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 Open Access
Monotone traveling waves for reactiondiffusion equations involving the curvature operator
 Maurizio Garrione^{1} and
 Luís Sanchez^{2}Email author
https://doi.org/10.1186/s136610150303y
© Garrione and Sanchez; licensee Springer. 2015
 Received: 29 August 2014
 Accepted: 5 February 2015
 Published: 28 February 2015
Abstract
We study the existence of monotone traveling waves \(u(t, x)=u(x+ct)\), connecting two equilibria, for the reactiondiffusion PDE \(u_{t} = (\frac{u_{x}}{\sqrt{1+u_{x}^{2}}} )_{x} + f(u)\). Assuming different forms for the reaction term \(f(u)\) (among which we have the socalled types A, B, and C), we show that, concerning the admissible speeds, the situation presents both similarities and differences with respect to the classical case. We use a first order model obtained after a suitable change of variables. The model contains a singularity and therefore has some features which are not present in the case of linear diffusion. The technique used involves essentially shooting arguments and lower and upper solutions. Some numerical simulations are provided in order to better understand the features of the model.
Keywords
 traveling waves
 lower and upper solutions
 mean curvature operator
MSC
 34C37
 35K57
 34B18
1 Introduction
The process described by (2) is a reactiondiffusion one, where the spatial diffusion is linear. The reaction term may have several forms, but we will always assume that it satisfies \(f(0)=0=f(1)\), meaning that no reaction is present if the gene is completely spread or not spread at all into the population. Accordingly, we will be interested in monotone traveling waves connecting 0 (at −∞) and 1 (at +∞) and taking values strictly included between them, i.e., using the terminology in [2], the socalled fronttype solutions. The existence of such connections turns out to be important since, for instance in the case of linear diffusion, under certain assumptions on the initial datum the solutions to the PDE (2) converge, for \(t \to+\infty\), to suitable combinations of traveling wavetype solutions (see, e.g., [3, 4]).
It is well known that in the classical case, namely for equation (1), different phenomena may arise, depending on the shape of f. In particular, if \(f(u) > 0\) for every \(u \in\,]0, 1[ \), then the admissible speeds form an unbounded interval \([c^{*}, +\infty[ \), where the value \(c^{*}\) takes the name of critical speed. On the contrary, it suffices to have \(f(u) \leq0\) in a neighborhood of 0 to force the admissible speed, if any, to be unique, using a monotonicity argument (see Lemma 3.8 below in our quite general context; we will use the name critical speed also in this case). In any case, it is immediate to see that the existence of a positive admissible speed is possible only if \(f(u) > 0\) in a neighborhood of \(u=1\).
We will study the features of the set of the admissible speeds for (4) for different classes of reaction terms, and provide, in some cases, a way of computing the critical speed. We shall see that our problem presents both similarities and differences with respect to the FKPP equation. The most significant feature, in particular, is the appearance here of a singularity (see Section 2) which makes the model more delicate to deal with than in the classical case, requiring some restrictions which are peculiar of (4) in order to find a regular solution. Indeed, elementary arguments show that a classical solution not always exists already for \(c=0\), and one should in principle admit discontinuous solutions which satisfy the equation in a suitable weak sense, as was observed, e.g., in [8, 9]. We remark that recent efforts have been devoted to the study of periodic and Dirichlet boundary value problems associated with (4), in the case when \(c=0\), leading to some interesting results (see, e.g., the works [7, 10–14]). Moreover, it has been recently seen that, contrary to the classical case, models similar to the one taken into account in the present paper may display the existence of discontinuous traveling wave solutions. We refer the reader to the works [15, 16], in presence of more general operators, in relation with the socalled porous medium equation. Although we will not deal with discontinuous solutions, in Section 5 we shall refer to discontinuous steady states.
The plan of the paper is as follows. In Section 2, we will explicitly obtain the first order reduction for (4) (for a similar procedure, see for instance [17]) and state some elementary lemmas which will find application from there on. Section 3 is devoted to the study of the admissible speeds. Referring to the sign of \(f(u)\) in \([0, 1]\), we will mainly take into account the following three model cases: f always positive (‘type A’), f positive in an open left neighborhood of 1 and 0 else (‘type B’), f positive in a left neighborhood of 1 and negative in the complementary open neighborhood of 0 (‘type C’). This terminology was introduced in [18], but these shapes of the reaction term were already considered in [19]. We will see that the picture for type C functions can be extended to a quite general setting. In Section 4 we will briefly discuss a different perspective on equation (4), interpreting it as a quasilinear version of a damped pendulum when \(f(u)=B \sin u\). In this context, the critical speed assumes the meaning of a critical damping and we will be able to recover, with some differences, the usual picture already present in the classical case, in the case that an additional torque is applied to the pendulum as well. Finally, in Section 5 we will discuss the relationships between the appearance of discontinuous steady states (i.e., discontinuous solutions to (4) for \(c=0\)) and the nonexistence of right moving traveling waves, inspired by the work [9].
The figures appearing in the paper have been drawn using the open source software available at the web address [20]; we warn the reader that the numerical method used for the simulations, also in view of the presence of a singularity in the considered equation, is quite sensitive to the initial data and may display some problems for certain values of the parameters.
2 Traveling waves: the first order model
Remark 2.1
Remark 2.2
 (i)
there exists \(k>0\) such that \(f(u)\le ku\) for every \(u\in[0,1]\),
 (ii)
there exists \(l>0\) such that \(f(u)\le l(1u)\) for every \(u\in[0,1]\)
Thus problem (7) embodies (5). When we refer to admissible speeds, we have (7) in mind; anyway the existence of solutions to (7) in practice depends on condition (i) (see Proposition 3.2 below).
Remark 2.3
2.1 Preliminary lemmas
In this subsection, we present some preliminary lemmas for the Cauchy problems (8) and (9). They mainly follow from elementary and wellknown facts about subsolutions and supersolutions, but for the sake of simplicity we prefer to prove them explicitly in our particular case.
The first lemma concerns some general considerations about the solution \(y_{c, f}^{}(u)\), for c, f fixed.
Lemma 2.4
Proof
We now observe that \(z(u) \equiv0\) is a lower solution for (8) in \([u_{0}, 1]\), so that \(y_{c, f}^{}(u) \geq0\) for every \(u \in[u_{0}, 1]\). On the other hand, if there existed \(u_{1} \in\,]u_{0}, 1]\) with \(y_{c, f}^{}(u_{1})=0\), then the differential equation in (8) would yield \((y_{c, f}^{})'(u_{1}) < 0\), implying that \(y_{c, f}^{}\) is decreasing in a neighborhood of \(u_{1}\), a contradiction. Hence, (11) is proved. □
With the following two lemmas, we now take into account some monotonicity issues for \(y_{c, f}^{}(u)\) with respect to c and f.
Lemma 2.5
Proof
On the other hand, if there existed \(u' \in\,]u_{0}, 1[ \) such that \(0 < y_{c_{2}, f}^{}(u')=y_{c_{1}, f}^{}(u')\) (recall that, in \(]u_{0}, 1[ \), the solution to (8) is strictly positive), then, since \(c_{1} < c_{2}\), using the differential equation in (8) we would have \((y_{c_{1}, f}^{})'(u)  (y_{c_{2}, f}^{})'(u) < 0\) in a neighborhood of \(u'\), which is impossible because \(y_{c_{1}, f}^{}(u)  y_{c_{2}, f}^{}(u) \geq0\) for every \(u \in\,]u_{0}, 1[ \). This gives the conclusion for the case \(y_{c_{2}, f}^{}(u_{0}) > 0\), as well. □
If \(u_{0}=0\), this yields in particular a strict monotonicity of the solutions, with respect to the constant c, along the open interval \(]0, 1[ \).
Lemma 2.6
Proof
On the other hand, if there existed \(v \in[\hat{u}, \tilde{u}[ \) such that \(y_{c, f_{1}}^{}(v)=y_{c, f_{2}}^{}(v)\), we would have \((y_{c, f_{1}}^{})'(v) < (y_{c, f_{2}}^{})'(v) \) which is impossible. □
We finally analyze the behavior of the maximum of \(y_{c, f}^{}(u)\), in dependence of a small c.
Lemma 2.7
Proof
In the case when \(\int_{u_{0}}^{1} f(u)\,du < 1\), we argue analogously to obtain that \(\mathcal{M} \geq\int_{u_{0}}^{1} f(u)\,du\), assuming at the beginning that \(y_{c_{n}, f}^{}(t_{n}) \leq\int_{u_{0}}^{1} f(u)\,du  \delta \). For the reverse inequality, we first observe that \(\mathcal{M} < 1\) since \(y_{c_{n}, f}^{}(t_{n}) < \max_{u \in[u_{0}, 1]}y_{0, f}^{}(u) < 1\) (in view of Lemma 2.5) and then reason as in (12). □
The last lemma concerns the solutions shot forward from 0.
Lemma 2.8
Proof
3 Traveling waves: the admissible speeds
We now turn to the search for solutions to (7), under some classical assumptions on f.
3.1 f of type A
Under some mild assumptions on f, we first give a lower bound for the admissible speeds.
Lemma 3.1
Let \(k:=f'(0)\) exist and let \(y(u)\) be a solution to (7), with \(y(u) > 0\) in a right neighborhood of 0. Then, \(c \geq2 \sqrt{f'(0)}\).
Proof
Given a reaction term f belonging to a quite large subset of \(\mathcal {A}\), the following proposition relates the corresponding set of the admissible speeds to some bound on f.
Proposition 3.2
Proof
Let us fix \(c \geq2\sqrt{M}\) and consider the (backward) Cauchy problem (8), which has the unique global, positive solution \(y_{c, f}^{}(u)\) in \(]0, 1[ \). If \(y_{c, f}^{}(0)=0\) we are done, so we assume \(0 < y_{c, f}^{}(0)\) (<1).
If \(y_{c, f}^{+}(u)\) blows up at a finite time \(u_{\infty}\in\,]0, 1[ \), then there exists \(\hat{u} \in\,]0, u_{\infty}[ \) such that \(y_{c, f}^{+}(\hat{u}) > \max_{u \in[0, 1]} y_{c, f}^{}(u)\). Since \(y_{c, f}^{+}\) is of class \(C^{1}\) and the graph of \(y_{c, f}^{}(u)\) disconnects the square \([0, 1]^{2}\) in the \((u, y)\)plane, this implies that \(y_{c, f}^{+}(\bar{u})=y_{c, f}^{}(\bar{u})\) for some \(\bar{u} \in\,]0, \hat {u}[ \), which is impossible by uniqueness.
If \(y_{c, f}^{+}(u)\) is globally defined in \([0, 1]\), the same happens for \(z(u)\); then, recalling (14), \(y_{c, f}^{+}(u)\) is always strictly positive in \(]0, 1[ \). Thus, either \(y_{c, f}^{+}(1) = 0\) and we are done, or we can argue similarly as in the previous case to deduce that \(y_{c, f}^{+}(\bar{u})=y_{c, f}^{}(\bar{u})\) at some point \(\bar{u} \in\,]0, 1[ \), which is impossible by uniqueness. This concludes the proof. □
It is worth observing that, differently from the classical and the Minkowski case, in our setting the family of functions giving the bound on f may be unbounded on \([0,1]\). We explicitly remark, moreover, that in order to perform the previous proof it is essential that \(y_{c, f}^{}(u)\) disconnects the square \([0, 1]^{2}\) in the \((u, y)\)plane, thus forming a kind of barrier which has to be crossed by the forward solution \(y_{c, f}^{+}(u)\) (also this point is different from the classical case).
We now focus on some characterizations of the socalled critical speed  i.e., the value \(c^{*}\) such that every \(c \geq c^{*}\) is admissible  for functions of type A. In the previous example (15), for instance, we have \(c^{*}=2 \sqrt{M}\).
Proposition 3.3
To characterize the critical speed, we now need the following analog of [21, Lemma 4.1].
Proposition 3.4
Proposition 3.5

if \(c=c^{*}\), then$$E(y)'(0)=\lambda^{+}(c); $$

if \(c > c^{*}\), then$$E(y)'(0)=\lambda^{}(c). $$
The proof goes as the one for [21, Proposition 4.2], with minor changes.
 (1)By analogy with the classical Fisher equation and [21, Example 1], letwith \(\beta> 0\). An explicit computation shows that$$f(u)=\frac{u(1u)}{\sqrt{1\beta u^{2} (1\sqrt{u})^{2}}}, $$is a solution for \(\beta=2/3\) and \(c=5/\sqrt{6}\). In this case, \(f'(0)=1\); since$$y(u)=1\sqrt{1\beta u^{2} (1\sqrt{u})^{2}} $$in view of Lemma 3.1 and Proposition 3.2 we conclude that \(c^{*}=2\). Hence, the solution found does not correspond to a critical speed, as it is possible to see also using Proposition 3.5:$$f(u) \leq\frac{u}{\sqrt{1u^{2}}}, $$$$E(y)'(0)=\frac{d}{du} \sqrt{\beta u^{2} (1 \sqrt{u})^{2}} \vert_{u=0} = \sqrt{\beta} = \sqrt{2/3} = \lambda_{}(5/\sqrt{6}). $$
 (2)By analogy with the Zeldovich equation, ifwe find a solution of the form$$f(u)=\frac{u^{2}(1u)}{\sqrt{1\beta(uu^{2})^{2}}}, $$for \(\beta=1/2\) and \(c=1/\sqrt{2}\). In this case, \(f'(0)=0\), and since$$y(u)=1\sqrt{1\beta\bigl(uu^{2}\bigr)^{2}} $$we conclude that \(c=1/\sqrt{2}\) is the critical speed. Notice that, in this case, the bound$$E(y)'(0)=\frac{d}{du} \sqrt{\beta} \bigl(uu^{2} \bigr) \vert_{u=0} = \sqrt{\beta} = 1/\sqrt{2} = \lambda _{+}(1/\sqrt{2}), $$for every \(u \in[0, 1]\) would provide only a rougher estimate of \(c^{*}\) via Proposition 3.2.$$f(u) \leq\frac{\beta u}{\sqrt{1\beta u^{2}}} $$
3.2 f of type B
Proposition 3.6
Let \(f \in\mathcal{B}\). Then there exists \(0 < c^{*} < 1/\theta\) such that (7) has a solution if and only if \(c=c^{*}\).
Proof
Similar to the classical case, the lower bound for the admissible speeds of a function f of type A can be reconstructed starting from a sequence of increasing approximations of type B. Precisely, we have the following proposition.
Proposition 3.7
Before going into the details of the proof, let us explicitly state that, if \(\theta_{n}>0\) is the greatest zero of \(f_{n} \) on \([0, 1[\), we assume that \(\theta_{n}\) is strictly decreasing and \(f_{n} \le f_{n+1}\).
Proof
The proof is similar to the one for the classical case. We divide it in four steps. To fix the notation, denote by \(y_{c, f}\), \(y_{c, f}^{}\), \(y_{c, f}^{+}\) the solutions to (7), (8), (9), respectively; for briefness, we write \(y_{n}\), \(y_{n}^{+}\), \(y_{n}^{}\) to denote \(y_{c_{n}, f_{n}}\), \(y_{c_{n}, f_{n}}^{+}\), \(y_{c_{n}, f_{n}}^{}\).
Step 1: \(c^{*}(f_{n})\) is increasing.
Step 2: for every n, \(c^{*}(f_{n}) \leq c^{*}(f)\).
From Step 1 and Step 2, it follows that \(c_{n}\) converges.
Step 3: there exists \(0 < H < 1\) such that \(\Vert y_{n} \Vert _{\infty}\leq H\) for every n.
Step 4: \(c^{*}(f_{n}) \nearrow c^{*}(f)\).
3.3 f of type C
Lemma 3.8
The proof is analogous to the one of Lemma 2.5, and for this reason we will omit it.
In the classical case, the behavior of functions of type B or type C with respect to the admissible speeds does not change. We are now going to see that, for the curvature case, the situation is somehow different, due to the fact that the ‘barrier’ \(\{y=1\}\) has to be avoided in order to obtain a classical solution to our problem.
Proposition 3.9
Proof
 (1)We first show that, if (19) is violated, then every solution shot from the left blows up at a finite time. This is an easy consequence of the fact that, for every \(c \geq0\),so that$$y' = \frac{c \sqrt{y(2y)}}{1y}  f(u) \geqf(u), $$is a subsolution from the left. Lemma 3.8 now ensures that$$z(u)=\int_{0}^{u} f(s)\,ds $$and since \(\int_{0}^{1} f^{}(u)\,du \geq1\), this shows that \(y_{c, f}^{+}(u)\) reaches the value 1 in a finite time, so that no regular solutions to (7) will exist.$$y_{c, f}^{+}(u) > z(u) \quad \text{for every } u \in\,]0, \theta[ \text{ s.t. } z(u) < 1, $$
 (2)Second, we show that if (18) is violated, then no positive speeds will be admissible for our problem. Without loss of generality, we can assume that \(\int_{0}^{1} f^{}(u)\,du < 1\). Set \(c=0\) and consider \(y_{0, f}^{+}(\theta)\), \(y_{0, f}^{}(\theta)\). It is clear that, since \(\int_{0}^{1} f^{+}(u)\,du \leq\int_{0}^{1} f^{}(u)\,du\),If we now increase c, from \(y_{0, f}^{}(\theta) > 0\) we infer, using Lemma 2.5, that for \(c > 0\) small we have$$y_{0, f}^{+}(\theta) = \int_{0}^{\theta}f(u) \,du \geq y_{0, f}^{}(\theta)=\int_{\theta}^{1} f(u)\,du. $$Hence, using again Lemma 2.5, the two curves cannot meet for any value of \(c > 0\) and our problem does not have a solution.$$y_{c, f}^{+}(\theta) \geq y_{0, f}^{+}(\theta) \geq y_{0, f}^{}(\theta) > y_{c, f}^{}(\theta). $$
 (3)
We finally show that if (18) and (19) hold, then there exists a unique admissible speed for f. To this aim, we focus again on \(y_{c, f}^{+}(\theta)\) and \(y_{c, f}^{}(\theta)\).
Case 1. \(\int_{\theta}^{1}f(u)\,du<1\). Setting \(\eta= \int_{0}^{1}f(u)\,du\), from Lemma 2.4, together with the observation after Lemma 3.1, \(y_{c, f}^{}(\theta) \) and \(y_{c, f}^{+}(\theta) \) are close respectively tofor \(c > 0\) small. Since for c large enough \(y_{c, f}^{+}(\theta) \) reaches the value 1, the result follows from continuity and monotonicity.$$\int_{\theta}^{1} f(u)\,du = \int_{0}^{\theta}f(u) \,du+\eta\quad \text{and}\quad \int_{0}^{\theta}f(u)\,du $$Case 2. \(\int_{\theta}^{1}f(u)\,du\ge1\). Let \(1>\xi_{0}\ge\theta \) be such that \(\int_{\xi_{0}}^{1}f(u)\,du=1\).
In order to apply the continuity argument, we claim that given any number \(a<1\) , there exists \(\gamma>0\) such that the inequality \(y_{c, f}^{}(\theta) \ge a\) holds whenever \(0< c<\gamma\). In fact, choose ε such thatBy continuous dependence, there exists \(c_{0}>0\) such that if \(0< c<c_{0}\) then \(y_{c, f}^{}(\xi) \ge1\sqrt{\varepsilon}\) for some \(\xi>\xi _{0}\). The solution to$$1\sqrt{\varepsilon}>a. $$is \(w_{c}(s)=1\sqrt{1(kcs)^{2}}\) with \(k=c\xi+\sqrt{1\varepsilon}\). Hence$$w'=cR(w),\qquad w(\xi)=1\sqrt{\varepsilon}$$so that \(\liminf_{c\to0}w_{c}(\theta)\ge1\sqrt{\varepsilon}\). On the other hand, since \(f>0\) on \(]\theta,1[\), the graph of the solution \(y_{c, f}^{}\) stays above the graph of \(w_{c}\) in \(]\theta,\xi[\). Hence the claim follows and the proof is complete.$$w_{c}(\theta)=1\sqrt{1\bigl(c(\xi\theta)+\sqrt{1\varepsilon} \bigr)^{2}}, $$
The picture highlighted in Proposition 3.9 is preserved up to changes in f on a neighborhood of θ, as shown by the following result of perturbative type.
Proposition 3.10
Proof
Indeed, the problem is that, on varying of c, \(y_{c, f}^{}\) could vanish in \(]0, 1[ \) before the equality \(y_{c, f}^{+}(u_{0}') = y_{c, f}^{}(u_{0}')\) has been obtained. Of course, the situation is determined by the interplay between the intervals of positivity and negativity of f and the corresponding values of the integral function: the farther it is from 0, the better. In the next paragraph we will briefly go through a possible result in presence of more general reaction terms; as will easily be realized, a general analysis is far from being feasible.
3.4 Remarks on more general reaction terms
We briefly review a couple of situations for more general reaction terms.
 (a)for every \(j=1, \ldots,r\).$$ \int_{I_{j}} f^{}(u)\,du < 1 $$(21)Indeed, if it were not the case, the solution \(y_{c, f}^{}(u)\) would vanish in \(]0, 1[ \), since, assuming that \(\int_{I_{j'}} f^{}(u)\,du \geq1\) for a certain \(j'\), the solution towould be a (backward) supersolution for our problem, vanishing in a certain point of \(I_{j'}\).$$\begin{cases} z'=f(t), \\ z(\theta_{j'})=y_{c, f}^{}(\theta_{j'}) < 1 \end{cases} $$
Condition (21) is necessary also in order to avoid that the solutions \(y_{c, f}^{+}\) shot forward touch the barrier \(y=1\) (both in the case f is positive in a neighborhood of 0, shooting from \(\theta_{1}\), and if f is negative in a neighborhood of 0, shooting from \(\theta_{0}\)). According to the terminology of Section 5 below, we are avoiding the appearance of discontinuous steady states.
 (b)In particular, a necessary condition is that \(\int_{0}^{1} f(u)\,du > 0\).$$F_{1}(u) > 0\quad \text{for every } u \in[0, 1]. $$
Proposition 3.11
Proof

for small values of c, we have \(y_{c, f}^{}(\alpha) > 0\), \(y_{c, f}^{}(0)> 0\);

for large values of c, the situation, restricted to \([\alpha, 1]\), is similar to the type A case, so that we have a heteroclinic connection between 1 and α.
Our equation seems to be quite sensitive to large values of the parameter, so that the reliability of the simulations should be considered with caution.
The variety of configurations which may appear suggests that a general analysis is far from being feasible without the expense of heavy assumptions.
4 A mechanical interpretation: a quasilinear damped pendulum
In this section we give a motivation to the second order equation (5) and the problem (7) that does not come from (2).
A numerical approach to the analogous classical pendulum equation was proposed in [27].
Now consider the dynamics for smaller values of the torque (\(D < B\)), taking into account the existence of solutions going monotonically to an equilibrium and (without loss of generality because of periodicity) restricting our interest to the intervals \([u_{S}, u_{S}+2\pi]\) or \([u_{U}, u_{U}+2\pi]\).

if \(A \geq A^{*}\), then the solution is an increasing connection between π and 2π;

if \(A < A^{*}\), then the solution oscillates around the stable equilibrium 2π, since by strict monotonicity it vanishes before reaching 3π (for \(c=0\) one has \(y_{0, f}(3\pi)=0\)).
Proposition 4.1
discriminating between different kinds of motions. In the preceding considerations we have \(A^{*}=A^{*}_{0}\). Via an analogue of Lemma 3.1 and Proposition 3.2, the fact that \(\sin u \leq u \leq u/(1u^{2})\) implies that \(A_{0}^{*}=2\sqrt{B}\).the critical value \(A_{D}^{*}\) assumes the meaning of a threshold damping,
It may be added that, in the absence of torque, for \(A \geq A_{0}^{*}\) the motion that starts from a rest position at \(u_{0}\in\,]\pi, 2\pi[\) ends monotonically at 2π as time goes to infinity. On the other hand, if \(A< A_{0}^{*}\) such motions approach 2π in an oscillatory manner.
In a similar way, the characterization of the possible motions for \(D > 0\) is now clear:

the branch shot forward from \(u_{U}\), which does not reach the threshold 1 in view of the sign of the derivative and stays over the connection \(y_{A_{D}^{*}}\) by monotonicity (thus existing globally) is either a positive, rotational motion (if A is small: the torque wins over damping), or it vanishes at some point inside the interval \(]u_{S}+2\pi, u_{U}+2\pi[\), being an oscillation around the stable equilibrium \(u_{S}+2\pi\);

the branch shot backward from \(u_{S} + 2\pi\) vanishes before reaching \(u_{U}\), thus giving rise to an oscillatory type motion (the less the value of the damping, the nearer the vanishing point will be to \(u_{S}+2\pi\)).
(b) If \(A \geq A_{D}^{*}\), the damping is too high to climb over \(u_{S}+2\pi \), and, on the other hand, the solution to the first order model cannot stop out of the equilibrium, so that we find the already mentioned increasing connections between \(u_{U}\) and \(u_{S} + 2\pi\).
5 Discontinuous steady states
In [9], the existence of traveling waves for reaction terms of type C was related with the appearance of discontinuous steady states. In this last section, we want to briefly look at our first order model from this perspective.
Of course, the solutions to \(y'=f(u)\) from the left and from the right are unique.
Definition 5.1
Let \(f \in\mathcal{C}\). We will say that \(y(u)\) is a discontinuous steady state for (32) if there exist \(\theta_{1} < \theta_{2} \in\,]0, 1[ \), with \(\theta_{1} \leq \theta\) and \(\theta_{2} \geq\theta\), such that y is defined and continuous in \([0,\theta_{1}]\cup[\theta_{2},1] \), \(y(u)\) satisfies (32) in \([0,\theta_{1}[\,\cup\,]\theta_{2},1] \) and \(y(\theta _{1})=y(\theta_{2})=1\). If such conditions are verified for \(\theta _{1}=\theta _{2}\), we will say that \(y(u)\) is a border steady state.
The definition corresponds to heteroclinic connections between 0 and 1 which ‘break’ at some time instant; with the notation of the previous section, this means that both \(y_{0, f}^{+}\) and \(y_{0, f}^{}\) reach the value 1 in a finite time. For a border steady state, coming back to the second order model, we would have a continuous heteroclinic connection between 0 and 1 having infinite derivative in a point.
We now show the validity of the following proposition, agreeing with [9].
Proposition 5.2
Proof
As usual, we write \(f(u)=f^{+}(u)f^{}(u)\), with \(f^{+}(u)=\max\{f(u), 0\}\) and \(f^{}(u)=\max\{f(u), 0\}\).
Assume first that \(\int_{0}^{1} f(u)\,du > 0\), namely \(\int_{0}^{1} f^{+}(u)\,du > \int_{0}^{1} f^{}(u)\,du\).
Case 1: \(\int_{0}^{1} f^{}(u)\,du < 1\). Then, as we have seen, there exists a unique positive admissible speed. Also, no discontinuous steady states appear, since the solution \(y_{0, f}^{+}\) shot from the left cannot reach 1.
Case 2: \(\int_{0}^{1} f^{}(u)\,du \ge1\). As we have seen, no traveling waves with positive speed appear. On the other hand, the integral of \(f^{+}\) is strictly greater than 1, so that there exist \(\theta_{1} < \theta_{2}\) with \(y_{0, f}^{+}(\theta_{1})=1=y_{0, f}^{}(\theta _{2})\), yielding the discontinuous steady state. Finally, no traveling waves with negative speed are possible, as well, since by monotonicity \(y_{c, f}^{}\) would reach the threshold 1 all the same in a finite time.
Finally, assume \(\int_{0}^{1} f(u)\,du = 0\), i.e., \(\int_{0}^{1} f^{}(u)\,du = \int_{0}^{1} f^{+}(u)\,du\).
Case 1: \(\int_{0}^{1} f^{}(u)\,du < 1\). A continuous steady state (or a traveling wave with \(c=0\)) appears.
Case 2: \(\int_{0}^{1} f^{}(u)\,du \geq1\) . We have a border steady state (if equality holds) or a discontinuous one (if the inequality is strict). □

if \(\beta^{2} \frac{(2\alpha) \alpha^{3}}{12} < 1\), we use the first part of the proof of Proposition 5.2, Case 1, to infer the existence of a traveling wave with positive speed;

if \(\beta^{2} \frac{(2\alpha) \alpha^{3}}{12} \geq1\), we find a discontinuous steady state as a consequence of Case 2 in the first part of the proof of Proposition 5.2.

if \(\beta^{2} \frac{12\alpha+ 2\alpha^{3}  \alpha^{4}}{12} < 1\), we find a traveling wave with negative speed;

if \(\beta^{2} \frac{12\alpha+ 2\alpha^{3}  \alpha^{4}}{12} \geq1\), then a discontinuous steady state emerges.
We finally remark that, in the case of a more general type of reaction like the one discussed in Section 3.4, nonexistence of discontinuous steady states together with nonexistence of admissible speeds could occur, since the solution shot from the right can vanish before the admissible speed has been reached.
Declarations
Acknowledgements
The first author wishes to acknowledge the support of the Project PTDC/MAT/113383/2009 during his stay in Lisbon. The second author is supported by Fundação para a Ciência e a Tecnologia, PEstOE/MAT/UI0209/2013.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
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