\ s>0&\mbox{ if }\ c<c^{*}_{n,r},\end{array}\right.$
(2.15)

see Proposition 4.2. We set (see Figure 2)

$c^{*}_{np}:=\max\left\\{c^{*}_{n,r},c^{*}_{p,l}\right\\}.$ (2.16)

In the next theorem, $s_{\pm}(\beta,c)$ is given by (2.8). Note that, in spite
of $\dot{D}(\beta)\geq 0$, $s_{\pm}(\beta,c)$ is well-defined since in virtue
of (2.16) and (2.13) (or (2.14)), we clearly have
$(h(\beta)-c)^{2}\geq(h(\beta)-c_{n,p}^{*})^{2}\geq 4\dot{D}(\beta)g(\beta)$.

######  Theorem 2.2.

Assume (f), (g), $\rm(D_{np})$. Then, Equation (1.1) admits a (unique up to
space shifts) wavefront, with speed $c$ and profile $\varphi$ satisfying
(1.3), if and only if $c\geq c^{*}_{np}$.

For such $c$, we have $\varphi^{\prime}{}<0$ if
$\varphi\in(0,1)\setminus\left\\{\beta\right\\}$. There exists a unique
$\xi_{\beta}\in\mathbb{R}$ such that $\varphi(\xi_{\beta})=\beta$ and

$\varphi^{\prime}{}({\xi_{\beta}}^{-})=\begin{cases}\frac{g\left(\beta\right)}{s_{-}(\beta,c)}\
&\mbox{ if }\ c>c^{*}_{p,l},\\\\[5.69054pt]
\frac{g\left(\beta\right)}{s_{+}(\beta,c^{*}_{p,l})}\ &\mbox{ if }\
c=c^{*}_{p,l}\ \mbox{ and }\ \dot{D}(\beta)>0,\\\ -\infty\ &\mbox{ if }\
c=c_{p,l}^{*}\ \mbox{ and }\ \dot{D}(\beta)=0,\end{cases}\\\
\varphi^{\prime}{}({\xi_{\beta}}^{+})=\begin{cases}\frac{g\left(\beta\right)}{s_{-}(\beta,c)}\
&\mbox{ if }\ c>c^{*}_{n,r},\\\\[5.69054pt]
\frac{g\left(\beta\right)}{s_{+}(\beta,c^{*}_{n,r})}\ &\mbox{ if }\
c=c^{*}_{n,r}\ \mbox{ and }\ \dot{D}(\beta)>0,\\\ -\infty\ &\mbox{ if }\
c=c_{n,r}^{*}\ \mbox{ and }\ \dot{D}(\beta)=0.\end{cases}$ (2.17)

At last, the following holds true:

  1. (i)

if $D(0)D(1)<0$ then $\varphi$ is classical in
$\mathbb{R}\setminus\left\\{\xi_{\beta}\right\\}$;

  2. (ii)

if $D(0)=0$, and either $c>h(0)$ or $c=h(0)$ and $\dot{D}(0)<0$, then
$\varphi$ is classical at $0$; if $D(1)=0$, and either $c>h(1)$ or $c=h(1)$
and $\dot{D}(1)<0$ then $\varphi$ is classical at $1$;

  3. (iii)

if $D(0)=0$ and $c<h(0)$, then $\varphi$ is sharp at $0$ (reached at some
$\xi_{0}>\xi_{\beta}$); if $D(1)=0$ and $c<h(1)$, then $\varphi$ is sharp at
$1$ (reached at some $\xi_{1}<\xi_{\beta}$). In these cases we have

$\displaystyle\varphi^{\prime}{}(\xi_{0}^{-})=\left\\{\begin{array}[]{ll}\frac{h(0)-c}{\dot{D}(0)}&\mbox{
if }\dot{D}(0)<0,\\\\[5.69054pt] -\infty&\mbox{ if
}\dot{D}(0)=0,\end{array}\right.\
\varphi^{\prime}{}(\xi_{1}^{+})=\left\\{\begin{array}[]{ll}\frac{h(1)-c}{\dot{D}(1)}&\mbox{
if }\dot{D}(1)<0,\\\\[5.69054pt] -\infty&\mbox{ if
}\dot{D}(1)=0.\end{array}\right.$

$\xi$$\varphi$$1$$\beta$$\varphi^{1}$$\varphi^{2}$$\varphi^{3}$$\varphi^{4}$$\varphi^{5}$
Figure 4: Some possible wavefronts joining $1$ with $0$ in case $\rm(D_{np})$;
Profiles are labelled according to the cases (1)–(4) of Remark 4.2;
$\varphi^{5}$ occurs in both cases (3) and (4). For simplicity we only
represented strictly monotone profiles.

About the regularity of $\varphi$ at $\xi_{\beta}$, see also Remark 4.2. From
(2.13) and (2.14), we have $c^{*}_{np}\geq h(\beta)$ and $c^{*}_{np}>h(\beta)$
if $\dot{D}(\beta)>0$. Hence, all quantities represented as
$s_{\pm}(\beta,\cdot)$ in (2.17) are negative and finite real values.

Theorem 2.2 above extends [2, Theorem 1] to the case of a non-zero convection
term $f$; if $f=0$, the estimates on $c_{np}^{*}$ deduced from (2.13) and
(2.14) improve those in [2], not only because of a flaw in the upper estimate
in $c^{*}$ (the analog of $c_{np}^{*}$) in [2], see formula (14) there, but
also because the more precise estimates from [6] and [25] are involved here
(and not in [2]). Moreover, by $c_{np}^{*}\geq h(\beta)$, with a reasoning as
above Theorem 2.1 we can further show, as in [2], that $c^{*}_{np}>0$ when
$f=0$. In Theorem 2.2 we also prove the existence of sharp profiles at either
$0$ or $1$ even if $c>c_{np}^{*}$, see case (iii) above. Again, this is due to
the presence of $f$. Indeed, from a formal point of view, by the estimates in
(iii) it follows that, if $f=0$, then to have sharp profiles one needs $c<0$,
while $c$ is always positive in this case.

We point out that, if $c^{*}_{p,l}>c^{*}_{n,r}$ and $\varphi$ is the profile
corresponding to $c=c^{*}_{np}=c^{*}_{p,l}$ given by Theorem 2.2, then
$\varphi^{\prime}{}(\xi_{\beta}^{-})\neq\varphi^{\prime}{}(\xi_{\beta}^{+})$.
The same conclusion holds if $c^{*}_{p,l}<c^{*}_{n,r}$ and
$c=c^{*}_{np}=c^{*}_{n,r}$. Both alternatives $c^{*}_{p,l}=c^{*}_{n,r}$ and
$c^{*}_{p,l}\neq c^{*}_{n,r}$ can indeed occur: explicit examples are shown in
Example 5.3. This suggests that Theorems 2.1 and 2.2 produce two separate
families of solutions. Moreover, in Example 5.2, we show that the thresholds
$c^{*}_{pn}$ of Theorem 2.1 and $c^{*}_{np}$ of Theorem 2.2 are essentially
different, in the sense that taking opposite diffusivities do not produce
necessarily $c^{*}_{pn}=c^{*}_{np}$. Roughly speaking, this is due essentially
because (2.4)–(2.6) and (2.13)–(2.14) are unrelated estimates.

Also notice the different role played by the sub-thresholds $c_{p,r}^{*}$,
$c_{n,l}^{*}$ and $c_{p,l}^{*}$, $c_{n,r}^{*}$: the former two discriminate
the existence of the semi-wavefronts, the latter two the regularity. Indeed,
the estimates (2.4), (2.6) concern the equilibrium points $0,1$ of $g$, while
the estimates (2.13), (2.14) only concern the point $\beta$ where $D$
vanishes. The values of $\varphi^{\prime}$ at $\xi_{\alpha}$ or $\xi_{\beta}$:
$\varphi^{\prime}(\xi_{\alpha})$ is uniquely determined (being possibly
$-\infty$), while we can have
$\varphi^{\prime}(\xi_{\beta}^{-})\neq\varphi^{\prime}(\xi_{\beta}^{+})$.
Moreover, in the case $\rm(D_{pn})$, a profile may be sharp only if
$c=c_{pn}^{*}$; in case $\rm(D_{np})$, a profile can be sharp also if
$c>c_{np}^{*}$. As a consequence of these facts, items (i)–(iii) in the two
theorems are similar but far from being symmetric.

##  3 Wavefronts with positive to negative diffusivities

In this section we assume condition $(\rm D_{pn})$. First, we discuss the
existence of semi-wavefronts to $0$ and from $1$ in Proposition 3.1 and in
Proposition 3.2, respectively. Then, by pasting their profiles, we provide the
proof of Theorem 2.1.

In the next proposition $s_{-}(\alpha,c)$ is given by (2.8).

######  Proposition 3.1.

Assume (f), (g), $(\rm D_{pn})$, $\eqref{Dg}_{1}$. Then, there exists
$c_{p,r}^{*}$ satisfying (2.4) such that Equation (1.1) has strict semi-
wavefronts to $0$, connecting $\alpha$ to $0$, if and only if $c\geq
c^{*}_{p,r}$. If $\varphi$ is the non-increasing profile of one of them, then

$\varphi^{\prime}{}(\xi)<0\ \mbox{ for any }\ 0<\varphi(\xi)<\alpha.$ (3.1)

For $c>c^{*}_{p,r}$, there exists $\beta(c)<0$ such that every profile is
uniquely determined (up to space shifts) by the value

$\left(D\left(\varphi\right)\varphi^{\prime}{}\right)(\xi_{\alpha}^{+})=:\ell\in[\beta(c),0],$
(3.2)

where $\xi_{\alpha}\in\mathbb{R}$ is such that $(\xi_{\alpha},+\infty)$ is the
maximal-existence interval of $\varphi$.

If $\varphi_{\ell}$ is the profile satisfying (3.2), then
$\varphi_{\ell}^{\prime}{}(\xi_{\alpha}^{+})=-\infty$ if
$\ell\in[\beta(c),0)$, while

$\varphi_{0}^{\prime}{}(\xi_{\alpha}^{+})=\begin{cases}\frac{g(\alpha)}{s_{-}(\alpha,c)}\
&\mbox{ if }\ \dot{D}(\alpha)<0\ \mbox{ or }\ c>h(\alpha),\\\ -\infty\ &\mbox{
if }\ \dot{D}(\alpha)=0\mbox{ and }c\leq h(\alpha).\end{cases}$ (3.3)

###### Proof.

We refer to Figure 5, solid lines. The existence of these semi-wavefronts, the
estimate of $c^{*}_{p,r}$ and condition (3.1) are discussed in [6, Theorem
2.2]. The proof of [6, Theorem 2.2] also includes the equivalence between the
presence of a semi-wavefront with speed $c$ and the solvability of the problem
(1.4)1 for the same $c$, with $z(\varphi)=D(\varphi)\varphi^{\prime}$.
Conditions (3.2) and (3.3) then come, respectively, from [6, Proposition 5.1]
and [6, (9.2)] (see also [6, Remark 9.2]) which are directly formulated for
problem (1.4)1. ∎

$\rho$$\alpha$$D$$1$$0$$g$$\xi$$\varphi$$\xi_{\alpha}$$\alpha$$1$ Figure 5:
$D$ 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.

######  Remark 3.1.

Every semi-wavefront $\varphi$ given in Proposition 3.1 satisfies
$\left(D\left(\varphi\right)\varphi^{\prime}{}\right)(\xi_{0}^{-})=0$, (see
e.g. [6, formula (9.19)]), where
$\xi_{0}=\sup\left\\{\xi>\xi_{\alpha}:\varphi(\xi)>0\right\\}\in(\xi_{\alpha},+\infty]$.

######  Proposition 3.2.

Assume (f), (g), $(\rm D_{pn})$ and $\eqref{Dg}_{2}$. There exists
$c^{*}_{n,l}\in\mathbb{R}$ satisfying (2.6) such that Equation (1.1) admits
strict semi-wavefronts from $1$, connecting $1$ to $\alpha$, with speed $c$,
if and only if $c\geq c^{*}_{n,l}$. If $\varphi$ is the non-increasing profile
of one of such fronts, then it holds that

$\varphi^{\prime}{}(\xi)<0\ \mbox{ if }\ \alpha<\varphi(\xi)<1.$ (3.4)

For $c>c^{*}_{n,l}$, there exists $\gamma(c)>0$ such that every profile is
uniquely determined (up to space shifts) by the value

$\left(D\left(\varphi\right)\varphi^{\prime}{}\right)(\xi_{\alpha}^{-})=:s\in[0,\gamma(c)],$
(3.5)

where $\xi_{\alpha}\in\mathbb{R}$ is such that $(-\infty,\xi_{\alpha})$ is the
maximal-existence interval of $\varphi$.

If $\varphi_{s}$ is the profile satisfying (3.5), then
$\varphi_{s}^{\prime}{}(\xi_{\alpha}^{-})=-\infty$ if $s\in(0,\gamma(c)]$,
while

$\varphi_{0}^{\prime}{}(\xi_{\alpha}^{-})=\begin{cases}\frac{g(\alpha)}{s_{-}(\alpha,c)}\
&\mbox{ if }\ \dot{D}(\alpha)<0\ \mbox{ or }\ c>h(\alpha),\\\ -\infty\ &\mbox{
if }\ \dot{D}(\alpha)=0\mbox{ and }c\leq h(\alpha).\end{cases}$ (3.6)

###### Proof.

We refer to Figure 5, dashed lines. Take, for $\varphi\in[0,1]$,

$\bar{D}\left(\varphi\right):=-D\left(1-\varphi\right),\
\bar{g}\left(\varphi\right):=g\left(1-\varphi\right),\
\bar{f}\left(\varphi\right):=f(1)-f\left(1-\varphi\right).$ (3.7)

Set $\bar{\alpha}:=1-\alpha$. Then $\bar{D},\bar{g},\bar{f}$ satisfy
${\rm(D_{pn})}$ with $\bar{\alpha}$ replacing $\alpha$, (g), (f) and
$\eqref{Dg}_{1}$. Set $\bar{h}=\dot{\bar{f}}$. We apply Proposition 3.1 to
deduce that there exists a real value $c^{*}$ satisfying

$\max\left\\{\bar{h}(0)+2\sqrt{D_{+}\bar{D}\bar{g}(0)},\sup_{(0,\bar{\alpha}]}\delta\left(\bar{f},0\right)\right\\}\leq
c^{*}\leq\sup_{(0,\bar{\alpha}]}\delta\left(\bar{f},0\right)+2\sqrt{\sup_{(0,\bar{\alpha}]}\delta\left(Dg,0\right)}$

such that there exist strict semi-wavefronts connecting $0$ to $\bar{\alpha}$,
with speed $c$, if and only if $c\geq c^{*}$. In the formula above, we recall
that $\delta\left(\cdot,\cdot\right)$ is defined in (2.3). Direct
manipulations show that the above chain of inequalities coincides with (2.6),
in virtue of (3.7). Fix $c\geq c^{*}$ and let $\varphi_{\bar{\alpha},0}$ be
the profile of a semi-wavefront connecting $\bar{\alpha}$ to $0$, having speed
$c$, given by Proposition 3.1. Moreover, assume that
$\varphi_{\bar{\alpha},0}$ is maximally defined in
$(\xi_{\bar{\alpha}},+\infty)$. Define then
$\varphi_{1,\alpha}:(-\infty,-\xi_{\bar{\alpha}})\to\mathbb{R}$ by

$\varphi_{1,\alpha}(\xi):=1-\varphi_{\bar{\alpha},0}\left(-\xi\right)\ \mbox{
for }\ \xi<\xi_{\alpha}:=-\xi_{\bar{\alpha}}.$ (3.8)

Notice that, because $\varphi_{\bar{\alpha},0}$ connects $\bar{\alpha}$ to
$0$, then (3.8) implies

$\lim_{\xi\to-\infty}\varphi_{1,\alpha}(\xi)=1\ \mbox{ and }\
\lim_{\xi\to\xi_{\alpha}^{-}}\varphi_{1,\alpha}(\xi)=\alpha,$

thus $\varphi_{1,\alpha}$ connects $1$ to $\alpha$. Also, from (3.8) and
$\bar{D}\left(\varphi_{\bar{\alpha},0}\right)\varphi_{\bar{\alpha},0}^{\prime}{}\in
L^{1}\left(\xi_{\bar{\alpha}},\infty\right)$, we have
$D\left(\varphi_{1,\alpha}\right)\varphi_{1,\alpha}^{\prime}{}\in
L^{1}\left(-\infty,\xi_{\alpha}\right)$. Since $\varphi_{\bar{\alpha},0}$
satisfies (1.2), with $\bar{D},\bar{g}$, $\bar{h}$, in
$(\xi_{\bar{\alpha}},+\infty)$ for $c\geq c^{*}$, then direct computations
show that $\varphi_{1,\alpha}$ satisfies (1.2) in $(-\infty,\xi_{\alpha})$
with the same $c$. Solutions are always meant in the sense of Definition 2.1.

Viceversa, let $\varphi$ be a profile of a strict semi-wavefront of (1.1),
connecting $1$ to $\alpha$ associated to some $c\in\mathbb{R}$, defined
maximally in $(-\infty,0)$. By setting

$\varphi_{\bar{\alpha},0}(\tau):=1-\varphi\left(-\tau\right)\ \mbox{ for }\
\tau>0,$ (3.9)

we obtain the profile of a strict semi-wavefront, from $\bar{\alpha}$ to $0$,
of Equation (1.2) (with $\bar{D}$, $\bar{g}$ and $\bar{f}$ as in (3.7))
associated to the speed $c$. By applying Proposition 3.1 we deduce that $c\geq
c^{*}$. The first part of the statement is hence proved, with
$c^{*}_{n,l}=c^{*}$ defined above.

Finally, (3.4)–(3.6) follow by (3.8) and Proposition 3.1, for
$\gamma(c):=-\beta(c)$. ∎

######  Remark 3.2.

According to (3.8) and Remark 3.1, every semi-wavefront $\varphi$ given in
Proposition 3.2 satisfies
$\left(D\left(\varphi\right)\varphi^{\prime}{}\right)\left(\xi_{1}^{+}\right)=0$,
where
$\xi_{1}:=\inf\left\\{\xi<\xi_{\alpha}:\varphi(\xi)<1\right\\}\in[-\infty,\xi_{\alpha})$.

_Proof of Theorem 2.1._ Let $c^{*}_{p,r}$ and $c^{*}_{n,l}$ be the thresholds
given in Proposition 3.1 and 3.2, respectively, and define $c^{*}_{pn}$ as in
(2.7).

First, we take $c\geq c^{*}_{pn}$ and prove that there is a wavefront to
equation (1.1) that satisfies (1.3) and has speed $c$. Associated to such a
value of $c$, from Proposition 3.1, there exists a strict semi-wavefront
$\varphi_{0}$ (according to the notation introduced in the statement of
Proposition 3.1) to $0$ with wave speed $c$, connecting $\alpha$ to $0$, such
that

$\left(D\left(\varphi_{0}\right)\varphi_{0}^{\prime}{}\right)(\xi_{\alpha}^{+})=0,$
(3.10)

see (3.2). In (3.10), the value $\xi_{\alpha}$, which is finite because we are
considering a strict semi-wavefront, is such that $\varphi_{0}$ is maximally
defined in $(\xi_{\alpha},+\infty)$. Analogously, from Proposition 3.2, we
have that there exists a semi-wavefront from $1$ with wave speed $c$, which
connects $1$ to $\alpha$ and such that its profile $\psi_{0}$ realizes

$\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}\right)(\xi_{\alpha}^{-})=0,$
(3.11)

see (3.5). Here, we assumed that $\psi_{0}$ is maximally defined in
$(-\infty,\xi_{\alpha})$; this is always possible unless of shifting
$\psi_{0}$. We refer to Remark 3.4 for the reasons of the choices of
$\varphi_{0}$ and $\psi_{0}$.

Let $\xi_{1},\xi_{0}\in\mathbb{R}$ be such that

$\displaystyle\xi_{0}:=\sup\left\\{\xi>\xi_{\alpha}:\varphi_{0}(\xi)>0\right\\}\in(\xi_{\alpha},+\infty],$
$\displaystyle\xi_{1}:=\inf\left\\{\xi<\xi_{\alpha}:\psi_{0}(\xi)<1\right\\}\in[-\infty,\xi_{\alpha}),$

see Figure 6. Define then the function $\varphi=\varphi(\xi)$ by

$\varphi(\xi)=\begin{cases}\varphi_{0}(\xi),\ \xi\geq\xi_{\alpha},\\\
\psi_{0}(\xi),\ \xi<\xi_{\alpha}.\end{cases}$ (3.12)

It is clear that $\varphi$ is well-defined and continuous in
$(-\infty,+\infty)$; moreover, $\varphi$ is non-increasing and connects $1$ to
$0$. Since both $\varphi_{0}$ and $\psi_{0}$ satisfy (1.2) in their domains,
then $\varphi$ satisfies (1.2) pointwise in
$(-\infty,\xi_{1})\cup(\xi_{1},\xi_{\alpha})\cup(\xi_{\alpha},\xi_{0})\cup(\xi_{0},+\infty)$.

$\xi$$\xi_{\alpha}$$\varphi$$\psi_{0}$$\varphi_{0}$$\varphi$$\xi_{0}$$\psi_{0}$$\alpha$$1$$\xi_{1}$$\varphi_{0}$
Figure 6: Construction of the profile $\varphi$ in the case
$\xi_{0},\xi_{1}\in\mathbb{R}$.

Formulas (3.10), (3.11) imply that $D(\varphi)\varphi^{\prime}{}$ is
continuously extended to $\xi=\xi_{\alpha}$; we have

$\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}\right)(\xi_{1}^{+})=0\ \mbox{
and }\
\left(D\left(\varphi_{0}\right)\varphi_{0}^{\prime}{}\right)(\xi_{0}^{-})=0.$
(3.13)

Formula $\eqref{e:xi1 xi0}_{2}$ follows from Remark 3.1 applied to
$\varphi_{0}$ and $\eqref{e:xi1 xi0}_{1}$ follows from Remark 3.2 applied to
$\psi_{0}$. Hence, we showed that $D(\varphi)\varphi^{\prime}{}\in L^{1}_{\rm
loc}(-\infty,+\infty)$. It remains to show that $\varphi$ is a solution of
(1.2) in $(-\infty,+\infty)$, according to Definition 2.1. To this purpose,
take $\zeta\in C_{0}^{\infty}(-\infty,\infty)$. Since $\varphi$ is a
distributional solution of (1.2) in both $(-\infty,\xi_{\alpha})$ and
$(\xi_{\alpha},+\infty)$, then we can reduce to test (2.1) when $\mathop{\rm
supp}\zeta\subseteq[\xi_{\alpha}-\delta,\xi_{\alpha}+\delta]\subset(\xi_{1},\xi_{0})$,
for some $\delta>0$. Hence, we need to test whether

$\int_{\xi_{\alpha}-\delta}^{\xi_{\alpha}+\delta}\left(D\left(\varphi\right)\varphi^{\prime}{}-f\left(\varphi\right)+c\varphi\right)\zeta^{\prime}{}-g\left(\varphi\right)\zeta\,d\xi=0.$
(3.14)

Clearly, the left-hand side of (3.14) equals

$\int_{\xi_{\alpha}-\delta}^{\xi_{\alpha}}\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}-f\left(\psi_{0}\right)+c\psi_{0}\right)\zeta^{\prime}{}-g\left(\psi_{0}\right)\zeta\,d\xi+\\\
\int_{\xi_{\alpha}}^{\xi_{\alpha}+\delta}\left(D\left(\varphi_{0}\right)\varphi_{0}^{\prime}{}-f\left(\varphi_{0}\right)+c\varphi_{0}\right)\zeta^{\prime}{}-g\left(\varphi_{0}\right)\zeta\,d\xi.$
(3.15)

Let us focus on the former addend of (3.15). We have:

$\int_{\xi_{\alpha}-\delta}^{\xi_{\alpha}}\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}-f\left(\psi_{0}\right)+c\psi_{0}\right)\zeta^{\prime}{}-g\left(\psi_{0}\right)\zeta\,d\xi=\\\
\lim_{\delta>\varepsilon\to
0^{+}}\int_{\xi_{\alpha}-\delta}^{\xi_{\alpha}-\varepsilon}\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}-f\left(\psi_{0}\right)+c\psi_{0}\right)\zeta^{\prime}{}-g\left(\psi_{0}\right)\zeta\,d\xi=\\\
\lim_{\delta>\varepsilon\to
0^{+}}\left(\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}\right)\left(\xi_{\alpha}-\varepsilon\right)-f\left(\psi_{0}\left(\xi_{\alpha}-\varepsilon\right)\right)-c\psi_{0}\left(\xi_{\alpha}-\varepsilon\right)\right)\zeta\left(\xi_{\alpha}-\varepsilon\right)=\\\
\left(D\left(\psi_{0}\right)\psi_{0}^{\prime}{}\right)\left(\xi_{\alpha}^{-}\right)\zeta\left(\xi_{\alpha}\right)-\left(f\left(\alpha\right)-c\alpha\right)\zeta\left(\xi_{\alpha}\right)=-\left(f\left(\alpha\right)-c\alpha\right)\zeta\left(\xi_{\alpha}\right).$
(3.16)

Similar arguments involving the latter addend of (3.15) lead to

$\int_{\xi_{\alpha}}^{\xi_{\alpha}+\delta}\left(D\left(\varphi_{0}\right)\varphi_{0}^{\prime}{}-f\left(\varphi_{0}\right)+c\varphi_{0}\right)\zeta^{\prime}{}-g\left(\varphi_{0}\right)\zeta\,d\xi=\\\
-\left(D\left(\varphi_{0}\right)\varphi_{0}^{\prime}{}\right)\left(\xi_{\alpha}^{+}\right)\zeta\left(\xi_{\alpha}\right)+\left(f(\alpha)-c\alpha\right)\zeta\left(\alpha\right)=\left(f(\alpha)-c\alpha\right)\zeta\left(\xi_{\alpha}\right).$
(3.17)

Putting together (3.16) and (3.17) proves (3.14).

Conversely, we prove that if there exists a wavefront with speed $c$, whose
profile $\varphi$ is non-increasing and satisfies (1.2)-(1.3), then $c\geq
c^{*}_{pn}$. Let $\xi_{\alpha}\in\mathbb{R}$ be such that
$\varphi(\xi_{\alpha})=\alpha$. Such a $\xi_{\alpha}$ obviously exists since
$\varphi$ is continuous and satisfies (1.3). Furthermore, we have

$\left\\{\varphi=\alpha\right\\}=\left\\{\xi_{\alpha}\right\\}.$ (3.18)

Indeed, otherwise there exists an open set
$J\subset\left\\{\varphi=\alpha\right\\}$ where $\varphi$ is constantly equal
to $\alpha$. Thus, in $J$, equation (1.2) for $\varphi$ reduces to
$g(\alpha)=0$, which is clearly forbidden by (g). Then, we proved (3.18).

The function $\varphi_{\alpha,0}:(\xi_{\alpha},+\infty)\to(0,\alpha)$, defined
by $\varphi_{\alpha,0}=\varphi$, is a strict semi-wavefront to $0$ with speed
$c$, connecting $\alpha$ to $0$, and the function
$\varphi_{1,\alpha}:(-\infty,\xi_{\alpha})\to(\alpha,1)$, defined by
$\varphi_{1,\alpha}=\varphi$, is a strict semi-wavefront from $1$, with speed
$c$, connecting $1$ to $\alpha$. From Propositions 3.1 and 3.2, both $c\geq
c^{*}_{p,r}$ and $c\geq c^{*}_{n,l}$ must occur. Hence, $c\geq c^{*}_{pn}$,
and the first part of Theorem 2.1 is proved.

To conclude the proof, we apply [6, Corollary 9.4] to $\varphi_{\alpha,0}$
(defined just above) and to $\varphi_{\bar{\alpha},0}$ (defined by (3.9)).
$\square$

######  Remark 3.3 (Estimates for $c^{*}_{pn}$).

We now explicitly provide estimates for the threshold $c^{*}_{pn}$ in (2.7).
Obviously, $c^{*}_{pn}$ inherits the bounds, from above and below, for both
$c^{*}_{p,r}$ and $c^{*}_{n,l}$. Such estimates are contained in Propositions
3.1 and 3.2. We hence have:

$\displaystyle c^{*}_{pn}$
$\displaystyle\geq\max\left\\{\sup_{(0,\alpha]}\delta(f,0),\sup_{[\alpha,1)}\delta(f,1),h(0)+2\sqrt{D_{+}Dg(0)},h(1)+2\sqrt{D_{-}Dg(1)}\right\\},$
$\displaystyle c^{*}_{pn}$
$\displaystyle\leq\max\left\\{\sup_{(0,\alpha]}\delta(f,0)+2\sqrt{\sup_{(0,\alpha]}\delta(Dg,0)},\sup_{[\alpha,1)}\delta(f,1)+2\sqrt{\sup_{[\alpha,1)}\delta(Dg,1)}\right\\}.$

If $f$ is identically zero, such estimates were given in [21, formula (14)].

######  Remark 3.4.

It is worth emphasizing that, given $c$ large enough, among the profiles in
the families

$\\{\varphi_{\ell}:\ell\in[\beta(c),0]\\}\quad\hbox{ and
}\quad\\{\psi_{s}:s\in[0,\gamma(c)]\\},$

given by Propositions 3.1 and 3.2 respectively, we can only benefit from
$\varphi_{0}$ and $\psi_{0}$ to construct a wavefront as in Theorem 2.1. In
particular, we can take advantage only of those profiles whose associated
functions $z$ (solving (1.4)) vanish at $\alpha^{\pm}$. In all the other
cases, the pasting of a profile $\varphi_{\ell}$ with a profile $\psi_{s}$
does not provide a solution (according to the distributional sense of
Definition 2.1) in a neighborhood of the matching point. Indeed, under these
assumptions, (3.16) and (3.17) read respectively as

$\displaystyle\int_{\xi_{\alpha}-\delta}^{\xi_{\alpha}}\left(D\left(\psi_{s}\right)\psi_{s}^{\prime}{}-f(\psi_{s})+c\psi_{s}\right)\zeta^{\prime}{}-g(\psi_{s})\zeta\,d\xi$
$\displaystyle=\left[s+c\alpha-f(\alpha)\right]\zeta(\xi_{\alpha}),$ (3.19)
$\displaystyle\int_{\xi_{\alpha}}^{\xi_{\alpha}+\delta}\left(D\left(\varphi_{\ell}\right)\varphi_{\ell}^{\prime}{}-f(\varphi_{\ell})+c\varphi_{\ell}\right)\zeta^{\prime}{}-g(\varphi_{\ell})\zeta\,d\xi$
$\displaystyle=\left[-\ell-c\alpha+f(\alpha)\right]\zeta(\xi_{\alpha}).$
(3.20)

Thus, in place of (3.14), putting together (3.19) and (3.20) gives

$\int_{\xi_{\alpha}-\delta}^{\xi_{\alpha}+\delta}\left(D\left(\varphi\right)\varphi^{\prime}{}-f(\varphi)+c\varphi\right)\zeta^{\prime}{}-g(\varphi)\zeta\,d\xi=\left(s-\ell\right)\zeta(\xi_{\alpha}),$
(3.21)

which vanishes for each arbitrary test function $\zeta$ if and only if
$s=\ell=0$.

In the first instance, this seems to be suggested by the fact that if,
formally,

$z(\alpha)=D\left(\varphi(\xi_{\alpha})\right)\varphi^{\prime}(\xi_{\alpha}^{-})=D(\alpha)\varphi^{\prime}(\xi_{\alpha}^{-})<0,$

then necessarily $\varphi^{\prime}{}(\xi_{\alpha}^{-})=-\infty$, because
$D(\alpha)=0$; the same remark holds if $w(\alpha)>0$. Nonetheless, the
failure of the pasting is not due to a possibly infinite derivative of the
profile at the matching point; indeed, Theorem 2.1 (ii) shows that the
wavefront $\varphi$ can well have infinite slope at $\xi_{\alpha}$, see
profile $\varphi^{2}$ in Figure 3.

##  4 Wavefronts with negative to positive diffusivities

In this section we assume condition $(\rm D_{np})$ and prove Theorem 2.2.

The existence and regularity of semi-wavefronts from $1$, connecting $1$ to
$\beta$, was obtained essentially in [10, Theorem 2.7] in the case $D(1)>0$
and in [8, Theorems 2.3 and 2.5] when $D(1)=0$. We collect these results in
the next proposition. The sharper estimate (2.13) for the threshold
$c^{*}_{p,l}$ comes from [6, Corollary 5.4 and Remark 5.5]. The symbols
$s_{\pm}(\alpha,c)$ are given by (2.8). We refer to Figure 7, solid lines.

######  Proposition 4.1.

Assume (f), (g), $\rm(D_{np})$. Then, for every $c\in\mathbb{R}$, Equation
(1.1) has a (unique up to shifts) strict semi-wavefront solution from $1$,
connecting $1$ to $\beta$, with speed $c$ and profile $\varphi$ defined in its
maximal-existence interval $(-\infty,\xi_{\beta})$, for some
$\xi_{\beta}\in\mathbb{R}$. It holds that $\varphi^{\prime}{}<0$ if
$\beta<\varphi<1$. There exists $c^{*}_{p,l}$ satisfying (2.13) such that
(2.12) holds true and we have

$\varphi^{\prime}{}(\xi_{\beta}^{-})=\begin{cases}\frac{g(\beta)}{s_{-}\left(\beta,c\right)}\
&\mbox{ if }\ c>c^{*}_{p,l},\\\\[5.69054pt]
\frac{g(\beta)}{s_{+}(\beta,c^{*}_{p,l})}\ &\mbox{ if }\ c=c^{*}_{p,l}\ \mbox{
and }\ \dot{D}(\beta)>0,\\\ -\infty\ &\mbox{ if }\ c=c^{*}_{p,l}\ \mbox{ and
}\ \dot{D}(\beta)=0\ \mbox{ or }\ c<c^{*}_{p,l}.\end{cases}$ (4.1)

Moreover, the following results hold.

  1. (i)

If $D(1)>0$, then $\varphi$ is classical.

  2. (ii)

If $D(1)=0$ and either $c>h(1)$ or $c=h(1)$ and $\dot{D}(1)<0$, then $\varphi$
is classical.

  3. (iii)

If $D(1)=0$ and $c<h(1)$, then $\varphi$ is sharp at $1$ (reached at some
$\xi_{1}<\xi_{\beta}$) with

$\varphi^{\prime}{}(\xi_{1}^{+})=\left\\{\begin{array}[]{ll}\frac{h(1)-c}{\dot{D}(1)}<0&\mbox{
if }\ \dot{D}(1)<0,\\\\[5.69054pt] -\infty&\mbox{ if }\
\dot{D}(1)=0.\end{array}\right.$

$\rho$$\beta$$D$$1$$g$$\xi$$\varphi$$\xi_{\beta}$$\beta$$1$ Figure 7: $D$
satisfies $(\rm D_{np})$. Left: the plots of $D$, $g$ in $[0,\beta]$ (dashed
lines) and $[\beta,1]$ (solid lines); right, a corresponding profile.

We discuss the existence of semi-wavefronts to $0$ in the next proposition.

######  Proposition 4.2.

Assume (f), (g), $\rm(D_{np})$. Then, for every $c\in\mathbb{R}$ Equation
(1.1) has a (unique up to space shifts) strict semi-wavefront solution to $0$,
connecting $\beta$ to $0$, with speed $c$ and profile $\varphi$ defined in its
maximal-existence interval $(\xi_{\beta},+\infty)$, for some
$\xi_{\beta}\in\mathbb{R}$. It holds that $\varphi^{\prime}{}<0$ if
$0<\varphi<\beta$. In addition, there exists $c^{*}_{n,r}\in\mathbb{R}$
satisfying (2.15) and we have

$\varphi^{\prime}{}(\xi_{\beta}^{+})=\begin{cases}\frac{g(\beta)}{s_{-}(\beta,c)}\
&\mbox{ if }\ c>c^{*}_{n,r},\\\\[5.69054pt]
\frac{g(\beta)}{s_{+}(\beta,c_{n,r}^{*})}\ &\mbox{ if }\ c=c^{*}_{n,r}\ \mbox{
and }\ \dot{D}(\beta)>0,\\\ -\infty\ &\mbox{ if }\ c<c^{*}_{n,r}\ \mbox{ or }\
c=c^{*}_{n,r}\ \mbox{ and }\ \dot{D}(\beta)=0.\end{cases}$ (4.2)

Moreover, the following results hold.

  1. (i)

If $D(0)<0$, then $\varphi$ is classical.

  2. (ii)

If $D(0)=0$ and either $c>h(0)$ or $c=h(0)$ and $\dot{D}(0)<0$ then $\varphi$
is classical.

  3. (iii)

If $D(0)=0$ and $c<h(0)$ then $\varphi$ is sharp at $0$ (reached at some
$\xi_{0}>\xi_{\beta}$) with

$\varphi^{\prime}{}(\xi_{0}^{-})=\left\\{\begin{array}[]{ll}\frac{h(0)-c}{\dot{D}(0)}<0&\mbox{
if }\ \dot{D}(0)<0,\\\\[5.69054pt] -\infty&\mbox{ if }\
\dot{D}(0)=0.\end{array}\right.$

###### Proof.

We refer to Figure 7, dashed lines. Let $\bar{D}$, $\bar{g}$ and $\bar{f}$ be
defined by (3.7). We already observed in the proof of Proposition 3.2 that
$\bar{g}$ and $\bar{f}$ still satisfy (g) and (f). In this case, instead,
$\bar{D}$ satisfies ${\rm(D_{np})}$ with $\bar{\beta}:=1-\beta$ replacing
$\beta$. Hence, Proposition 4.1 applied to $\bar{D}$, $\bar{g}$ and $\bar{f}$
informs us that strict semi-wavefronts with speed $c\in\mathbb{R}$ connecting
$1$ to $\bar{\beta}$ exist for every $c$. Let $\varphi_{1,\bar{\beta}}$ be a
profile of one of such fronts, defined in $(-\infty,\xi_{\bar{\beta}})$. Set
$\xi_{\beta}:=-\xi_{\bar{\beta}}$ and

$\varphi_{\beta,0}(\xi):=1-\varphi_{1,\bar{\beta}}(-\xi)\ \mbox{ for }\
\xi>\xi_{\beta}.$ (4.3)

With arguments analogous to those in the proof of Proposition 3.2, it turns
out that $\varphi_{\beta,0}$ is the profile of a desired strict semi-wavefront
of (1.1), connecting $\beta$ to $0$. Hence, the first part of the statement is
proved. By Proposition 4.1 applied to $\bar{D}$, $\bar{g}$ and $\bar{f}$, in
the interval $[\bar{\beta},1]$, we obtain that there exists a real value, say
$\bar{c}^{*}$, satisfying

$\max\left\\{\sup_{(\bar{\beta},1]}\delta(\bar{f},\bar{\beta}),\
\bar{h}(\bar{\beta})+2\sqrt{\dot{\bar{D}}(\bar{\beta})\bar{g}(\bar{\beta})}\right\\}\leq\\\
\bar{c}^{*}\leq\sup_{(\bar{\beta},1]}\delta(\bar{f},\bar{\beta})+2\sqrt{\sup_{\varphi\in(\bar{\beta},1]}\frac{1}{\varphi-\bar{\beta}}\int_{\bar{\beta}}^{\varphi}\frac{\bar{D}(s)\bar{g}(s)}{s}\,ds}$

such that

$\left(\bar{D}(\varphi_{1,\bar{\beta}})\varphi_{1,\bar{\beta}}^{\prime}{}\right)(\xi_{\bar{\beta}}^{-})=\left\\{\begin{array}[]{ll}0&\mbox{
if }\ c\geq\bar{c}^{*},\\\ \ell<0&\mbox{ if }\
c<\bar{c}^{*},\end{array}\right.$

and

$\varphi_{1,\bar{\beta}}^{\prime}{}(\xi_{\bar{\beta}}^{-})=\left\\{\begin{array}[]{ll}\frac{\bar{g}(\bar{\beta})}{\bar{s}_{-}(\bar{\beta},c)}&\mbox{
if }\ c>\bar{c}^{*},\\\\[5.69054pt]
\frac{\bar{g}(\bar{\beta})}{\bar{s}_{+}\left(\bar{\beta},\tilde{c}^{*}\right)}&\mbox{
if }\ c=\bar{c}^{*}\ \mbox{ and }\
\dot{\bar{D}}(\bar{\beta})>0,\\\\[5.69054pt] -\infty&\mbox{ if }\
c=\bar{c}^{*}\ \mbox{ and }\ \dot{\bar{D}}(\bar{\beta})=0\ \mbox{ or }\
c<\bar{c}^{*}.\end{array}\right.$ (4.4)

Here, $\bar{s}_{\pm}$ is defined as $s_{\pm}$ in (2.8) but with $\bar{D}$,
$\bar{g}$ and $\bar{h}$ instead of the non-subscripted ones. Define
$c^{*}_{n,r}=\bar{c}^{*}$. The former of the two previous formulas yields to
(2.15). Analogously, (4.4) implies (4.2). Finally, (i)–(iii) follow from the
application of (i)–(iii) of Proposition 4.1 to $\varphi_{1,\bar{\beta}}$ and
(4.3)-(3.7). The proof is then concluded. ∎

_Proof of Theorem 2.2._ For each $c\in\mathbb{R}$, Propositions 4.1, 4.2
provide the existence of a strict semi-wavefront $\varphi_{1,\beta}$ from $1$,
connecting $1$ to $\beta$, and a strict semi-wavefront $\varphi_{\beta,0}$ to
$0$, connecting $\beta$ to $0$. Let $c^{*}_{np}$ be as in (2.16) and take
$c\geq c^{*}_{np}$. Proposition 4.1 informs us that $\varphi_{1,\beta}$
satisfies $\eqref{e:threshold cm-dpde condition2}_{1}$ while Proposition 4.2
implies that $\varphi_{\beta,0}$ realizes $\eqref{e:threshold cm-dpde
condition}_{1}$. In addition, from the uniqueness up to shifts of both
$\varphi_{1,\beta}$ and $\varphi_{\beta,0}$, we can suppose that their
maximal-existence intervals are one the complement of the other and that the
two of them have the finite extremum at the same $\xi_{\beta}\in\mathbb{R}$.
Thus, by proceeding as in the proof of Theorem 2.1, we conclude that gluing
together $\varphi_{1,\beta}$ and $\varphi_{\beta,0}$ at $\xi_{\beta}$ produces
the desired wavefront. This concludes the if part of the statement.

Suppose now that $\varphi$ is a profile of a wavefront connecting $1$ to $0$
associated to some $c\in\mathbb{R}$. Necessarily, there exists a unique
$\xi_{\beta}\in\mathbb{R}$ such that $\varphi(\xi_{\beta})=\beta$. Let
$\varphi_{+}$ be defined by $\varphi_{+}(\xi)=\varphi(\xi)$, for
$\xi\in(-\infty,\xi_{\beta})$. Here, the index “+” stands for the positive
sign of $D\left(\varphi_{+}\right)$ in its domain. The function $\varphi_{+}$
is a semi-wavefront from $1$, connecting $1$ to $\beta$. Analogously,
$\varphi_{-}$ defined by $\varphi_{-}(\xi)=\varphi(\xi)$, for
$\xi\in(\xi_{\beta},+\infty)$ is a semi-wavefront to $0$, connecting $\beta$
to $0$. The function $\varphi$ is a solution of Equation (1.2), according to
Definition 2.1. Thus, if $\zeta\in C^{\infty}_{0}\left(-\infty,+\infty\right)$
is a test function with $\mathop{\rm
supp}\zeta\subseteq\left[\xi_{\beta}-\delta,\xi_{\beta}+\delta\right]$ then we
have

$\int_{\xi_{\beta}-\delta}^{\xi_{\beta}+\delta}\left(D\left(\varphi\right)\varphi^{\prime}{}-f\left(\varphi\right)+c\varphi\right)\zeta^{\prime}{}-g\left(\varphi\right)\zeta\,d\xi=0.$
(4.5)

Observe that both $\varphi_{+}$ and $\varphi_{-}$ are classical solution of
(1.2) in $(-\infty,\xi_{\beta})$ and $(\xi_{\beta},+\infty)$, respectively,
because in these domains $\pm D\left(\varphi_{\pm}\right)>0$. Since
$\int_{\xi_{\beta}-\delta}^{\xi_{\beta}+\delta}=\lim_{\delta>\varepsilon\to
0^{+}}\int_{\xi_{\beta}-\delta}^{\xi_{\beta}-\varepsilon}+\int_{\xi_{\beta}+\varepsilon}^{\xi_{\beta}+\delta}$,
from (4.5) and integration by parts we obtain

$\left(D\left(\varphi_{+}\right)\varphi_{+}^{\prime}{}\right)\left(\xi_{\beta}^{-}\right)\zeta(\xi_{\beta})=0\
\mbox{ and }\
\left(D\left(\varphi_{-}\right)\varphi_{-}^{\prime}{}\right)\left(\xi_{\beta}^{+}\right)\zeta(\xi_{\beta})=0.$
(4.6)

Therefore, $\eqref{e:threshold cm-dpde condition}_{1}$ for $\varphi_{+}$ and
$\eqref{e:threshold cm-dpde condition2}_{1}$ for $\varphi_{-}$ both hold true.
As a consequence, by applying Proposition 4.1 to $\varphi_{+}$ and Proposition
4.2 to $\varphi_{-}$ we deduce that $c\geq c^{*}_{p,l}$ and $c\geq
c^{*}_{n,r}$, from which $c\geq c^{*}_{np}$.

Finally, the values of $\varphi^{\prime}(\xi_{\beta}^{\pm})$ follow from each
possible combination of (4.1) applied to $\varphi_{+}$ and (4.2) applied to
$\varphi_{-}$. Analogously, Parts (i)–(iii) follow from putting together
(i)–(iii) of Proposition 4.1 applied to $\varphi_{+}$ and (i)–(iii) of
Proposition 4.2 applied to $\varphi_{-}$. $\square$

######  Remark 4.1.

In Remark 3.3 we deduced estimates from above and below for $c^{*}_{pn}$.
Bounds for $c^{*}_{np}$ of Theorem 2.2 can be obtained similarly, starting now
from (2.13), (2.14) and (2.16).

######  Remark 4.2.

From (2.17), we can make explicit the a priori regularity of $\varphi$ at
$\xi_{\beta}$ depending on the value of $c$ and the relative order between the
thresholds (see also Figure 4). We have:

  1. (1)

if $c>c^{*}_{np}$, we have
$\varphi^{\prime}{}({\xi_{\beta}}^{-})=\varphi^{\prime}{}({\xi_{\beta}}^{+})\in(-\infty,0)$;

  2. (2)

if $c=c^{*}_{np}=c^{*}_{n,r}=c^{*}_{p,l}$, we have
$\varphi^{\prime}{}({\xi_{\beta}}^{+})=\varphi^{\prime}{}({\xi_{\beta}}^{-})\in(-\infty,0)$
if $\dot{D}(\beta)>0$ and $\varphi^{\prime}(\xi_{\beta}^{\pm})=-\infty$ if
$\dot{D}(\beta)=0$;

  3. (3)

if $c=c^{*}_{np}=c_{n,r}^{*}>c^{*}_{p,l}$, we have
$\varphi^{\prime}{}({\xi_{\beta}}^{-})>\varphi^{\prime}{}({\xi_{\beta}}^{+})$,
with $\varphi^{\prime}{}({\xi_{\beta}}^{-})\in(-\infty,0)$ and
$\varphi^{\prime}{}({\xi_{\beta}}^{+})\in[-\infty,0)$, with
$\varphi^{\prime}{}({\xi_{\beta}}^{+})=-\infty$ if and only if
$\dot{D}(\beta)=0$;

  4. (4)

if $c=c^{*}_{np}=c_{p,l}^{*}>c^{*}_{n,r}$, we have
$\varphi^{\prime}{}({\xi_{\beta}}^{+})>\varphi^{\prime}{}({\xi_{\beta}}^{-})$,
with $\varphi^{\prime}{}({\xi_{\beta}}^{+})\in(-\infty,0)$ and
$\varphi^{\prime}{}({\xi_{\beta}}^{-})\in[-\infty,0)$, with
$\varphi^{\prime}{}({\xi_{\beta}}^{-})=-\infty$ if and only if
$\dot{D}(\beta)=0$.

##  5 Examples

In this section we provide some examples about Theorems 2.1 and 2.2.

######  Example 5.1.

This example shows that $\eqref{e:phip(xialpha)}_{2}$ can indeed occur when a
front goes from a positive- towards a negative-diffusivity region and the term
$f$ is not identically zero. Consider $D$, $g$ and $f$ defined by

$D(\varphi)=\begin{cases}\varphi\left(\varphi-\frac{1}{2}\right)^{2}\ &\mbox{
if }\ \varphi\in[0,1/2],\\\
-\left(1-\varphi\right)\left(\frac{1}{2}-\varphi\right)^{2}\ &\mbox{ if }\
\varphi\in(1/2,1],\end{cases}\ g(\varphi)=\begin{cases}\varphi\ &\mbox{ if }\
\varphi\in[0,1/2],\\\ 1-\varphi\ &\mbox{ if }\ \varphi\in(1/2,1],\end{cases}$

and

$f(\varphi)=\begin{cases}\frac{\varphi^{3}}{3}+\frac{3}{4}\varphi^{2}-\frac{1}{2}\varphi\
&\mbox{ if }\ \varphi\in[0,1/2],\\\
\frac{\varphi^{3}}{3}-\frac{7}{4}\varphi^{2}+2\varphi-\frac{5}{8}\ &\mbox{ if
}\ \varphi\in(1/2,1].\end{cases}$

Note, with these choices, $D$ satisfies ${\rm(D_{pn})}$ with $\alpha=1/2$ and
$\dot{D}(\alpha)=0$ while (g) and (f) hold for $g$ and $f$. Moreover,
$h(\alpha)=1/2$. From direct inspection, the function
$z(\varphi):=\varphi\left(\varphi-1/2\right)$, for $0\leq\varphi\leq 1/2$,
satisfies (1.4)1 with $c=0<h(\alpha)$. By integrating the formal identity
$z(\varphi)=D(\varphi)\varphi^{\prime}{}$, the profile $\varphi_{\alpha,0}$ of
a semi-wavefront connecting $\alpha$ to $0$, with speed $c=0$, can be
determined. In particular, the following Cauchy problem:

$\varphi^{\prime}{}=\frac{1}{\varphi-1/2}\ \mbox{ and }\
\varphi(0)=\frac{1}{4},$

is solved by $\varphi_{\alpha,0}(\xi):=\frac{1}{2}-\sqrt{\frac{1}{16}+2\xi}$,
for any $\xi\geq-\frac{1}{32}$, and $\varphi>0$ for $\xi<\frac{3}{32}$. Since
$g(0)=0$, by setting $\varphi_{\alpha,0}(\xi)=0$ for $\xi\geq\frac{3}{32}$ we
have that $\varphi_{\alpha,0}:(-\frac{1}{32},+\infty)\to[0,1/2)$ is the
desired wave profile associated to the speed $c=0$. Moreover, with the
notations of Theorem 2.1, $\xi_{\alpha}=-\frac{1}{32}$ and
$\varphi_{\alpha,0}^{\prime}{}\left(\xi_{\alpha}^{+}\right)=-\infty$. Similar
arguments lead to conclude that

$\varphi(\xi):=\begin{cases}1\ &\mbox{ if }\ \xi\leq-5/32,\\\
\frac{1}{2}+\sqrt{-1/16-2\xi}\ &\mbox{ if }\ -5/32<\xi<-1/32,\\\
\frac{1}{2}-\sqrt{1/16+2\xi}\ &\mbox{ if }\ -1/32\leq\xi<3/32,\\\ 0\ &\mbox{
if }\ \xi\geq 3/32,\end{cases}$

is the profile of a (sharp at both $0$ and $1$) wavefront of (1.1) with speed
$c=0$, such that $\varphi^{\prime}\left(\xi_{\alpha}^{\pm}\right)=-\infty$.
Observe, from Part (ii) of Theorem 2.1 it must occur $c^{*}_{pn}=0$.

######  Example 5.2.

We show that $c^{*}_{pn}$ in Theorem 2.1 and $c^{*}_{np}$ in Theorem 2.2 are
essentially different: opposite diffusivities do not produce necessarily
$c^{*}_{pn}=c^{*}_{np}$. To this aim, define

$\ g(\varphi):=\left\\{\begin{array}[]{ll}\varphi^{2}&\mbox{ if }\
\varphi\in\left[0,1/2\right],\\\\[5.69054pt] \left(1-\varphi\right)^{2}&\mbox{
if }\ \varphi\in\left(1/2,1\right],\end{array}\right.\
f(\varphi):=\left\\{\begin{array}[]{ll}\varphi^{2}\left(\varphi-1\right)&\mbox{
if }\ \varphi\in\left[0,1/2\right],\\\\[5.69054pt]
\varphi(1-\varphi)^{2}-1/4&\mbox{ if }\
\varphi\in\left(1/2,1\right].\end{array}\right.$

The functions $g$ and $f$ satisfy (g) and (f). Let $D_{1}$ and $D_{2}$ be
defined by

$D_{1}(\varphi):=\left\\{\begin{array}[]{ll}\varphi\left(1/2-\varphi\right)&\mbox{
if }\ \varphi\in\left[0,1/2\right],\\\\[5.69054pt]
-\left(1-\varphi\right)\left(\varphi-1/2\right)&\mbox{ if }\
\varphi\in\left(1/2,1\right],\end{array}\right.\
D_{2}(\varphi)=-D_{1}(\varphi)\ \mbox{ for }\ \varphi\in[0,1].$

With these choices, $D_{1}$ satisfies $\rm(D_{pn})$ with $\alpha=1/2$ and
$D_{2}$ satisfies $\rm(D_{np})$ with $\beta=1/2$. From (2.4) and
$h(0)=\dot{f}(0)=0$ we have $c^{*}_{p,r}\geq 0$. Direct computations show that
the function $z=z(\varphi)$ defined by

$z(\varphi):=\varphi^{2}\left(\varphi-1/2\right),\
\varphi\in\left[0,1/2\right],$ (5.1)

solves $\dot{z}=h-D_{1}g/z$ in $(0,1/2)$, $z<0$ in $(0,1/2)$ and $z(0)=0$.
From the proof of [6, Theorem 2.2] we have $c^{*}_{p,r}\leq 0$. Then,
$c^{*}_{p,r}=0$ follows at once by (2.4). Also, the symmetry of the
coefficients implies that $c^{*}_{n,l}=c^{*}_{p,r}=0$. Thus,

$c^{*}_{pn}=0.$

Observe that we actually need to involve $z$ in (5.1) since none of the
intervals given by (2.4) and (2.6), even in the sharper form involving (2.5),
reduce to the point $\\{0\\}$.

Starting from the formal identity
$\dot{z}(\varphi)=D(\varphi)\varphi^{\prime}{}$, (5.1), we can compute the
profile $\varphi$ of the (unique up to shifts) wavefront associated to
$c^{*}_{pn}$ in the current case. We have:

$\varphi(\xi)=\begin{cases}\frac{1}{4}e^{-\xi}\ &\xi>\log(1/2),\\\\[5.69054pt]
1-e^{\xi}\ &\xi\leq\log(1/2),\end{cases}$

where in the interval $\xi\leq\log(1/2)$ we make use of the symmetry of the
problem.

On the other hand, consider now $D_{2}$, $g$ and $f$ together. We have

$c^{*}_{p,l}\geq\max\left\\{\sup_{(\frac{1}{2},1]}\delta\left(f,1/2\right),\
h(1/2)+2\sqrt{\dot{D_{2}}(1/2)g(1/2)}\right\\}>0,$

since $h(1/2)=-1/4$ and $2\sqrt{\dot{D_{2}}(1/2)g(1/2)}=1/\sqrt{2}$. It
follows necessarily that

$c^{*}_{np}\geq c^{*}_{p,l}>0.$

Then, we proved that replacing $D$, which satisfies $\rm(D_{pn})$, with $-D$,
which then satisfies $\rm(D_{np})$, one can get $c_{np}^{*}>c_{pn}^{*}$.

######  Example 5.3.

In Theorem 2.2 the case $c^{*}_{p,l}\neq c^{*}_{n,r}$ reveals the existence of
unusual non-regular fronts, where $\varphi=\beta$, while in the case
$c^{*}_{p,l}=c^{*}_{n,r}$ this is not possible. We now show an example in
either case.

First, assume that $D$ and $g$ satisfy $\rm(D_{np})$-(g) and are such that
$Dg$ is convex in $(0,\beta)$ and concave in $(\beta,1)$. For instance, we can
take $\beta=\frac{1}{2}$, $D(\varphi)=\varphi-1/2$ and
$g(\varphi)=\varphi(1-\varphi)$. We plainly have

$\sup_{\varphi\in[0,\beta)}\frac{1}{\beta-\varphi}\int_{\varphi}^{\beta}\frac{(Dg)(s)}{s-\beta}\,ds=\dot{D}(\beta)g(\beta)\
\mbox{ and }\
\sup_{\varphi\in(\beta,1]}\frac{1}{\varphi-\beta}\int_{\beta}^{\varphi}\frac{(Dg)(s)}{s-\beta}\,ds=\dot{D}(\beta)g(\beta).$

Suppose also that $f=0$ in $[0,1]$. Under these assumptions, inequalities
(2.13) and (2.14) become indeed equalities:

$c^{*}_{n,r}=c^{*}_{p,l}=c_{np}^{*}=2\sqrt{\dot{D}(\beta)g(\beta)}.$

Second, consider $D\in C^{1}\left[0,1\right]$ such that $D<0$ in $(0,1/2)$ and
$D(\varphi)=\left(\varphi-1/2\right)^{3}$, for $\varphi\in[1/2,1]$; assume
that $g$ satisfies (g) with $g(\varphi)=1-\varphi$, for $\varphi\in[1/2,1]$,
and let $f$ be defined by

$f(\varphi):=\left\\{\begin{array}[]{ll}0&\mbox{ if }\
\varphi\in[0,\frac{1}{2}),\\\
\left(\varphi-\frac{1}{2}\right)^{2}\left(\varphi-\frac{3}{2}\right)&\mbox{ if
}\ \varphi\in[\frac{1}{2},1].\end{array}\right.$

For $1/2\leq\varphi\leq 1$, set

$z(\varphi):=\left(\varphi-\frac{1}{2}\right)^{2}\left(\varphi-1\right).$

Such a $z$ solves (1.4)2 with $\alpha=1/2$ and $c=0$ in $(1/2,1)$. Moreover,
since it also holds $z(1/2)=0$, then $0\geq c^{*}_{p,l}$ (see e.g. the proof
of [6, Theorem 2.2]). The left-hand side of (2.13) implies that
$c^{*}_{p,l}\geq 0$, because $h(1/2)=\dot{f}(1/2)=0$. Thus, $c^{*}_{p,l}=0$.
On the other hand, if we have $h=0$ constantly in $(0,1/2)$, then
$c^{*}_{n,r}>0$ (as already observed in Introduction, for the case of non-
negative $D$ we refer to [6, Remark 6.4] and reference therein; the case when
$D$ is negative is treated similarly). Thus, in this case we have

$c^{*}_{np}=c^{*}_{n,r}>c^{*}_{p,l}.$

Similarly, one can provide examples where
$c^{*}_{np}=c^{*}_{p,l}>c^{*}_{n,r}$.

##  6 The case when $D$ vanishes more than once

In this final section we briefly outline how to cope with the case when $D$
changes sign more than once, focusing on the simplest of these cases. More
precisely we consider one of the following assumptions, see Figure 8:

  * $(\rm D_{pnp})$

$D\in C^{1}[0,1]$, $D>0$ in $(0,\alpha)\cup(\beta,1)$ and $D<0$ in
$(\alpha,\beta)$, with $\alpha<\beta$;

  * $(\rm D_{npn})$

$D\in C^{1}[0,1]$, $D<0$ in $(0,\beta)\cup(\alpha,1)$ and $D>0$ in
$(\beta,\alpha)$, with $\beta<\alpha$.

$\rho$$D$$\alpha$$\beta$$1$$(\rm D_{pnp})$$\rho$$D$$\beta$$\alpha$$1$$(\rm
D_{npn})$ Figure 8: Typical plots of the functions $D$.

About the case $(\rm D_{pnp})$ we refer to [14, 20], where $g$ is,
respectively, monostable and bistable; [3] for the case $(\rm D_{pnp})$ and
monostable $g$, but with a specific quadratic diffusivity $D$. The case $(\rm
D_{npn})$ has never been considered.

######  Theorem 6.1.

Assume (f), $\rm(D_{pnp})$, (g) and $\eqref{Dg}_{1}$. Then, there exists
$c^{*}_{pnp}\in\mathbb{R}$ such that Equation (1.1) admits a (unique up to
space shifts) wavefront, with speed $c$ and profile $\varphi$ satisfying
(1.3), if and only if $c\geq c^{*}_{pnp}$.

######  Theorem 6.2.

Assume (f), $\rm(D_{npn})$, (g) and $\eqref{Dg}_{2}$. Then, there exists
$c^{*}_{npn}\in\mathbb{R}$ such that Equation (1.1) admits a (unique up to
space shifts) wavefront, with speed $c$ and profile $\varphi$ satisfying
(1.3), if and only if $c\geq c^{*}_{npn}$.

In both theorems one can easily deduce more informations on the profiles, as
in Theorems 2.1 and 2.2. We leave these details to the reader.

_Proof of Theorem 6.1._ We divide our problem in three sub-problems
corresponding to the three connected components of $\left\\{D\neq 0\right\\}$.
To the intervals $(0,\alpha)$ and $(\beta,1)$ we apply Propositions 3.1 and
4.1, respectively. Indeed, the results of Sections 3 and 4 hold under lighter
assumptions, involving only that part of the conditions corresponding to the
interval under consideration.

Let ${c}_{p,r}^{*}$ and ${c}_{p,l}^{*}$ be the thresholds appearing in
Propositions 3.1 and 4.1, respectively. Hence, associated to the same speed
$c$, both a semi-wavefront connecting $\alpha$ and $0$, with profile
$\varphi_{\alpha,0}$ satisfying (3.10) and a semi-wavefront connecting $\beta$
to $1$, whose profile is $\varphi_{1,\beta}$ satisfying $\eqref{e:threshold
cm-dpde condition}_{1}$, are given, if and only if
$c\geq\max\\{{c}^{*}_{p,r},{c}_{p,l}^{*}\\}$.

We claim that there exists $c^{*}_{\alpha\beta}\in\mathbb{R}$ such that the
following holds: if and only if $c\geq c^{*}_{\alpha\beta}$, Equation (1.1)
admits a strict TW, connecting $\beta$ to $\alpha$, whose non-increasing
profile $\varphi_{\beta,\alpha}\in(\alpha,\beta)$ is defined in some interval
$(\xi_{\beta},\xi_{\alpha})$, where $-\infty<\xi_{\beta}<\xi_{\alpha}<+\infty$
are such that $\varphi_{\beta,\alpha}(\xi_{\beta}^{+})=\beta\ \mbox{ and }\
\varphi_{\beta,\alpha}(\xi_{\alpha}^{-})=\alpha$ and also

$\left(D\left(\varphi_{\beta,\alpha}\right)\varphi_{\beta,\alpha}^{\prime}{}\right)\left(\xi_{\beta}^{+}\right)=0=\left(D\left(\varphi_{\beta,\alpha}\right)\varphi_{\beta,\alpha}^{\prime}{}\right)\left(\xi_{\alpha}^{-}\right).$

Then by gluing together $\varphi_{1,\beta}$, $\varphi_{\beta,\alpha}$ and
$\varphi_{\alpha,0}$ (modulo shifts) we obtain the front of (1.1) satisfying
(1.3), in virtue of (6), as in the proof of Theorem 2.1. Clearly, we define

$c^{*}_{pnp}=\max\left\\{c^{*}_{p,r},c_{p,l}^{*},c^{*}_{\alpha\beta}\right\\}.$

This proves the if part of Theorem 6.1. Viceversa, assume that a wavefront
$\varphi$ of (1.1), associated to some speed $c$, is given. As in the proof of
the only if parts of Theorems 2.1 and 2.2, $\varphi$ is decomposed into a
semi-wavefront which connects $1$ to $\beta$ whose profile satisfies (2.12), a
strict TW connecting $\beta$ to $\alpha$ and a strict semi-wavefront
connecting $\alpha$ to $0$. By Propositions 3.1 and 3.2, the only if part of
our claim implies $c\geq c^{*}_{pnp}$. Thus, Theorem 6.1 follows from the
claim. We refer to Figure 9.

$\xi$$\varphi$$1$$\beta$$\xi_{\beta}$$\alpha$$\xi_{\alpha}$$\varphi_{1,\beta}$$\varphi_{\alpha,\beta}$$\varphi_{\alpha,0}$
Figure 9: The pasting of the profiles.

We prove the claim. Let $\bar{D}$, $\bar{g}$, $\bar{f}$ and $\bar{\alpha}$ be
as in (3.7) and let $\bar{h}$ be the derivative of $\bar{f}$, which gives
$\bar{h}(\varphi)=h(1-\varphi)$. Set $\bar{\beta}=1-\beta$. Let $c$ be a real
value and consider the following equation in the unknown
$\psi=\psi(\xi)\in[\bar{\beta},\bar{\alpha}]$:

$\left(\bar{D}(\psi)\psi^{\prime}{}\right)^{\prime}{}+\left(c-\bar{h}(\psi)\right)\psi^{\prime}{}+\bar{g}(\psi)=0.$
(6.1)

We are interested in a non-increasing $\psi$ satisfying (6.1) in some interval
$(\xi_{\bar{\alpha}},\xi_{\bar{\beta}})$, with
$-\infty<\xi_{\bar{\alpha}}<\xi_{\bar{\beta}}<+\infty$, such that
$\psi({\xi_{\bar{\alpha}}}^{+})=\bar{\alpha}$,
$\psi({\xi_{\bar{\beta}}}^{-})=\bar{\beta}$ and

$\left(\bar{D}(\psi)\psi^{\prime}{}\right)({\xi_{\bar{\alpha}}}^{+})=0=\left(\bar{D}(\psi)\psi^{\prime}{}\right)({\xi_{\bar{\beta}}}^{-}).$

By using a standard argument (see e.g. [24]) based on the monotonicity of
$\psi$ and the definition of $z(\psi):=\bar{D}(\psi)\psi^{\prime}{}$, the
existence of a $\psi$ as above, but with not necessarily finite
$\xi_{\bar{\alpha}}$ and $\xi_{\bar{\beta}}$, is equivalent to the solvability
of the problem

$\begin{cases}\dot{z}(\psi)=\bar{h}(\psi)-c-\frac{\bar{D}(\psi)\bar{g}(\psi)}{z(\psi)},\
&\psi\in(\bar{\beta},\bar{\alpha}),\\\ z(\psi)<0\
&\psi\in(\bar{\beta},\bar{\alpha}),\\\
z(\bar{\beta})=z(\bar{\alpha})=0.\end{cases}$

Since $\bar{D}>0$ in $(\bar{\beta},\bar{\alpha})$,
$\bar{D}(\bar{\beta})=\bar{D}(\bar{\alpha})=0$, $\bar{g}>0$ in
$[\bar{\beta},\bar{\alpha}]$, by applying [6, Proposition 4.2], such a problem
is solvable if and only if $c\geq c^{*}$, for some $c^{*}\in\mathbb{R}$. Also,
since (by arguing as in [6, Lemma 9.1] and [10, proof of Theorem 2.5] where
analogous assumptions were considered) we have

$\lim_{\psi\to\bar{\alpha}^{-}}\frac{\bar{D}(\psi)}{z(\psi)}\in\mathbb{R}\
\mbox{ and }\
\lim_{\psi\to\bar{\beta}^{+}}\frac{\bar{D}(\psi)}{z(\psi)}\in\mathbb{R},$

then $\xi_{\bar{\alpha}}$ and $\xi_{\bar{\beta}}$ are finite because $\psi$ is
bounded and $\psi^{\prime}{}(\xi)<0$ for $\xi$ which tends to either
${\xi_{\bar{\alpha}}}^{+}$ or ${\xi_{\bar{\beta}}}^{-}$. Finally, setting
$c_{\alpha\beta}^{*}:=c^{*}$ concludes the proof of the claim since the
desired $\varphi_{\beta,\alpha}$ exists if and only if there exists $\psi$, by
$\varphi_{\beta,\alpha}(\xi):=1-\psi(-\xi)$.

$\square$

######  Remark 6.1.

The analog of Theorem 6.1 when $f=0$ was given in the first part of [14,
Theorem 4.2] with slightly different notation. We point out that in [14]
wavefronts are necessarily classical at $1$ (called classical or sharp of type
(I), there). Instead, when $D(1)=0$ and $c<h(1)$ the wavefronts of Theorem 6.1
can be sharp at $1$, too, by applying Part $(iii)$ of Proposition 4.1 to
$\varphi_{1,\beta}$ of the proof of Theorem 6.1. Note, $c<h(1)$ cannot happen
if $f=0$.

We also observe that, as in [14, Theorem 4.2], wavefronts are sharp at $0$
only if $D(0)=0$ and $c=c^{*}_{pnp}$. Moreover, if $D(0)=0$ and
$c=c^{*}_{pnp}$, generalizing [14, (4.1)], it is possible to provide a
sufficient condition on $f$ and $Dg$ in order to make wavefronts of Theorem
6.1 always classical at $0$.

_Proof of Theorem 6.2._ We proceed in the spirit of the proof of Theorem 6.1.
We consider separately the intervals where $D$ has constant sign. From
Proposition 3.2 we deduce that a semi-wavefront connecting $1$ to $\alpha$,
with speed $c$ and profile satisfying (3.5) with $s=0$, exists if and only if
$c\geq c^{*}_{n,l}$. Proposition 4.2 implies that there exists a semi-
wavefront connecting $\beta$ to $0$ with speed $c$ and profile satisfying
$\eqref{e:threshold cm-dpde condition2}_{1}$ if and only if $c\geq
c^{*}_{n,r}$. About the interval $(\beta,\alpha)$, we argue as in the proof of
Theorem 6.1, but considering Equation (1.2), directly, rather than Equation
(6.1). We have that a strict TW connecting $\alpha$ to $\beta$ with speed $c$
and profile
$\varphi_{\alpha,\beta}:(\xi_{\alpha},\xi_{\beta})\to(\beta,\alpha)$
satisfying $\varphi_{\alpha,\beta}(\xi_{\alpha}^{+})=\alpha$,
$\varphi_{\alpha,\beta}(\xi_{\beta}^{-})=\beta$ and such that
$\left(D(\varphi_{\alpha,\beta})\varphi_{\alpha,\beta}\right)(\xi)$ tends to
$0$ if either $\xi\to\xi_{\alpha}^{+}$ and $\xi\to\xi_{\beta}^{-}$, exists if
and only if $c\geq c^{*}_{\beta\alpha}$. To conclude the proof, we set
$c^{*}_{npn}:=\max\\{c_{n,l}^{*},c^{*}_{n,r},c^{*}_{\beta\alpha}\\}$.
$\square$

######  Remark 6.2.

Similarly to Remark 6.1, we can deduce information on the regularity of
wavefronts given in Theorem 6.2.

## Acknowledgments

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM) and acknowledge financial support from this institution.

## References

  * [1] D. G. Aronson, H. F. Weinberger.  Multidimensional nonlinear diffusion arising in population genetics.  Adv. in Math., 30(1):33–76, 1978.
  * [2] L. Bao, Z. Zhou.  Traveling wave in backward and forward parabolic equations from population dynamics.  Discrete Contin. Dyn. Syst. Ser. B, 19(6):1507–1522, 2014.
  * [3] L. Bao, Z. Zhou.  Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term.  Discrete Contin. Dyn. Syst. Ser. S, 10(3):395–412, 2017.
  * [4] N. Bellomo, M. Delitala, V. Coscia.  On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling.  Math. Models Methods Appl. Sci., 12(12):1801–1843, 2002.
  * [5] N. Bellomo, C. Dogbe.  On the modeling of traffic and crowds: a survey of models, speculations, and perspectives.  SIAM Rev., 53(3):409–463, 2011.
  * [6] D. Berti, A. Corli, L. Malaguti.  Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection reaction equations.  Electron. J. Qual. Theory Differ. Equ., Paper No. 66, 34, 2020.
  * [7] L. Bruno, A. Tosin, P. Tricerri, F. Venuti.  Non-local first-order modelling of crowd dynamics: a multidimensional framework with applications.  Appl. Math. Model., 35(1):426–445, 2011.
  * [8] A. Corli, L. di Ruvo, L. Malaguti.  Sharp profiles in models of collective movements.  NoDEA Nonlinear Differential Equations Appl., 24(4):Paper No. 40, 31 pp., 2017.
  * [9] A. Corli, L. di Ruvo, L. Malaguti, M. D. Rosini.  Traveling waves for degenerate diffusive equations on networks.  Netw. Heterog. Media, 12(3):339–370, 2017.
  * [10] A. Corli, L. Malaguti.  Semi-wavefront solutions in models of collective movements with density-dependent diffusivity.  Dyn. Partial Differ. Equ., 13(4):297–331, 2016.
  * [11] A. Corli, L. Malaguti.  Viscous profiles in models of collective movement with negative diffusivity.  Z. Angew. Math. Phys., 70(2):Art. 47, 22 pp., 2019.
  * [12] A. Corli, L. Malaguti.  Wavefronts in traffic flows and crowds dynamics.  Lecture Notes INDAM, to appear, 2020.
  * [13] D. A. DiCarlo, R. Juanes, L. Tara, T. P. Witelski.  Nonmonotonic traveling wave solutions of infiltration into porous media.  Water Resources Res., 44:1–12, 2008.
  * [14] L. Ferracuti, C. Marcelli, F. Papalini.  Travelling waves in some reaction-diffusion-aggregation models.  Adv. Dyn. Syst. Appl., 4(1):19–33, 2009.
  * [15] M. Garavello, K. Han, B. Piccoli.  Models for vehicular traffic on networks.  American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016\.
  * [16] M. Garavello, B. Piccoli.  Traffic flow on networks.  American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006\.
  * [17] B. H. Gilding, R. Kersner.  Travelling waves in nonlinear diffusion-convection reaction.  Birkhäuser Verlag, Basel, 2004.
  * [18] D. Horstmann, K. J. Painter, H. G. Othmer.  Aggregation under local reinforcement: from lattice to continuum.  European J. Appl. Math., 15(5):546–576, 2004.
  * [19] B. S. Kerner, V. V. Osipov.  Autosolitons.  Kluwer Academic Publishers Group, Dordrecht, 1994.
  * [20] M. Kuzmin, S. Ruggerini.  Front propagation in diffusion-aggregation models with bi-stable reaction.  Discrete Contin. Dyn. Syst. Ser. B, 16(3):819–833, 2011.
  * [21] P. K. Maini, L. Malaguti, C. Marcelli, S. Matucci.  Diffusion-aggregation processes with mono-stable reaction terms.  Discrete Contin. Dyn. Syst. Ser. B, 6(5):1175–1189, 2006.
  * [22] P. K. Maini, L. Malaguti, C. Marcelli, S. Matucci.  Aggregative movement and front propagation for bi-stable population models.  Math. Models Methods Appl. Sci., 17(9):1351–1368, 2007.
  * [23] L. Malaguti, C. Marcelli.  Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations.  J. Differential Equations, 195(2):471–496, 2003.
  * [24] L. Malaguti, C. Marcelli.  Finite speed of propagation in monostable degenerate reaction-diffusion-convection equations.  Adv. Nonlinear Stud., 5(2):223–252, 2005.
  * [25] C. Marcelli, F. Papalini.  A new estimate of the minimal wave speed for travelling fronts in reaction-diffusion-convection equations.  Electron. J. Qual. Theory Differ. Equ., 10:1–13, 2018.
  * [26] P. Nelson.  Synchronized traffic flow from a modified Lighthill-Whitham model.  Phys. Review E, 61:R6052–R6055, 2000.
  * [27] V. Padrón.  Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation.  Trans. Amer. Math. Soc., 356(7):2739–2756, 2004.
  * [28] M. D. Rosini.  Macroscopic models for vehicular flows and crowd dynamics: theory and applications.  Springer, Heidelberg, 2013.