Abstract
This paper is concerned with combustion transition fronts in \(\mathbb {R}^{N}\)\((N\ge 1)\). Firstly, we prove the existence and the uniqueness of the global mean speed which is independent of the shape of the level sets of the fronts. Secondly, we show that the planar fronts can be characterized in the more general class of almost-planar fronts. Thirdly, we show the existence of new types of transitions fronts in \(\mathbb {R}^{N}\) which are not standard traveling fronts. Finally, we prove that all transition fronts are monotone increasing in time, whatever shape their level sets may have.
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1 Introduction
This paper investigates reaction diffusion equations of the type
where \(N\in \mathbb {N}\), \(u_{t}=\frac{\partial u}{\partial t}\) and \(\varDelta \) denotes the Laplace operator with respect to the space variables \(x\in \mathbb {R}^{N}\). The nonlinear reaction term f(u) is of the “ignition temperature” type, that is, \(f:[0,1]\rightarrow \mathbb {R}\) is a \(C^{1}\) function such that
Such a profile can be derived from the Arrhenius kinetis with a cut-off for low temperatures and from the law of mass action. The real number \(\theta \) is the ignition temperature, below which no reaction happens.
In any dimension \(N\ge 1\), standard planar traveling fronts are solutions of the type
where e is any given unit vector of \(\mathbb {R}^{N}\), \(c_{f}\in \mathbb {R}\) is the propagation speed and \(\phi _{f}:\mathbb {R}\rightarrow [0,1]\) is the propagation profile, such that
The profile \(\phi _{f}\) is then a heteroclinic connection between the state 0 and the stable state 1. The level sets of such traveling fronts are parallel hyperplanes which are orthogonal to the direction of the propagation e. These fronts are invariant in the moving frame with speed \(c_{f}\) in the direction e. It is well known [1] that such front exists and is unique up to translation. Besides, the speed \(c_{f}\) is positive which has the sign of \(\int _{0}^{1}f(s)ds\) [5] and the function \(\phi _{f}\) is decreasing.
In \(\mathbb {R}^{N}\) with \(N\ge 2\), propagating wave fronts contains more types of traveling fronts except planar traveling fronts, such as V-shaped traveling fronts in two-dimensional spaces, pyramidal traveling fronts with non-axisymmetric shape in three-dimensional spaces and conical-shaped axisymmetric traveling fronts in high-dimensional spaces. The profiles of these fronts are still invariant in a moving frame with constant speed. But they have non-planar level sets. For instance, (1) admits the conical-shaped fronts of the type
where \(x'=(x_1,\ldots ,x_{N-1})\) and \(|x'|=(x_1^2+\cdots +x_{n-1}^2)^{1/2}\) whose profiles are invariant and which have non-planar level sets. For the existence, uniqueness, stability and other qualitative properties of these non-planar traveling fronts, we refer to [7, 8, 12,13,14, 24, 25, 33,34,35,36] and the references therein.
As we introduced above, Eq. (1) admits many types of traveling fronts. However, they have some common properties. For instance, the solutions u converge to the equilibrium states 0 or 1 far away from their moving or stationary level sets, uniformly in time. Their common properties led us to ask whether it is possible to introduce a more general notion of traveling fronts to include all types of waves. Berestycki and Hamel [3, 4] give an affirmative answer. They introduce the general notion of transition fronts. Before we describe the definition of transition fronts, we firstly introduce some notions. For any two subsets A and B of \(\mathbb {R}^{N}\) and for \(x\in \mathbb {R}^{N}\), we set
and \(d(x,A)=d(\{x\},A)\), where \(\vert \cdot \vert \) is the Euclidean norm in \(\mathbb {R}^{N}\). Let \((\varOmega _{t}^{-})_{t\in \mathbb {R}}\) and \((\varOmega _{t}^{+})_{t\in \mathbb {R}}\) be two families of open nonempty subsets of \(\mathbb {R}^{N}\), which satisfy
and
Notice that the condition (5) implies that the interface \(\varGamma _{t}\) is not empty for every \(t\in \mathbb {R}\).
Definition 1
(See [3, 4]) For problem (1), a transition front connecting 0 and 1 is a classical solution \(u:\mathbb {R}\times \mathbb {R}^{N}\rightarrow (0,1)\) for which there exist some sets \((\varOmega _{t}^{\pm })_{t\in \mathbb {R}}\) and \((\varGamma _{t})_{t\in \mathbb {R}}\) satisfying (5) and (6), and, for every \(\varepsilon >0\), there exists \(M\ge 0\) such that
Furthermore, u is said to have a global mean speed \(\varLambda ~(\ge 0)\) if
Remark 1
Notice that, for a given transition front u connecting 0 and 1, the sets \((\varOmega _{t}^{\pm })_{t\in \mathbb {R}}\) and \((\varGamma _{t})_{t\in \mathbb {R}}\) are not uniquely determined. In fact, for any sets \((\widetilde{\varGamma }_{t})_{t\in \mathbb {R}}\), if
then the family \((\widetilde{\varGamma }_{t})_{t\in \mathbb {R}}\) with corresponding sets \((\widetilde{\varOmega }_{t}^{\pm })_{t\in \mathbb {R}}\) also satisfies (5), (6) and (7). That is, the solution u is also a transition front connecting 0 and 1 with the families \((\widetilde{\varOmega }_{t}^{\pm })_{t\in \mathbb {R}}\) and \((\widetilde{\varGamma }_{t})_{t\in \mathbb {R}}\).
Notice furthermore that for any transition front u connecting 0 and 1, the interfaces \((\varGamma _{t})_{t\in \mathbb {R}}\) have uniformly bounded local oscillations, that is
In fact, it is shown in Lemma 3 and Remark 3 of [10], in the case of reaction–diffusion equations (1) with nonlinearity f satisfying \(f(u)>0\) for \(u\in (1-\delta ,1)\), where \(0<\delta <1\). Obviously, the assumptions of nonlinear reaction term f in this paper (see (2)) satisfy the above condition with \(\delta =1-\theta \).
In [3, 4, 11], the authors have showed that all the known standard traveling fronts (planar and non-planar traveling fronts) are transition fronts in the sense of Definition 1. In particular, Hamel [11] proved that for Eq. (1) with bistable nonlinearity there exist new types of transition fronts in \(\mathbb {R}^{N}\) which are not invariant in any frame as time runs. This property is different from standard traveling fronts which are invariant in a moving frame with constant speed. It also shows the broadness of Definition 1. In recent years, many papers have been devoted to the investigation of the existence and stability of transition fronts. For bistable transition fronts, we refer to [3, 4, 10, 11]. For Fisher-KPP transition fronts, the readers can see [15, 16, 21,22,23, 28, 31, 38]. Transition fronts for equations with combustion nonlinearity, the investigations mainly focus on the case of the heterogeneous equations in \(\mathbb {R}\), see [19, 20, 29, 30, 32, 37, 39, 40]. In this paper, we prove that even the homogeneous combustion equation (1) in \(\mathbb {R}^{N}\)\((N\ge 1)\) also has many deep properties, such as the existence of new transition fronts and general estimates shared by all transition fronts.
The first main result of this paper proves the existence and uniqueness of the global mean speed for any transition fronts connecting the state 0 and the stable state 1, regardless of the shape of the level sets of the transition fronts.
Theorem 1
For problem (1), any transition front u connecting 0 and 1 has a global mean speed \(\varLambda \). Furthermore, this global mean speed \(\varLambda \) is equal to \(c_{f}\).
The second result of this paper gives a characterization of the planar fronts \(\phi _{f}(x\cdot e-c_{f}t)\) among the more general class of almost-planar transition fronts introduced in [4], and defined as follows.
Definition 2
(See [4, 11]) A transition front u in the sense of Definition 1 is called almost-planar if, for every \(t\in \mathbb {R}\), the set \(\varGamma _{t}\) can be chosen as the hyperplane
for some vector \(e_{t}\) of the unit sphere \(\mathbb {S}^{N-1}\) and some real number \(\xi _{t}\).
From the definition, we can easily see that the level sets of almost-planar fronts are in some sense close to hyperplanes, even if they are not a priori assumed to be planar. The following theorem shows that planar fronts \(\phi _{f}(x\cdot e-c_{f}t)\) for problem (1) fall within the more general class of almost-planar fronts.
Theorem 2
For problem (1), any almost-planar transition front u connecting 0 and 1 is planar, that is, there exist a unit vector e of \(\mathbb {R}^{N}\) and a real number \(\xi \) such that
Thirdly, we show the broadness of transition fronts. In other words, we prove the existence of new types transition fronts of the Eq. (1), which are not invariant as time runs in any moving frame. Recall that the profiles of standard traveling fronts are invariant in a moving frame with constant speed.
Theorem 3
Let \(N\ge 2\). The problem (1) admits transition fronts u connecting 0 and 1 which satisfy the following property: there is no function \(\varPhi :\mathbb {R}^{N}\rightarrow (0,1)\) (independent of t) for which there would be some families \((R_{t})_{t\in \mathbb {R}}\) and \((x_{t})_{t\in \mathbb {R}}\) of rotations and points in \(\mathbb {R}^{N}\) such that \(u(t,x)=\varPhi (R_{t}(x-x_{t}))\) for all \((t,x)\in \mathbb {R}\times \mathbb {R}^{N}\).
Finally, we establish the time monotonicity of the transition front u.
Theorem 4
For problem (1), any transition front u connecting 0 and 1 is monotone increasing in time t. That is, \(u_{t}>0\) for all \((t,x)\in \mathbb {R}\times \mathbb {R}^{N}\).
In fact, in order to prove Theorem 4, it is sufficient to prove that the transition front u is an invasion of the state 0 by the state 1, in the sense that the sets \((\varOmega ^{\pm }_t)_{t\in \mathbb {R}}\) can be chosen so that
since it is easy to check that the problem (1) and the nonlinearity f satisfy all assumptions of [4, Theorem 1.11]. Similar to [10], it follows from Theorem 1 and Lemma 1 (see Sect. 2) that u is an invasion in the above sense with the families \(\left( \widehat{\varOmega }_{t}^{\pm }\right) _{t\in \mathbb {R}}\) and \(\left( \widehat{\varGamma }_{t}\right) _{t\in \mathbb {R}}\), where for some constant \(\tau _{0}>0\),
Now we give a brief stated on the methods of our proofs. Firstly, in order to prove the existence and the uniqueness of the global mean speed of the transition fronts connecting 0 and 1, we need introduce two radially symmetric functions and show their dynamical properties, see Lemmas 1 and 2 below. Secondly, using the one-dimensional stability of the planar front and parabolic Liouville type result of Berestycki and Hamel [3, Theorem 3.1], we show that the planar fronts can be characterized by the more general class of almost-planar transition fronts. Thirdly, by mixing three planar fronts moving in three different directions, we show that the new transition fronts exist in dimension \(N=2\). And by trivially extending the two-dimensional solutions in the variables \(x_{3},\ldots ,x_{N}\), we obtain that the new transition fronts exist in all dimensions \(N\ge 3\).
Here we would like to point out that the main results of this paper (Theorems 1, 2, 3 and 4) are similar to those established for Eq. (1) with bistable nonlinearity by Hamel [11] and Guo and Hamel [10], where the reaction term \(f:[0,1]\rightarrow \mathbb {R}\) is a \( C^1\) function such that
But in this paper we treat the combustion case, in particular, the reaction term f satisfies \(f(u)=0\) for any \(u\in [0,\theta ]\) with some \(\theta \in (0,1)\), which is essentially different from the assumption \(f^\prime (0)<0\) in the bistable case. Since the signs of \(f'(0)\) and \(f'(1)\) play important roles in the estimates of speeds and constructing the super-sub solutions, some new difficulties occur in the combustion reaction diffusion equations. To overcome these difficulties, we need some new techniques and establish some new estimates. See Lemmas 1, 2, 4 and Proposition 1 below for more details.
The rest of this paper is organized as follows. Section 2 proves the existence and the uniqueness of the global mean speed among all transition fronts. That is, we give the proof of Theorem 1. In Sect. 3, we prove Theorem 2. That is, we give a characterization of the planar fronts among the more general class of almost-planar transition fronts. In Sect. 4, we construct new types transition fronts. That is, we are devoted to the proof of Theorem 3.
2 The Global Mean Speed
In this section, we prove that any transition front of the Eq. (1) has a global mean speed and this speed is unique. We first introduce auxiliary notations for some radially symmetric functions and we show some of their dynamical properties. The following two key properties, Lemmas 1 and 2 below, will provide a sharp lower bound and a upper bound for the speed of the interfaces \(\varGamma _{t}\) of any transition front connecting 0 and 1 for the problem (1), respectively.
In the following, let \(\theta<\beta <1\). For any \(R>0\), let \(v_{R}^{f}\) denote the solution of the Cauchy problem
with initial value
Lemma 1
There is \(R>0\) such that the following holds: for any \(\varepsilon \in (0,c_{f}]\), there is \(T_{\varepsilon }>0\) such that
In fact,
Proof
Let g be any given \(C^{1}([0,1])\) function which satisfies
and
where \(c_{g}\) is the wave speed of the planar front \(\phi _{g}\) which satisfies (3) with the nonlinearity g. In fact, g is of the bistable type. Such fronts exist, see [2, 9, 17]. It is easy to see that \(f\ge g\) on [0, 1]. Then the comparison principle implies that
For the solution \(v_{R}^{g}\) of the equation (10)-(11) with replacing f by g, it follows from Lemma 4.1 of [11] that we have
Inequalities (14) and (15), together with the above formula, yield that (13) holds. This completes the proof. \(\square \)
Lemma 2
For any \(\varepsilon >0\), there exist some positive real numbers \(\alpha _{\varepsilon }\), \(T_{\varepsilon }\) and \(R_{\varepsilon }\) such that for all \(R\ge R_{\varepsilon }\), the solution \(w_{R}\) of the following Cauchy problem
with initial value
satisfies
Proof
Let \(\delta \) be chosen so that
Since \(\phi ''_{f}(s)\sim \nu e^{-c_{f}s}\) as \(s\rightarrow +\infty \) with \(\nu >0\), one can choose \(C>0\) such that
Since \(\phi '_{f}\) is negative and continuous on \(\mathbb {R}\), there is \(\kappa >0\) such that
Set \(L=\max \limits _{u\in [0,1]}\vert f'(u)\vert \). For every \(\varepsilon >0\), let
Choose \(D_{\varepsilon }>0\) such that
Let \(\rho _{\alpha _{\varepsilon }}\) be the solution of the following ordinary differential equation
Since \(\alpha _{\varepsilon }\in (0,\theta )\), f is Lipschitz-continuous and \(f\equiv 0\) on \([0,\theta ]\), then \(\rho _{\alpha _{\varepsilon }}(t)\equiv \alpha _{\varepsilon }\) by the existence and uniqueness of solution of the ordinary differential equation. It follows from the maximum principle and (2) that for any \(R>0\),
Then the following inequality holds
Thus for the above equation, the assumptions of the initial value yield
Therefore, if \(0<B\le R\) and \(\vert x\vert \le R-B\), one infers that
Thus, take a real number \(T>0\) and there exists \(B>0\) such that for all \(R\ge B\) and \(\vert x\vert \le R-B\),
whence
It is elementary to check that for every \(\varepsilon >0\), there is a \(C^{2}\) function \(h_{\varepsilon }:[0,+\infty )\rightarrow \mathbb {R}\) satisfying the following properties:
Notice in particular that
We choose \(T_{\varepsilon }>T>0\) such that
and \(R_{\varepsilon }>0\) such that
In the sequel, R is arbitrary real number such that
For all \((t,x)\in \mathbb {R}\times \mathbb {R}^{N}\), we set
where
In the set \(\varSigma =\left[ T,\frac{R}{c_{f}+\varepsilon }\right] \times \mathbb {R}^{N}\), let us then check that \(\overline{W}\) is a supersolution for the problem satisfied by \(w_{R}\).
Since \(f(1)=0\), it is sufficient to check that
Since \(\phi _{f}\) is of class \(C^{2}\) and h vanishes in the neighborhood of 0, then \(\overline{W}(t,x)=\phi _{f}(\bar{\xi }(t,x))+2\alpha _{\varepsilon }\) is of class \(C^{2}\) in the set where \(\overline{W}(t,x)<1\).
In this paragraph, let (t, x) be any point in \(\varSigma \) such that \(\overline{W}(t,x)<1\). Since \(\phi ^{\prime \prime }_{f}+c_{f}\phi '_{f}+f(\phi _{f})=0\) in \(\mathbb {R}\), then by (22) and \(\phi '_{f}\le 0\), there holds
Firstly, if \(\bar{\xi }(t,x)\le -C\), then (18) and the definition of \(\overline{W}\) yield \(1-\delta \le \phi _{f}(\bar{\xi }(t,x))\le \overline{W}(t,x)<1\). Whence by (17), one gets
In addition, it follows from (25) and (26) that the inequalitues \(\bar{\xi }(t,x)\le -C\) and \(T\le t\le \frac{R}{c_{f}+\varepsilon }\) yield
From the properties (22), the inequality \(h_{\varepsilon }(\vert x\vert )\ge H_{\varepsilon }\) implies that \(h'_{\varepsilon }(\vert x\vert )= 1\). Therefore, if \(\bar{\xi }(t,x)\le -C\), then \(\phi '_{f}\le 0\) implies
Secondly, if \(\bar{\xi }(t,x)\ge C\), then by (18), \(\phi _{f}(\bar{\xi }(t,x))\le \delta \). Thus,
Since \(f=0\) on \([0,\theta ]\), \(\phi ''_{f}\ge 0\) on \([C,+\infty )\) from (18), \(0\le h'_{\varepsilon }(\vert x\vert )\le 1\) on \([0,+\infty )\) and \(\phi '_{f}\le 0\) on \(\mathbb {R}\), one gets that, if \(\bar{\xi }(t,x)\ge C\), then
Lastly, if \(-C\le \bar{\xi }(t,x)\le C\), then
recall that \(L=\max \limits _{u\in [0,1]}\vert f'(u)\vert \). It follows from (24) and (26) that \(\bar{\xi }(t,x)\le C\) and \(T\le t\le \frac{R}{c_{f}+\varepsilon }\) imply
Thus by (22), \(h'_{\varepsilon }(\vert x\vert )=1\). Consequently, it follows from the definition of \(\alpha _{\varepsilon }\) and (19) that
On the other hand, at the time T, it follows from (21), (24), (26) and the definition of \(\overline{W}\) that
If \(\vert x\vert \ge R-B\), then \(h_{\varepsilon }(\vert x\vert )\ge \vert x\vert \ge R-B\) from (23), whence \(\bar{\xi }(T,x)\le -D_{\varepsilon }\) and
from (20) and the fact that \(w_{R}\le 1\) on \((0,+\infty )\times \mathbb {R}^{N}\). Thus
As a conclusion, the maximum principle implies that, for all \(T\le t\le \frac{R}{c_{f}+\varepsilon }\) and \(x\in \mathbb {R}^{N}\),
For all \(T_{\varepsilon }\le t\le \frac{R}{c_{f}+\varepsilon }\) and \(\vert x\vert \le R-(c_{f}+\varepsilon )t\), there hold
and
from (24). Thus, (20) yields \(\phi _{f}(\bar{\xi }(t,x))\le \alpha _{\varepsilon }\). Whence, if \(T_{\varepsilon }\le t\le \frac{R}{c_{f}+\varepsilon }\) and \(\vert x\vert \le R-(c_{f}+\varepsilon )t\), then
This completes the proof. \(\square \)
Proof of Theorem 1
Let u be any transition front of problem (1) which connects the equilibrium points 0 and 1. For any \(\varepsilon \in (0,c_{f}]\), let \(\alpha _{\varepsilon }\) be defined as in Lemma 2 and \(\theta<\beta <1\). It follows from Definition 1 that there is \(M\ge 0\) such that
Let \(R>0\) be as in Lemma 1. Without loss of generality, one can assume that \(R\ge M\) (since the functions \(v_{R}^{f}\) are nondecreasing with respect to the parameter \(R>0\)). By (6), there exists a real number \(r>0\) such that
Our goal is to prove
For this purpose, we divide our proof into two steps. In the first step, we prove inequality for the \(\liminf \). At the second step, we show inequality for the \(\limsup \).
Step 1. the lower estimate We show that
We assume that (29) does not hold, then one has
for some \(\varepsilon >0\) small enough. Thus, there exist two sequences \((t_{k})_{k\in \mathbb {N}}\) and \((s_{k})_{k\in \mathbb {N}}\) such that \(\vert t_{k}-s_{k}\vert \rightarrow +\infty \) as \(k\rightarrow +\infty \) and
Without loss of generality, we assume that \(t_{k}<s_{k}\) for all \(k\in \mathbb {N}\). The definition of the distance \(d(\varGamma _{t_{k}},\varGamma _{s_{k}})\) implies that there exist two sequences \((x_{k})_{k\in \mathbb {N}}\) and \((z_{k})_{k\in \mathbb {N}}\) in \(\mathbb {R}^{N}\) such that
First of all, by (28), there exists a sequence \((y_{k}^{+})_{k\in \mathbb {N}}\) of points in \(\mathbb {R}^{N}\) such that
Thus, for every \(k\in \mathbb {N}\) and \(y\in B(y_{k}^{+},R)\), one has \(y\in \varOmega _{t_{k}}^{+}\) and \(d(y,\varGamma _{t_{k}})\ge R\ge M\), whence \(u(t_{k},y)\ge \beta \) from (27). By (11), one has
Thus the maximum principle yields
Let \(T_{\varepsilon }\) be defined as in Lemma 1, thus Lemma 1 yields that for every \(k\in \mathbb {N}\),
Next, it follows from (28) that there exists a sequence \((y_{k}^{-})_{k\in \mathbb {N}}\) of points in \(\mathbb {R}^{N}\) such that
Property (27) implies that
Finally, notice that for all \(k\in \mathbb {N}\),
Thus, it follows from \(s_{k}-t_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \) that \(s_{k}\ge t_{k}+T_{\varepsilon }\) for k large enough and
Choose \(t=s_{k}\) and \(x=y_{k}^{-}\) in (31) for k large enough. Thus,
But \(\alpha _{\varepsilon }<\beta \) contradicting (32). Therefore, the assumption (30) cannot hold. That is,
Step 2: the upper estimate We show that
Let us assume by contradiction that
for some \(\varepsilon >0\) small enough. Then there exist two sequences \((t_{k})_{k\in \mathbb {N}}\) and \((s_{k})_{k\in \mathbb {N}}\) of real numbers such that \(\vert t_{k}-s_{k}\vert \rightarrow +\infty \) as \(k\rightarrow +\infty \) and
Without loss of generality, one can assume that \(t_{k}<s_{k}\) for all \(k\in \mathbb {N}\). For each \(k\in \mathbb {N}\), pick a point \(z_{k}\) on \(\varGamma _{s_{k}}\). It follows from (28) that there are two sequences \((y_{k}^{\pm })_{k\in \mathbb {N}}\) of points in \(\mathbb {R}^{N}\) such that
Thus, by (27), one has
It follows from \(d(z_{k},\varGamma _{t_{k}})>(c_{f}+3\varepsilon )( s_{k}-t_{k})>0\) for k large enough that there holds
We claim that \(B(z_{k},(c_{f}+3\varepsilon )( s_{k}-t_{k}))\subset \varOmega _{t_{k}}^{-}\) for k large enough. If not, up to extraction of a subsequence,
Since \(s_{k}-t_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \), then for k large enough,
recall that \(R>0\) is defined as in Lemma 1. Thus for k large enough,
from (27). It follows from (11) that one has
By the maximum principle, one gets
Let \(T_{\varepsilon '}>0\) be defined as in Lemma 1 with \(\varepsilon '=\frac{c_{f}}{2}\). The inequality (37) and Lemma 1 yield that, for k large enough,
Since \(c_{f}>0\) and \(s_{k}-t_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \), then for k large enough,
Thus, the previous inequality imples \(u(s_{k},y_{k}^{-})\ge \beta \) for k large enough. This is in contradiction with (35). Whence for k large enough,
Since \(s_{k}-t_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \), then for k large enough,
Thus by (27), one has
It follows from the definition of \(w_{(c_{f}+2\varepsilon )( s_{k}-t_{k})}\) (as in Lemma 1) that for k large enough,
Thus the maximum principle implies
Since \(s_{k}-t_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \), then for k large enough, one has \((c_{f}+2\varepsilon )(s_{k}-t_{k})\ge R_{\varepsilon }\) and
where \(T_{\varepsilon }>0\) and \(R_{\varepsilon }\ge (c_{f}+\varepsilon )T_{\varepsilon }>0\) are given by Lemma 2 so that (16) is valid for all \(R\ge R_{\varepsilon }\). Choose \(R=(c_{f}+2\varepsilon )( s_{k}-t_{k})\), \(t=s_{k}-t_{k}\) and \(x=y_{k}^{+}-z_{k}\) in (16) for k large enough, one can obtain
This is in contradiction with (35). Whence the conclusion (33) follows.
Combining with the Step 1 and Step 2, the proof of Theorem 1 is thereby complete.
3 Almost-Planar Fronts
In this section, we characterize the planar fronts \(\phi _{f}(x\cdot e-c_{f}t)\) for Eq. (1) among the more general class of almost-planar fronts. The proof of Theorem 2 mainly uses the one-dimensional stability of the planar front \(\phi _{f}\) [18] and the parabolic Liouville type result of Berestycki and Hamel [3, Theorem 3.1]. Before the proof, we first give some auxiliary lemmas.
Lemma 3
Let \(u:\mathbb {R}\times \mathbb {R}^{N}\rightarrow [0,1]\) be a solution of (1) for which there are a real number \(t_{0}\in \mathbb {R}\) and a unit vector \(e\in \mathbb {S}^{N-1}\) such that
Then property (38) holds at every time \(t_{1}>t_{0}\) with the same vector e.
Proof
Since for the case \(\inf \limits _{x\in \mathbb {R}^{N},x\cdot e\le -A}u(t_{0},x)\rightarrow 1\) as \(A\rightarrow +\infty \), the proof of Lemma 3 is similar to [11, Lemma 3.1], we only give the proof for the case \(\sup \limits _{x\in \mathbb {R}^{N},x\cdot e\ge A}u(t_{0},x)\rightarrow 0\) as \(A\rightarrow +\infty \).
For any \(\delta \in (0,1)\), let \(v^{\delta }\) be the solution of the following one-dimensional Cauchy problem
Let \(\rho ^{\delta }:\mathbb {R}\rightarrow (0,1)\) be the solution of the following ordinary differential equation
Then by the maximum principle, one has
Thus
where \(L=\max \limits _{u\in [0,1]}\vert f'(u)\vert \). Then the maximum principle and standard parabolic estimates imply that for each \(t>0\), \(v^{\delta }(t,\cdot )\) is decreasing in \(\mathbb {R}\), \(v^{\delta }(t,-\infty )=1\) and \(v^{\delta }(t,+\infty )=\rho ^{\delta }(t)\).
Assume that \(\sup \limits _{x\in \mathbb {R}^{N},x\cdot e\ge A}u(t_{0},x)\rightarrow 0\) as \(A\rightarrow +\infty \). Let \(\varepsilon \in (0,\theta )\) be arbitrary. Then there exists a constant M such that
Thus it follows from the maximum principle that
and whence
Since \(\varepsilon \in (0,\theta )\), f is \(C^{1}\) on [0, 1] and \(f=0\) on \([0,\theta ]\), then the existence and uniqueness of solution of the ordinary differential equation yield that \(\rho ^{\varepsilon }(t)\equiv \varepsilon \) for all \(t\ge 0\). Therefore, one has
This completes the proof. \(\square \)
The following corollary can be obtained immediately from Lemma 3.
Corollary 1
Let \(u:\mathbb {R}\times \mathbb {R}^{N}\rightarrow [0,1]\) be a solution of (1) such that, for every time \(t\in \mathbb {R}\), there is a unit vector \(e_{t}\in \mathbb {S}^{N-1}\) such that
Then \(e_{t}=e\) is independent of time t.
Let u be an almost-planar transition front connecting 0 and 1, in the sense of Definition 2, for problem (1). That is, there exist some families \((e_{t})_{t\in \mathbb {R}}\) in \(\mathbb {S}^{N-1}\) and \((\xi _{t})_{t\in \mathbb {R}}\) in \(\mathbb {R}\) such that
for every \(t\in \mathbb {R}\). Up to changing \(e_{t}\) into \(-e_{t}\), (5) and Definition 1 yields that (39) holds for every \(t\in \mathbb {R}\). It follows from Corollary 1 that \(e_{t}=e\) is a constant vector, whence
for all \(t\in \mathbb {R}\).
In Sect. 2, we have already proved that any transition front connecting equilibrium points 0 and 1 has a global mean speed \(c_{f}\). Here, for almost planar fronts, one has that
Then for any \(\gamma \in (0,1)\), there exists a constant \(K>0\) large enough such that
For \(n\in \mathbb {Z}\), we define \(\widetilde{\xi }_{t}\) such that
It follows from (9) that one has
Thus, one gets
Moreover, one has
Now we mollify the function \(\widetilde{\xi }_{t}\) to make it smooth. Define \(\eta \in C^{\infty }(\mathbb {R})\) by
where the constant \(C>0\) is selected so that \(\int _{\mathbb {R}}\eta dz=1\). For each \(\epsilon >0\), set \(\eta _{\epsilon }(z)=\frac{1}{^{\epsilon }}\eta \left( \frac{z}{\epsilon }\right) \). Let
such that
and hence, u(t, x) is still an almost-planar front with sets
and
from Remark 1. Let \(\alpha \) and \(\beta \) be two given real numbers such that
where we recall that \(\theta \) is defined in (2). By the Definition 1, there is \(M>0\) such that
Lemma 4
For any \(\gamma \in (0,1)\),
in \(\varSigma =\left\{ (t,x)\in \mathbb {R}\times \mathbb {R}^{N}\big \vert x\cdot e-\xi _{t}^{\epsilon }\ge M\right\} \).
Proof
Let \(\bar{u}(t,x)=\theta e^{-\gamma c_{f}(x\cdot e-\xi _{t}^{\epsilon }-M)}\). On \(\partial \varSigma =\left\{ (t,x)\in \mathbb {R}\times \mathbb {R}^{N}\big \vert x\cdot e-\xi _{t}^{\epsilon }= M\right\} \), it follows from (44) and (45) that
Define
Since u is bounded, \(\varepsilon _{*}\) is a well-defined real number and \(\varepsilon _{*}\ge 0\). Furthermore, one has
In particular,
One only has to prove that \(\varepsilon _{*}=0\).
Assume by contradiction that \(\varepsilon _{*}>0\). Then there exists a sequence \((\varepsilon _{n})_{n\in \mathbb {N}}\) of positive real numbers and a sequence of points \((t_{n},x_{n})_{n\in \mathbb {N}}\) in \(\varSigma \) such that
We claim that the sequence \((x_{n}\cdot e-\xi _{t_{n}}^{\epsilon })_{n\in \mathbb {N}}\) is bounded. Assume not, up to extraction of some sequence, one has
But
This gives a contradiction. Thus, the sequence \((x_{n}\cdot e-\xi _{t_{n}}^{\epsilon })_{n\in \mathbb {N}}\) is bounded.
It follows from (9) that for any \(\sigma >0\), there holds
Since \((\widetilde{\varGamma }_t)_{t\in \mathbb {R}}\) are all parallel hyperplanes, it then follows that for any fix \(\tau >0\), there exists a sequence \((\tilde{x}_{n})_{n\in \mathbb {N}}\) such that
By (6), there exist \(r>0\) and a sequence \((y_{n})_{n\in \mathbb {N}}\) such that
Then there exists a sequence \((z_{n})_{n\in \mathbb {N}}\) such that
Since \(d(y_{n},z_{n})\le y_{n}\cdot e-\xi _{t_{n}-\tau }^{\epsilon }\le d(\tilde{x}_{n},y_{n})\le r\) and since the sequence \((d(\tilde{x}_{n},x_{n}))_{n\in \mathbb {N}}\) is bounded, then the sequence \((d(x_{n},z_{n}))_{n\in \mathbb {N}}\) is bounded.
Choose \(\rho >0\) so that
which is possible since \(\bar{u}\) and u have bounded derivatives. Choose \(K\in \mathbb {N}\setminus \{0\}\) so that
For each \(n\in \mathbb {N}\), then there exists a sequence of points \((X_{n,0},X_{n,1},\ldots ,X_{n,K})\) in \(\varSigma \) such that
For each \(n\in \mathbb {N}\) and \(0\le i\le K-1\), set
Since \(w(t_{n},x_{n})\rightarrow 0\) as \(n\rightarrow +\infty \), (49) and (50) yield that \(w<\varepsilon _{*}\) in \(E_{n,0}\) for large n. It follows from (47) and the connectivity of \(E_{n,0}\) that \(E_{n,0}\subset \varSigma \) for large n.
By the definition of \(\varSigma \) and \(\bar{u}\), one has \(0\le \bar{u}\le \theta \) in \(\varSigma \). Then from (2) and (43), one has
in \(\varSigma \). On the other hand, \(u-\varepsilon _{*}<u\le \alpha <\theta \) in \(\varSigma \). Assumption (2) implies that \(u-\varepsilon _{*}\) is a subsolution of (1) in \(\varSigma \). Since f is of class \(C^{1}\), the function w satisfies inequations of the type
for n large enough, where the sequence \((\Vert b\Vert _{L^{\infty }(E_{n,0})})_{n\in \mathbb {N}}\) is bounded. Since \(w(t_{n},X_{n,0})=w(t_{n},x_{n})\rightarrow 0\) as \(n\rightarrow +\infty \), it follows from the linear parabolic estimates that
An immediate induction yields \(w\left( t_{n}-\frac{i\tau }{K},X_{n,i}\right) \rightarrow 0\) as \(n\rightarrow +\infty \) for each \(i=1,\ldots ,K\). In particular, for \(i=K\),
But \(z_{n}\in \overline{\widetilde{\varOmega }_{t_{n}-\tau }^{-}}\) and \(z_{n}\cdot e-\xi _{t_{n}-\tau }^{\epsilon }=M\) for all \(n\in \mathbb {N}\). As a consequence, for all \(n\in \mathbb {N}\), \(w\left( t_{n}-\tau ,z_{n}\right) >\varepsilon _{*}\) from (47).
One has reached a contradiction, which means that \(\varepsilon _{*}=0\). Thus,
for all \((t,x)\in \varSigma \). This completes the proof. \(\square \)
Proof of Theorem 2
For any fixed \(\gamma \in (0,1)\), let \(\underline{v}_{\beta }\) and \(\bar{v}_{\alpha }\) be the solution of the one-dimensional Cauchy problem
with initial condition
and \(\bar{v}_{\alpha }(0,y)\in C(\mathbb {R},[0,1])\),
respectively. It follows from (45) and Lemma 4 that for every \(t_{0}\in \mathbb {R}\) and \(x\in \mathbb {R}^{N}\),
Thus,
for all \(t>t_{0}\) and \(x\in \mathbb {R}^{N}\), from the maximum principle. By [18], there exist two constants \(\underline{\omega }>0\) and \(\bar{\omega }>0\) such that
and
for some \(\underline{A}>0\), \(\bar{A}>0\), \(\underline{\xi }\in \mathbb {R}\) and \(\bar{\xi }\in \mathbb {R}\). In particular, since \(\phi _{f}(-\infty )=1\) and \(\phi _{f}(+\infty )=0\), there exist \(T>0\) and \(B>0\) such that, for all \(s\ge T\),
It follows from (54) that for all \(t_{0}<t_{0}+T\le t\),
By (45) and (55), for all \(t_{0}<t_{0}+T\le t\), we have
By fixing \(t=0\), one gets that \(\limsup \limits _{t_{0}\rightarrow -\infty }\vert \xi _{t_{0}}^{\epsilon }-c_{f}t_{0}\vert \le \vert \xi _{0}^{\epsilon }\vert +2M+B\). For any arbitrary \(t\in \mathbb {R}\), letting \(t_{0}\rightarrow -\infty \) in (56) then leads to
Thus, by Definition 1 and (40), our solution \(u:\mathbb {R}\times \mathbb {R}^{N}\rightarrow (0,1)\) of (1) satisfies
It follows from Theorem 3.1 of [3] and the uniqueness of the planar fronts that there exists \(\xi \in \mathbb {R}\) such that \(u(t,x)=\phi _{f}(x\cdot e-c_{f}t+\xi )\) for all \((t,x)\in \mathbb {R}\times \mathbb {R}^{N}\). This completes the proof of Theorem 2.
4 Existence of Non-standard Transition Fronts
In this section, we prove Theorem 3. That is, we prove the existence of new kinds of transition fronts, which are not invariant in any moving frame. We first consider the case \(N=2\) and construct two-dimensional transition fronts satisfying the conclusion of Theorem 3. The conclusion in \(\mathbb {R}^{N}\) with \(N>2\) will be then obtained immediately by trivially extending the constructed two-dimensional fronts in variables \(x_{3},\ldots ,x_{N}\). Now we first give some preliminaries.
For the standard planar traveling fronts \(\phi _{f}\), it is well known that there exist some positive constants \(\lambda _{1}\), \(C_{0}\), \(C_{1}\) and \(C_{2}\) such that
Fix an angle \(\alpha \) such that \(\frac{\pi }{4}<\alpha <\frac{\pi }{2}\). Consider the quasilinear parabolic equation
It follows from Propositions 1.1 and 2.5 of [27] that for any \(c>c_{f}\), there exists a unique solution \(\varphi (x;c)\) of (60) with asymptotic lines \(y=|x|\cot \alpha \) satisfying
Lemma 5
(Brazhnik [6], Ninomiya and Taniguchi [24, 26, 27]) There exist positive constants \(\gamma _{1}\), \(k_{i}\)\((i=1,2,3)\) and \(\omega _{\pm }\) such that
for any \(x\in \mathbb {R}\), where
By Lemma 5, it is easy to obtain that
and that there exists a constant \(a>0\) such that
It follows from [12, 36] that there exists a unique V-shaped traveling front \(\phi (x_{1},x_{2}-ct)\) (Fig. 1) of the problem (1) in \(\mathbb {R}^{2}\) satisfying the following properties: \(0<\phi <1\) in \(\mathbb {R}^{2}\), \(\phi \) is of class \(C^{2}(\mathbb {R}^{2})\), \(c=\frac{c_{f}}{\sin \alpha }\) and
The profiles (left figure) and the contour lines (right figure) of the V-shaped traveling front
Furthermore, for any \(\beta _{1}\in (0,1)\), there exist two positive constants \(\varepsilon _{0}^{+}(\beta _{1})\) and \(\alpha _{0}^{+}(\beta _{1},\varepsilon )\) so that, for \(0<\varepsilon <\varepsilon _{0}^{+}(\beta _{1})\) and \(0<\vartheta <\alpha _{0}^{+}(\beta _{1},\varepsilon )\),
Fix \(\beta _{1}\in (0,1)\), \(0<\varepsilon <\varepsilon _{0}^{+}(\beta _{1})\) and \(0<\vartheta <\alpha _{0}^{+}(\beta _{1},\varepsilon )\). Now we show that \(\phi \) is asymptotically planar along the directions \((\pm \sin \alpha ,\cos \alpha )\). This property plays an important role in the proof of Theorem 3.
Proposition 1
There exist two positive constants \(\rho _{1}\) and \(\omega _{1}\) such that
Proof
Let
Choose \(\mu \in (0,1)\) such that
Fix a real number \(\omega _{1}\) such that
Now we divide our proof into three cases.
Case 1 when \(x_{2}\le 0\), by (58), one has
Case 2 when \(x_{2}>0\) and \(x_{2}\sin \alpha -\vert x_{1}\vert \cos \alpha <0\), one has
from \(\frac{1}{\sqrt{1+\vert \varphi '(x_{1})\vert ^{2}}}>\frac{c_{f}}{c}\) and \(|x_{1}|\cot \alpha \le \varphi (x_{1})\) for any \(x_{1}\in \mathbb {R}\). Thus it follows from (59), Lemma 5 and \(c=\frac{c_{f}}{\sin \alpha }\) that
Case 3 when \(x_{2}>0\) and \(x_{2}\sin \alpha -\vert x_{1}\vert \cos \alpha \ge 0\), one has \(x_{2}>\vert x_{1}\vert \cot \alpha \). It follows from (57), (59), Lemma 5 and \(c=\frac{c_{f}}{\sin \alpha }\) that
Combining the above three cases, the proof of Proposition 1 is thereby complete. \(\square \)
It follows from Proposition 1 and the Schauder interior estimates that there exist two positive constants \(\rho _{2}\) and \(\omega _{2}\) such that
Whence
for all \(x_{1}\ge 0,~x_{2}\in \mathbb {R}\). Since the standard planar traveling fronts \(\phi _{f}(s)\) converges exponentially fast to 0 and 1 as \(s\rightarrow \pm \infty \), Proposition 1 yields that the V-shaped traveling front \(\phi \) also converges exponentially fast to 0 and 1 as \(x_{2}-\vert x_{1}\vert \cot \alpha \rightarrow \pm \infty \). By the Schauder interior estimates, there exist two positive constants \(\rho _{3}\) and \(\omega _{3}\) such that
It follows from Corollary 3.3 and Lemma 3.4 of [36] that
and that \(\phi \) is decreasing in any direction \((\cos \hat{\alpha },\sin \hat{\alpha })\) such that \(\pi /2-\alpha<\hat{\alpha }<\pi /2+\alpha \), see also [12]. In particular, the function \(\phi \) is nonincreasing along the directions \((\pm \sin \alpha ,\cos \alpha )\).
The profiles (left figure) and the contour lines (right figure) of function \(\psi \)
Define
which rotates the function \(\phi \) with angle \(\alpha -\frac{\pi }{2}\) clockwise. Then the function \(\psi \) (Fig. 2) is decreasing in any direction \((\cos \hat{\beta },\sin \hat{\beta })\) with \(0<\hat{\beta }<2\alpha \). In particular, \(\psi \) is nonincreasing in the horizontal direction (1, 0) and it converges to the planar front \(\phi _{f}(x_{2})\) along this direction. Set
Since \(\phi (x_{1},x_{2}-ct)\) solves the Eq. (1) in \(\mathbb {R}^{2}\), then the \(C^{2}(\mathbb {R}\times \mathbb {R}^{2})\) function \(\underline{v}\) also satisfies (1) in \(\mathbb {R}^{2}\). Moreover, the definition of \(\underline{v}\) yields \(\underline{v}_{t}(t,x_{1},x_{2})>0\) and \(\underline{v}_{x_{1}}(t,x_{1},x_{2})\le 0\) in \(\mathbb {R}\times \mathbb {R}^{2}\).
Now we consider the following Neumann boundary value problem in half-space \(H=\{(x_{1},x_{2})\in \mathbb {R}^{2},x_{1}<0\}\)
It is easy to see that the function \(\underline{v}\) is a subsolution of (69).
In the following lemma, we construct a supersolution which looks like the function \(\underline{v}\) for very negative times, up to some exponentially small terms.
Lemma 6
There exist some constants \(\sigma >0\), \(\delta >0\) and \(T<0\) such that the function
is a supersolution of (69) for \(t\le T\) and \((x_{1},x_{2})\in \overline{H}\).
Proof
Let
where \(\omega _{2}\) and \(\omega _{3}\) are given in (64) and (65). Choose \(\delta \) such that
It follows from (61) that there exists a real number \(A>0\) such that
Equation (66) implies that there exists a constant \(\kappa >0\) such that
Choose \(\sigma >0\) such that
Set
Let \(T<0\) be such that
Similar to the Lemma 5.1 of Hamel [11] combining with Proposition 1, we can prove that \(\bar{v}_{x_{1}}\ge 0\) on \((-\infty ,T]\times \partial H\) in the region where \(\bar{v}<1\).
Since \(f(1)=0\), it is sufficient to show that
on the region \((t,x_{1},x_{2})\in (-\infty ,T]\times \overline{H}\) such that \(\bar{v}<1\). Since \(\underline{v}\) satisfies (1) in \(\mathbb {R}^{2}\) and \(\delta <1\), thus
For simplicity, by (68), we can set
where
Firstly, if \(\xi _{2}(t,x_{1},x_{2})\le \vert \xi _{1}(x_{1},x_{2})\vert \cot \alpha -A\), then (72) implies that
It follows from (71), (75) and \(\underline{v}_{t}>0\) that one has
Secondly, if \(\xi _{2}(t,x_{1},x_{2})\ge \vert \xi _{1}(x_{1},x_{2})\vert \cot \alpha +A\), then it follows from (71), (72), \(x_{1}\le 0\) and \(t\le T<0\) that
Since \(f=0\) on \([0,\theta ]\) and \(\underline{v}_{t}>0\), then
Lastly, if \(-A\le \xi _{2}(t,x_{1},x_{2})-\vert \xi _{1}(x_{1},x_{2})\vert \cot \alpha \le A\), then
and
It follows from (74) and \(x_{1}\le 0\) that
Combining with (76), (77) and (78), one has \(\overline{\mathscr {L}}(t,x_{1},x_{2})\ge 0\) for all \((t,x_{1},x_{2})\in (-\infty ,T]\times \overline{H}\) such that \(\bar{v}(t,x_{1},x_{2})<1\). This completes the proof. \(\square \)
Proof of Theorem 3
It follows from the positivity of \(\underline{v}_{t}\) and the definition of \(\bar{v}\) that \(\underline{v}(t,x_{1},x_{2})<\bar{v}(t,x_{1},x_{2})\) in \(\mathbb {R}\times \overline{H}\). For any \(n\in \mathbb {N}\) such that \(n>\vert T\vert \), let \(v^{n}\) be the solution of the Cauchy problem associated to (69) for times \(t>-n\), with initial condition
Since \((\underline{v},\bar{v})\) is a couple of sub-supersolution of the problem (69), the maximum principle implies that
for all \(-n<t\le T\) and \((x_{1},x_{2})\in \overline{H}\) and that
In particular, for every \((t,x_{1},x_{2})\in \mathbb {R}\times \overline{H}\), the sequence \((v^{n}(t,x_{1},x_{2}))_{n>\max (\vert T\vert ,\vert t\vert )}\) is nondecreasing. Furthermore, since \(\underline{v}_{t}>0\), (79) and the maximum principle yield that \(v^{n}\) is increasing with respect to time t in \(\overline{H}\).
It follows from monotone convergence and standard parabolic estimates up to the boundary that the functions \(v^{n}\) converge to a solution v of (69) as \(n\rightarrow +\infty \) in \(C_{loc}^{1,2}(\mathbb {R}\times \overline{H})\). Furthermore, one has
and
In particular, since for each fixed \((x_{1},x_{2})\in \overline{H}\), the function \(\bar{v}(t,x_{1},x_{2})\rightarrow 0<1\) as \(t\rightarrow -\infty \), then it follows from (80) and the strong maximum principle that \(0<v<1\) in \(\mathbb {R}\times \overline{H}\).
Now we construct a solution u of (1) in \(\mathbb {R}^{2}\). Define u in \(\mathbb {R}\times \mathbb {R}^{2}\) as
Since v satisfies (69) in the half-plane H with Neumann boundary conditions, then u is a classical time-global solution of (1) in the whole plane \(\mathbb {R}^{2}\). Furthermore, \(0<u<1\) in \(\mathbb {R}\times \mathbb {R}^{2}\),
and
Therefore, by the definition of \(\underline{v}\) and (62), one has
for all \((t,x_{1},x_{2})\in \mathbb {R}\times \mathbb {R}^{2}\). And it follows from the definition \(\bar{v}\) and Proposition 1 that
for all \(t\le T\) and \((x_{1},x_{2})\in \mathbb {R}^{2}\).
For \(t\le 0\), let
and
where the superscript l (resp. r) stands for left (resp. right). Define
Thus, for every \(t\in \mathbb {R}\), \(\varGamma _{t}\) can be written as a graph \(\varGamma _{t}=\{(x_{1},x_{2})\in \mathbb {R}^{2};x_{2}=\hat{\varphi }_{t}(x_{1})\}\), where \(\hat{\varphi }_{t}:\mathbb {R}\rightarrow \mathbb {R}\) is a Lipschitz-continuous function. For all \(t\in \mathbb {R}\), define
Obviously, the sets \((\varOmega _{t}^{\pm })_{t\in \mathbb {R}}\) and \((\varGamma _{t})_{t\in \mathbb {R}}\) satisfy the general properties (5) and (6).
Similar to the proof of Lemma 5.2 of [11], the function u is a transition front connecting 0 and 1 for problem (1) in \(\mathbb {R}^{2}\) with the sets \((\varOmega _{t}^{\pm })_{t\in \mathbb {R}}\) and \((\varGamma _{t})_{t\in \mathbb {R}}\).
Now we prove that the solution u is not invariant as time runs with any moving frame. That is, it satisfies the conclusion of Theorem 3. Assume by contradiction that there exist a function \(\varPhi :\mathbb {R}^{2}\rightarrow (0,1)\) and some families \((R_{t})_{t\in \mathbb {R}}\) and \((X_{t})_{t\in \mathbb {R}}=(x_{1,t},x_{2,t})_{t\in \mathbb {R}}\) of rotations and points in \(\mathbb {R}^{2}\) such that
Then there is \(M\ge 0\) such that
which is contradicted with the definitions of the sets \(\varGamma _{t}\) defined as (81) and (82). Whence, Theorem 3 holds in \(\mathbb {R}^{2}\).
Now, we extend the transition front u trivially in \(\mathbb {R}^{N}\)\((N\ge 3)\). Let
Obviously, the function \(\widetilde{u}\) is a transition front connecting 0 and 1 for problem (1) in \(\mathbb {R}^{N}\) with the sets
and satisfies the desired conclusion. This completes the proof of Theorem 3.
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Acknowledgements
The first author and the second author would like to thank Professor François Hamel of Aix-Marseille University for the valuable discussions. They was supported by the China Scholarship Council. The third author was supported by NNSF of China (11371179, 11731005) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2016-ct12, lzujbky-2017-ct01). This work has been partly carried out in the framework of the ANR DEFI Project NONLOCAL (ANR-14-CE25-0013), of Archimèdes Labex (ANR-11-LABX-0033), of the A*MIDEX Project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR), and of the ERC Project ReaDi - Reaction–Diffusion Equations, Propagation and Modelling, Grant Agreement n. 321186 funded by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013).
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Bu, ZH., Guo, H. & Wang, ZC. Transition Fronts of Combustion Reaction Diffusion Equations in \(\mathbb {R}^{N}\). J Dyn Diff Equat 31, 1987–2015 (2019). https://doi.org/10.1007/s10884-018-9675-x
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DOI: https://doi.org/10.1007/s10884-018-9675-x