Abstract
Let \(n\ge 2\) be a given integer. In this paper, we assert that an n-dimensional traveling front converges to an \((n-1)\)-dimensional entire solution as the speed goes to infinity in a balanced bistable reaction–diffusion equation. As the speed of an n-dimensional axially symmetric or asymmetric traveling front goes to infinity, it converges to an \((n-1)\)-dimensional radially symmetric or asymmetric entire solution in a balanced bistable reaction–diffusion equation, respectively. We conjecture that the radially asymmetric entire solutions obtained in this paper are associated with the ancient solutions called the Angenent ovals in the mean curvature flows.
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1 Introduction
In this paper we study a reaction–diffusion equation
where \(n\ge 2\) is a given integer, and \(u_{0}\) is a given bounded and uniformly continuous function from \({\mathbb {R}}^{n}\) to \({\mathbb {R}}\). The following is the standing assumptions of \(W\in C^{3}[-1,1]\) in this paper
Here \(W'\) and \(W''\) are the first and second derivatives of W, respectively. Equation (1.1) is called the Allen–Cahn equation, the scalar Ginzburg–Landau equation or the Nagumo equation if
A nonlinear term \(-W'(u)\) with (1.3) is called a bistable one. It is called balanced if \(W(-1)=W(1)\), and is called imbalanced if \(W(-1) \ne W(1)\). In this paper we assume that \(-W'(u)\) is balanced. We write the solution of (1.1)–(1.2) as \(u(x,t;u_{0})\). Under the assumption of W stated above, there exists \(\Phi \) that satisfies
This \(\Phi \) is called the one-dimensional standing front, and is explicitly given by
Under assumptions (1.4) and (1.3), there exists \(\Phi \) if and only if (1.5) holds true. Let
Now we write \(r=|\varvec{x}'|\). Let \(c\in (k,\infty )\) be arbitrarily given. We put \(z=x_{n}-ct\) and \(w(\varvec{x}',z,t)=u(\varvec{x}',x_{n},t)\) for \((\varvec{x}',z)\in {\mathbb {R}}^{n}\) and \(t>0\). Then we have
We write z simply as \(x_{n}\). Then we have
We write the solution of (1.6)–(1.7) as \(w(\varvec{x},t;u_{0})\).
If \(v\in C^{2}({\mathbb {R}}^{n})\) satisfies
for \(c\in {\mathbb {R}}\), \(v(\varvec{x}',x_{n}-ct)\) becomes a traveling wave or a traveling front of (1.1). We call (1.8) the profile equation of \(v(\varvec{x})\). We call v a traveling profile and call c its speed. Sometimes we denote a traveling front \(v(\varvec{x}',x_{n}-ct)\) simply by (c, v). Now \(x_{n}\) is the traveling direction of (c, v). For a multidimensional traveling front, a traveling direction of (c, v) might not be uniquely determined. We say that a traveling front is axisymmetric if we can choose a traveling direction such that v is axisymmetric with respect to the traveling direction. We say that a traveling front is axially asymmetric if v is axially asymmetric with respect to every traveling direction.
If a function \(U(\varvec{x}',t)\) satisfies
\(U(\varvec{x}',t)\) is called an entire solution in \({\mathbb {R}}^{n-1}\), where
A traveling front solution to (1.9) is itself an entire solution to (1.9). Now we say that \(U(\varvec{x}',t)\) is radially symmetric or spherically symmetric with respect to \(\varvec{a}'\in {\mathbb {R}}^{n-1}\) if U is a function of \(|\varvec{x}'-\varvec{a}'|\) and \(t\in {\mathbb {R}}\). If \(U(\varvec{x}',t)\) is not radially symmetric with respect to any \(\varvec{a}'\in {\mathbb {R}}^{n-1}\), we say that \(U(\varvec{x}',t)\) is radially asymmetric.
Traveling fronts to (1.1) have been studied by [6, 10, 11, 31,32,33,34] in \({\mathbb {R}}^{n}\) for \(n\ge 2\). Axisymmetric traveling fronts have been studied by [6, 32], and axially asymmetric traveling fronts have been studied by [31]. A one-dimensional entire solution is studied by Chen, Guo and Ninomiya [7] and del Pino and Gkikas [9]. It is an interesting question whether there exists a relation between traveling fronts in \({\mathbb {R}}^{n}\) and entire solutions in \({\mathbb {R}}^{n-1}\). In this paper we show that a traveling profile of (1.8) in \({\mathbb {R}}^{n}\) converges to an entire solution of (1.9) in \({\mathbb {R}}^{n-1}\) as the speed c goes to infinity. Then, using this fact, we show the existence of a radially symmetric entire solution and a radially asymmetric entire solution of (1.9) as the limits of an axisymmetric traveling front and an axially asymmetric traveling front of (1.8), respectively.
Now we define \(s_{*}\in (-1,1)\) by
and fix \(\theta _{0}\in (s_{*},1)\) with \(-W'(\theta _{0})>0\). Now we assert the existence of a radially symmetric entire solution as follows.
Theorem 1
(Radially symmetric entire solution) Let \(R_{0}\in (0,\infty )\) be arbitrarily given. One has \(U_{\textrm{sym}}(|\varvec{x}'|,t)=U(\varvec{x}',t)\) for \(\varvec{x}'\in {\mathbb {R}}^{n-1}\) such that one has (1.9) with
Here \(r=|\varvec{x}'|\). One has
As \(t\rightarrow -\infty \), \(U_{\textrm{sym}}(r,t)\) converges to \(-1\) on every compact set in \([0,\infty )\). For any fixed \(t\in {\mathbb {R}}\), one has
See Fig. 1 for \(\{\varvec{x}'\in {\mathbb {R}}^{n-1}\,|\, U(\varvec{x}',0)=\theta _{0}\}\) for U in Theorem 1. Now we assert the existence of radially asymmetric entire solution.
Theorem 2
(Radially asymmetric entire solution) Let
be arbitrarily given. Then there exists U that satisfies
with
One has
As \(t\rightarrow -\infty \), \(U(\varvec{x}',t)\) converges to \(-1\) on every compact set in \({\mathbb {R}}^{n-1}\). For any fixed \(t\in {\mathbb {R}}\), one has
See Fig. 2 for \(\{\varvec{x}'\in {\mathbb {R}}^{n-1}\,|\,U(\varvec{x}',0)=\theta _{0}\}\) for U in Theorem 2. For a reaction–diffusion equation with an imbalanced bistable reaction term, traveling fronts have been studied by [16,17,18, 23, 24, 27,28,29,30, 34] and so on, and and entire solutions have been studied by [3, 5, 13,14,15, 21, 22, 37] and so on. See [25] for a relation between traveling fronts in \({\mathbb {R}}^{n}\) and entire solutions in \({\mathbb {R}}^{n-1}\) for \(n\ge 2\).
This paper and [25] suggest that an n-dimensional traveling fronts converges to an \((n-1)\)-dimensional entire solution as the speed goes to infinity in various kind of reaction–diffusion equations.
This paper is organized as follow. In Sect. 3, we summarize the properties of n-dimensional traveling fronts with a speed \(c\in (0,\infty )\). In Sect. 4, we study n-dimensional traveling fronts as the speed \(c\in (0,\infty )\) goes to infinity and obtain entire solutions as the limits. We prove the existence of radially symmetric entire solutions and radially asymmetric entire solutions to (1.1).
2 Discussions
Let \(U(\varvec{x}',t)\) be given by Theorem 1 or Theorem 2. When
for \(\varepsilon >0\), the motion of a level set \(\{\varvec{x}'\in {\mathbb {R}}^{n-1} \,|\, U(\varvec{x}',t)=0\}\) is approximated by a mean curvature flow in the limit of \(\varepsilon \rightarrow 0\). See [4] for instance. Axisymmetric or axially asymmetric traveling fronts in Theorem 4 or in Theorem 8 are closely related to those in mean curvature flows studied by Wang [35]. For a mean curvature flow, a curve or a surface evolves with time. If a curve or a surface is defined for all \(t\in (-\infty , t_{0})\) with some \(t_{0}\in {\mathbb {R}}\), it is called an ancient solution. A typical example is ancient solutions studied by Angenent [1] and Angenent, Daskalopoulos and Sesum [8] for a two-dimensional plane, and they are called the Angenent ovals or the paper clip solutions. Ancient solutions have been studied by White [36], Haslhofer and Hershkovits [19] and Angenent, Daskalopoulos and Sesum [2] for a space whose dimension is three or more. The author conjectures as follows.
Conjecture 1
Let W be given by (2.1). Let \(U(\varvec{x}',t)\) be an entire solution in Theorem 1 or in Theorem 2. Let \(T_{0}\in {\mathbb {R}}\) be uniquely given by \(U(\varvec{0}',T_{0})=0\) in Theorem 1 or in Theorem 2. Let \(0< \gamma _{1}< \gamma _{2}< \infty \) be arbitrarily given. Then, as \(\varepsilon \rightarrow 0\), a level set \(\{\varvec{x}'\in {\mathbb {R}}^{n-1} \,|\, U(\varvec{x}',t)=0\}\) converges to an ancient solution of a mean curvature flow for \(t\in [T_{0}-\gamma _{2},T_{0}-\gamma _{1}]\).
The study on this convergence will give an important insight on entire solutions in a reaction–diffusion equation and on ancient solutions in a mean curvature flow. Note that a paper clip solution lies between two parallel lines for all time till it extinguishes. As \(t\rightarrow -\infty \), an axially asymmetric entire solution U in Theorem 2 converges to \(-1\) on every given compact set in \({\mathbb {R}}^{2}\) for \(n=3\). Thus \(\{\varvec{x}'\in {\mathbb {R}}^{n-1} \,|\, U(\varvec{x}',t)=0\}\) cannot lie between two parallel lines as \(t\rightarrow -\infty \). This means that axially asymmetric entire solutions in Theorem 2 are novel propagation phenomena.
3 Properties of n-dimensional traveling fronts with various speeds
We extend W as a function of class \(C^{3}\) in an open interval that includes \([-1,1]\). Let
Let \(\delta _{*}\in (0,1/4)\) be small enough such that \([-1-2\delta _{*},1+2\delta _{*}]\) is included in the open interval and one has
Now we put
and introduce a positive constant \(\sigma \) by
Throughout this paper we assume
and
Then \(u(\varvec{x},t)=u(\varvec{x},t;u_{0})\) satisfies (1.1) with
Now the Schauder estimates [34, Proposition 2.9, Lemma 2.6] give
Here \(K_{*}\in (0,\infty )\) is a constant depending only on \((W,\delta _{*}, n)\) and is independent of \(u_{0}\). We use
Lemma 3
Assume \(c\in {\mathbb {R}}\) and \(v\in C^{2}({\mathbb {R}}^{n})\) satisfy (1.8) and (3.2). Then one has
Here \(K_{*}\) is a constant in (3.3)–(3.5), and is independent of \((c,v) \in {\mathbb {R}}\times C^{2}({\mathbb {R}}^{n})\).
Proof
By putting \(u(\varvec{x}',x_{n},t)=v(\varvec{x}',x_{n}-ct)\), u satisfies (1.1) with
Then (3.3), (3.4) and (3.6) give
which give
Thus we obtain (3.7) and (3.8). This completes the proof. \(\square \)
Now we state properties of axisymmetric traveling fronts as follows.
Theorem 4
(Axisymmetric traveling fronts [6, 32]) Let \(c\in (0,\infty )\) be arbitrarily given. There exists \(V_{c}(\varvec{x}',x_{n})=V_{\textrm{sym}}(|\varvec{x}'|, x_{n})\) such that \((c,V_{c})\) satisfies the profile equation (1.8), \(V_{\textrm{sym}}(0,0)=\theta _{0}\) and
For every \(\theta \in (-1,1)\), one has
Here \(r=|\varvec{x}'|\).
Remark 1
As far as the author knows, the uniqueness of \(V_{\textrm{sym}}\) in Theorem 4 is yet to be studied. Here we denote a traveling front that satisfies Theorem 4 by \(V_{\textrm{sym}}\). It depends on \(c\in (0,\infty )\).
Now we state properties of axially asymmetric traveling fronts in [31] as follows.
Theorem 5
[31] Let
be arbitrarily given and let
Let \(c\in (0,\infty )\) and \(\zeta \in (0,\infty )\) be arbitrarily given. There exists \(V(\varvec{x})=V(\varvec{x};\varvec{\alpha }',c)\) that satisfies (1.8) with \(V(\varvec{0})=\theta _{0}\) and
where a positive number \(r_{j}\) \((1\le j\le n-1)\) satisfies
For every \(\theta \in (-1,1)\) one has
Let \(c\in (0,\infty )\) be given and let \(V_{c}\in C^{2}({\mathbb {R}}^{n})\) satisfy (1.8). Under some condition, we assert that a level set \(\{(\varvec{x}',x_{n})\,|\, V_{c}(\varvec{x}',x_{n})=\theta _{0}\}\) is a graph on the whole space \({\mathbb {R}}^{n-1}\) in the following proposition.
Proposition 6
Fix \(\theta _{0}\in (s_{*},1)\) with \(-W'(\theta _{0})>0\) arbitrarily. For any fixed \(c\in (0,\infty )\), let \(V_{c}\in C^{2}({\mathbb {R}}^{n})\) satisfy \(V_{c}(\varvec{0}',0)=\theta _{0}\), (1.8), (3.2), (3.14) and
Then, for arbitrarily given \(\mu _{0}\in (0,\infty )\), one has
Now we will make preparation to prove this proposition. For \(\mu _{0}\in (0,\infty )\), we define
Hereafter we simply write \(\Omega _{n-1}(\mu _{0})\) as \(\Omega _{n-1}\). Let \(c\in (0,\infty )\) be arbitrarily given. Taking an initial function
we consider \(w(\varvec{x}',x_{n},t;w_{0})\) as a solution of (1.6). Since \(w(\varvec{x}',x_{n},t;w_{0})\) is independent of \(x_{n}\), we simply write \(w(\varvec{x}',t;w_{0})\). Now \(w(\varvec{x}',t;w_{0})\) satisfies
Note that \(w(\varvec{x}',t;w_{0})\) is independent of \(c\in (0,\infty )\).
Lemma 7
Let \(w_{0}\) be given by (3.18). Then \(w(\varvec{x}',t;w_{0})\) satisfies
Proof
First we prove this lemma for \(n=2\). Let \(n=2\). Let \(\delta \in (0,\delta _{*}]\) be given. There exists \(T_{1}\in (0,\infty )\) such that we have
Note that \(w(\varvec{x},t;\theta _{0})\) depends only on t and is independent of \(\varvec{x}\). By applying [34, Theorem 5.8], there exists \(r_{1}\in (0,\infty )\) with
Combining these inequalities together, we have
Taking \(S_{1}\in (0,\infty )\) large enough, we have
which yields
Since \(w_{0}\) is symmetric in \(x_{1}\), we have
for \(x_{1}\in {\mathbb {R}}\) and \(t\ge 0\). Because the left-hand side of the above inequality is a subsolution, it is monotone increasing in \(t\ge 0\) and we can define
for \(x_{1}\in {\mathbb {R}}\). Now \(v_{\infty }\) satisfies
due to [26, 34]. Now we will show \(v_{\infty }\equiv 1\). For this purpose, we begin with
We define
and will show \(\Lambda =-\infty \). We will get a contradiction assuming \(\Lambda \in (-\infty ,S_{1}+\sigma \delta ]\). Then we have
Using (3.20) and the strong maximum principle, we have
Now we take \(R'\in (1+|\Lambda |,\infty )\) large enough such that we have
Then, we take \(h\in (0,1/(2\sigma ))\) small enough such that we have
If \(|x_{1}|\ge R'\), we have
Using
we have
Then, using
we find
Thus we get
Combining the two estimates stated above together, we obtain
Then we have
Sending \(t\rightarrow \infty \), we obtain
This contradicts the definition of \(\Lambda \). Thus we obtain \(\Lambda =-\infty \) and \(v_{\infty }\equiv 1\).
We will prove the lemma by induction. Let N be any integer with \(N\ge 3\). We prove this lemma for \(n=N\) assuming that it holds true for all n with \(n<N\). We have \(\varvec{x}'=(x_{1},\dots ,x_{N-1})\). Recall
Now \(w(\varvec{x}',t;w_{0})\) satisfies
where
Now we have \(\varvec{x}''=(x_{1},\dots ,x_{N-2})\in {\mathbb {R}}^{N-2}\) and
If \(|x_{N-1}|\ge \mu _{0}\), we have
Using
we have
By the assumption of the induction, Lemma 7 holds true for \(n=2,\dots ,N-1\). Let \(\delta \in (0,\delta _{*}]\) be given. There exists \(T_{2}\in (0,\infty )\) such that we have
By applying [34, Theorem 5.8] again, we have
by taking \(r_{2}\in (0,\infty )\) large enough. Thus we obtain
Then we can start the argument to prove Lemma 7 for \(n=2\) replacing (3.19) by (3.21), and we obtain
Thus Lemma 7 holds true for \(n=N\). Now it holds true for all \(n\ge 2\). This completes the proof. \(\square \)
Proof of Proposition 6
Assuming the contrary, we have
for some \(\mu _{0}\in (0,\infty )\). Since \(V_{c}(\varvec{x}',x_{n})\) is monotone non-increasing in \(x_{n}\), we have
Here \(w_{0}\) is given by (3.18). Taking the both sides as initial functions in (1.6), we have
Letting \(t\rightarrow \infty \) and applying Lemma 7, we obtain \(V_{c}\equiv 1\), which contradicts (3.13), (3.15), and (3.16). Now we complete the proof of Proposition 6. \(\square \)
Now we modify Theorem 5 in a form that is more useful for our discussion. Let
be arbitrarily given. In [31], we study an imbalanced reaction–diffusion equation
Then \(\Phi \) is the planar front with its speed k. The profile equation for a profile v with its speed \(c\in (0,\infty )\) is given by
For sufficiently small \(k>0\), say, \(k\in (0,k_{0})\) for \(k_{0}\in (0,c)\), we define a pyramidal traveling front solution \(v_{k}\) to (3.23) associated with a pyramid
for \(a_{j}\in [0,\infty )\) (\(2\le j\le n-1\)). For
one can define
For pyramidal traveling fronts, one can see [34] for instance. Now we have \(z_{k}\in {\mathbb {R}}\) with
Hereafter we write \(v_{k} (\varvec{x}',x_{n} + z_{k})\) simply as \(v_{k}(\varvec{x}',x_{n})\). Now we have
and
For every \(\eta \in (0,\infty )\), taking
given by [31, Lemma 1], we obtain
with
Using
we obtain a positive number \(\zeta _{k}\) with
Thus we have
with a positive number \(\zeta _{k}\). Then we define
for all \((\varvec{x}',x_{n})\) in every compact set in \({\mathbb {R}}^{n}\). Now we have \(V(\varvec{0}',0)=\theta _{0}\) and
See [31] for detailed arguments.
Now we state axially asymmetric traveling fronts as follows.
Theorem 8
(Axially asymmetric traveling fronts) Let \(c\in (0,\infty )\) be arbitrarily given. Let \(\{R_{j}\}_{1\le j\le n-1}\) satisfy (3.22). There exists \(V(\varvec{x})=V(\varvec{x}; c)\) that satisfies (1.8) with \(V(\varvec{0})=\theta _{0}\) and
with a positive number \(\zeta \). For every \(\theta \in (-1,1)\) one has
See Fig. 3 for the level set \(\{(\varvec{x}',x_{n})\,|\, V(\varvec{x}',x_{n})=0\}\) of V in Theorem 8.
Remark 2
The uniqueness of V in Theorem 8 is yet to be studied. It is an open problem to show V in Theorem 8 equals \(V_{\textrm{sym}}\) in Theorem 4 if \(R_{1}=R_{2}=\dots =R_{n-1}\).
Proof
Using (3.25) and Proposition 6, we obtain
It suffices to show
Assume \(\limsup _{k\rightarrow 0}\zeta _{k}=\infty \). Then we have \(\zeta =\infty \) and
Using
and applying Proposition 6, we have
This contradicts (3.27). Next we assume \(\liminf _{k\rightarrow 0}\zeta _{k}=0\). Then we have
Then we find
Thus the maximum principle gives
for \(1\le j\le n-1\). Then V is independent of \(x_{j}\) for \(1\le j\le n-1\) and is a function of \(x_{n}\), that is, \(V(x_{n}-ct)\) is a one-dimensional traveling front solution to (1.1). Since a one-dimensional traveling front solution to (1.1) and its speed is uniquely determined, we obtain \(c=0\). This contradicts \(c\in (0,\infty )\).
Then, taking a subsequence if necessary, we can define \(\zeta \in (0,\infty )\) with
Then V given by (3.24) satisfies Theorem 8. See [31] for detailed arguments. \(\square \)
Let \(\theta \in (-1,1)\) be arbitrarily given. We define \(R=R_{\theta }\) by
and have
For any given \((\xi _{1},\dots ,\xi _{n-1})\in {\mathbb {R}}^{n-1}\), we define \({\mathcal {D}}={\mathcal {D}}_{\theta }\) by
We have
where
Let \(c\in (0,\infty )\) be arbitrarily given and let \(V\in C^{2}({\mathbb {R}}^{n})\) satisfy (1.8), that is,
with (3.2), (3.13), (3.15), (3.14) and (3.16). Now V in Theorem 8 satisfies these assumptions. For any \(\theta \in (-1,1)\), we define \(q_{\theta }(\varvec{x}')\) by
Then we have \(q_{\theta }\in C^{1}({\mathbb {R}}^{n-1})\). If \(z_{0}\in {\mathbb {R}}\) satisfies \(q_{\theta }(\varvec{0}') < z_{0}\), we can uniquely determine \(x_{n}^{\theta }\in (0,\infty )\) with
and have
The following proposition plays an important role when we take the limits of traveling fronts as \(c\rightarrow \infty \).
Proposition 9
Let \(c\in (0,\infty )\) be arbitrarily given and let \(\theta \in (-1,1)\) be arbitrarily given. Assume \(V\in C^{2}({\mathbb {R}}^{n})\) satisfies (3.30), (3.2), (3.13), (3.14), (3.15) and (3.16). Then one has
where R is defined by (3.28). The right-hand side is independent of \(c\in (0,\infty )\).
We write the right-hand side of (3.32) as \(A(\theta ,R)^{2}\left| {\mathcal {D}}\right| \) with
Then (3.32) is written as
Let \(s_{1}\) be arbitrarily given with
The volume of \({\mathcal {D}}\) is given by \((2R) ^{n-1}\), and the surface area of the boundary of \({\mathcal {D}}\) is given by \(2(n-1)(2R)^{n-2}\). Using (3.28), we have
for every \((\xi _{1},\dots ,\xi _{n-1})\in {\mathbb {R}}^{n-1}\).
We define
Let \(\varvec{\nu }=(\nu _{1},\dots ,\nu _{n})\) be the outward normal vector on \(\partial \Omega \). We have
where
Now we have
Multiplying (3.30) by \(-\textrm{D}_{n}V\), we have
Integrating the both hand sides over \(\Omega \) and using the Gauss divergence theorem, we get
Here \(\textrm{d}S\) is the surface element of \(\partial \Omega \). Using
we get
Similarly, using
we get
Using \(\nu _{n}=0\) on \(\varGamma _{\textrm{f}}\), we have
We have
Now we calculate
Using
Then we obtain
Sending \(s_{1}\rightarrow -1\), we obtain
Now we use the following lemma.
Lemma 10
One has
Proof
Differentiating
by \(x_{j}\), we have
Then we have
Then we have
where
Since \(\varvec{\nu }\) is the outward normal vector at \(\partial \Omega \), we have
Thus we obtain
Now we complete the proof. \(\square \)
Now we give a proof for Proposition 9.
Proof of Proposition 9
Combining (3.34) and Lemma 10, we have
This completes the proof. \(\square \)
Now we show the following assertion.
Lemma 11
Under the same assumption of Proposition 9, one has
Here
Proof
Let \(\delta \in (0,\delta _{*}]\) be given. As was mentioned in the proof of Lemma 7, there exists \(T_{1}\in (0,\infty )\) such that we have
Let \((\varvec{x}_{0}',z_{0})\) belongs to
Then we have
By applying [34, Theorem 5.7],
if \(\nu \in (0,\infty )\) is large enough. Then we have
if \(\nu \in (0,\infty )\) is large enough. Since we can take \(\delta \in (0,\delta _{*}]\) arbitrarily small, the lemma follows from this inequality. \(\square \)
The following proposition asserts that Proposition 6 holds true for every \(\theta \in (-1,1)\). That is, for a traveling front \(V_{c}\), every level set \(\{(\varvec{x}',x_{n})\,|\, V_{c}(\varvec{x}',x_{n})=\theta \}\) is a graph on the whole space \({\mathbb {R}}^{n-1}\).
Proposition 12
Let \(c\in (0,\infty )\) be arbitrarily fixed. Let \(V_{c}(\varvec{x})\) satisfy
(1.8), (3.2), (3.13), (3.14) and (3.15). Then, for any given \(\mu _{0}\in (0,\infty )\) and for any given \(\theta \in (-1,1)\), one has
Proof
Now we define
Proposition 6 implies \((s_{*},1)\subset \Theta \). We define \(\theta _{\infty }=\inf \Theta \), and will show \(\theta _{\infty }=-1\). To do this, we will get a contradiction assuming \(\theta _{\infty }\in (-1,s_{*}]\). Then we choose \(\{\theta _{j}\}_{j\ge 1}\subset \Theta \) with
Now we define
Now \(V_{c}\) satisfies Proposition 9. We write the right-hand side of (3.32) as \(A(\theta ,R_{\theta })^{2}|{\mathcal {D}}_{\theta }|\), that is,
We define
We set
Using Proposition 9, we have \(\varvec{\eta }_{j}'\in {\mathcal {D}}_{\theta _{j}}\) with
Here \({\mathcal {D}}_{\theta _{j}}\) is given by (3.29) with \(\varvec{\xi }'\) in (3.36) and \(R=R_{\theta _{j}}\). Using Lemma 3, we can have \(\epsilon _{0}\in (0,1/2)\) with
where \(\varepsilon _{0}\) is independent of \(j\ge 1\). Then we define \(\varvec{x}(t)\) by
Then, using
we have
Now we have
where \(\Omega _{n-1}\) is given by (3.17). Using an assumption (3.15), we have
with \(x_{n}\in {\mathbb {R}}\) for \(j\ge 1\). This contradicts the definition of \(\theta _{\infty }\). Thus we obtain \(\theta _{\infty }=-1\). This completes the proof. \(\square \)
4 Proof of Theorems 1 and 2, and the limits of traveling fronts as the speeds go to infinity
Let \(\varvec{\alpha }'\) in (3.12) be arbitrarily fixed with (3.11). Let \(R_{1}\in (0,\infty )\) be arbitrarily fixed. Let \(V_{c}(\varvec{x})=V(\varvec{x};\varvec{\alpha }',c)\) be given by Theorem 8 for every \(c\in (0,\infty )\). Now \(V_{c}\) satisfies
with
Now we take \(\zeta _{c}\in (0,\infty )\) that depends on \(c\in (0,\infty )\) such that we have
When \(R_{0}\in (0,\infty )\) is arbitrarily given and we consider \(V_{\textrm{sym}}\) given by Theorem 4, we define \(\zeta _{c}\in (0,\infty )\) by
We take the limit of \(V_{c}\) as \(c\rightarrow \infty \). We have
where
Using Lemma 3, we have
Using
we obtain
Using Lemma 3, we have
where \(K_{*}\in (0,\infty )\) is independent of \((c,V_{c})\). Combining (4.1) and (4.2), we obtain
Indeed, we define \(m_{c}\in [0,\infty )\) by
and have
for some \((\varvec{y}',y_{n})\in {\mathbb {R}}^{n}\). Using (4.2), we have
and have
Combining this inequality and (4.1), we get \(\lim _{c\rightarrow \infty }m_{c}=0\), that is, (4.3).
Now we introduce
that is, \(x_{n}=\zeta _{c}-ct\). Then we define
Now \(u_{c}\) satisfies
Now we have
Then we find
Now we introduce
on every compact set in \({\mathbb {R}}^{n}\). The heat kernel in \({\mathbb {R}}^{n-1}\) is given by
Let \(t_{\textrm{init}}\in {\mathbb {R}}\) be arbitrarily given. Using (4.4), we get
for \(t>t_{\textrm{init}}\). Taking the limit of \(c\rightarrow \infty \) for the both sides, we find
for \(t>t_{\textrm{init}}\), which gives
for \(t>t_{\textrm{init}}\). Since \(t_{\textrm{init}}\in {\mathbb {R}}\) is arbitrary, we obtain
with
Thus the limit of an n-dimensional traveling front \(V_{c}\) gives an \((n-1)\)-dimensional entire solution U as \(c\rightarrow \infty \). The gradient in \({\mathbb {R}}^{n-1}\) is given by
The properties of U is as follows.
Proposition 13
Let
be arbitrarily given. Then U given by (4.5) satisfies
with
One has
Here \(L_{*}\in (0,\infty )\) is a constant depending only on (W, n).
Proof
The proof follows from the argument stated above. For the proof of the Schauder estimate (4.10) and (4.11), see [34, Proposition 2.9] for instance. \(\square \)
Now we introduce the following useful lemma.
Lemma 14
(Parabolic Harnack inequality) Let \(t_{1}\) and \(t_{2}\) satisfy \(-\infty<t_{1}<t_{2}<\infty \). Let \(a\in (0,\infty )\) be arbitrarily given. Let \({\mathcal {D}}\) be given by (3.29). Assume
for some \(1\le j\le n-1\). Then one has
where a constant C depends only on \((R, n, a, M, t_{2}-t_{1})\) and is independent of \((\xi _{1},\dots ,\xi _{n-1})\in {\mathbb {R}}^{n-1}\) and \(t_{1}\in {\mathbb {R}}\).
Proof
For every \(1\le j\le n-1\), we have
where \(\Delta '\) is defined by (1.10). Then, using (4.8), we obtain
for every \(1\le j\le n-1\). Here M is given by (3.1). Then this lemma follows from a general theory on the parabolic Harnack inequality [12, Chapter 7, Theorem 10]. \(\square \)
Now we show that U converges to 1 uniformly in \(\varvec{x}'\in {\mathbb {R}}^{n-1}\) as \(t\rightarrow \infty \).
Lemma 15
Under the same assumption of Proposition 13, one has
Proof
Using Proposition 13, we have
Let \(\Omega _{n-1}(R_{n-1})\) be defined by (3.17). Then we have
Now Lemma 7 gives
This completes the proof. \(\square \)
Combining Proposition 13 and Lemma 15, we obtain
Lemma 16
Under the same assumption of Proposition 13, let \(\mu _{0}\in (0,\infty )\) be arbitrarily given. Then one has
Proof
Assume the contrary. Then we have
Let \(\Omega _{n-1}\) be given by (3.17). We have
Let \(t_{0}\in {\mathbb {R}}\) be arbitrarily given. We have
Lemma 7 implies
Thus, for any given \(\delta \in (0,1-\theta _{0})\), we have \(T\in (0,\infty )\) that depends only on \(\delta \) such that we have
Since \(t_{0}\in {\mathbb {R}}\) can be chosen arbitrarily, we have
This contradicts (4.7). Now we complete the proof. \(\square \)
Let \(\theta \in (-1,1)\) be arbitrarily given. For \(\varvec{x}'\in {\mathbb {R}}^{n-1}\), we define \(h_{\theta }(\varvec{x}')\) by
if it exists. Lemma 15 and Lemma 16 imply that \(h_{\theta }(\varvec{x}')\) exists for every \(\varvec{x}'\in {\mathbb {R}}^{n-1}\) and every \(\theta \in [\theta _{0},1)\).
Lemma 17
Under the same assumption of Proposition 13, let R be given by (3.28), and let \({\mathcal {D}}\) be given by (3.29) for every \(\varvec{\xi }'\in {\mathbb {R}}^{n-1}\). Then one has
Let \(\theta \in (-1,\theta _{0})\) be arbitrarily given. If \(h_{\theta }(\varvec{x}')\) is defined for all \(\varvec{x}'\in {\mathcal {D}}\), one has
Proof
Proposition 9 implies
for any \(c\in (0,\infty )\). Combining this inequality and (4.3), we obtain (4.12) as \(c\rightarrow \infty \). Consequently, Proposition 9 implies
for any \(c\in (0,\infty )\). Combining this inequality and (4.3), we obtain (4.13) as \(c\rightarrow \infty \). \(\square \)
Now we assert
for any given \(1\le j\le n-1\). Indeed, in view of (4.8), we have \(\textrm{D}_{j}U \equiv 0\) in \({\mathbb {R}}^{n}\) if \(\textrm{D}_{j}U\) takes zero at some point with \(x_{j}>0\) by the maximum principle. Then, using (4.6), we have \(U(\varvec{0}',0)=\theta _{0}\). Combining this equality and (4.6) for all \(1\le j\le n-1\), we have \(\nabla ' U\equiv 0\) in \({\mathbb {R}}^{n}\). This contradicts (4.12) in Lemma 17. Thus (4.14) holds true.
Now we assert the following lemma.
Lemma 18
For every \(\theta \in (-1,\theta _{0})\), one has
for every \(\varvec{x}'\in {\mathbb {R}}^{n-1}\). That is, \(h_{\theta }(\varvec{x}')\) is defined for every \(\varvec{x}'\in {\mathbb {R}}^{n-1}\).
Proof
For arbitrarily given \(\mu _{4}\in (0,\infty )\), we consider \((\mu _{4},\dots ,\mu _{4})\in {\mathbb {R}}^{n-1}\). We take \(\varvec{\xi }'\in {\mathbb {R}}^{n-1}\) such that \({\mathcal {D}}\) given by (3.29) satisfies
Here \(\Omega _{n-1} (\mu _{4})\) is given by (3.17). Lemma 17 and (4.11) imply that there exists \(\varepsilon _{0}\in (0,\theta _{0})\) such that we have \(\varvec{x}_{0}'\in {\mathcal {D}}\) with
Then, using Proposition 13, we have
Since \(\mu _{4}\in (0,\infty )\) can be taken arbitrarily large, we complete the proof. \(\square \)
Lemma 18 implies that, as \(t\rightarrow -\infty \), \(U(\varvec{x}',t)\) converges to \(-1\) on every compact set in \({\mathbb {R}}^{n-1}\). Now we state the following assertion.
Lemma 19
Let U be given by (4.5) and let \(\theta \in (-1,1)\) be arbitrarily given. For any \((\xi _{1},\dots , \xi _{n-1})\in {\mathbb {R}}^{n-1}\), let \({\mathcal {D}}\) be given by (3.29). Then one has
Let \(a\in (0,\infty )\) be arbitrarily given. One has
Proof
Equation (4.15) follows from Lemmas 17 and 18. Equation (4.16) follows from (4.15). It suffices to prove (4.17) in
Let \(R=R_{\theta }\) be given by (3.28). We define
Let \({\mathcal {D}}_{\max }\) be
Let \(\tau \in {\mathbb {R}}\) satisfy
Let \(L_{*}\) be as in Proposition 13. We have
For some
we have
that is,
Here
We find
Thus, for some \(1\le j_{0} \le n-1\), we have
Now we have \((\xi _{1},\dots ,\xi _{n-1})\in {\mathcal {D}}_{\max }\). Using the parabolic Harnack inequality in Lemma 14, we obtain
Here \(C_{0}\in (0,\infty )\) is a constant depending only on \((M,n, R_{\max }, \theta , L_{*})\) and is independent of \((\xi _{1},\dots ,\xi _{n-1},\tau )\in {\mathbb {R}}^{n}\). Now we proved (4.17) and this completes the proof. \(\square \)
Proofs of Theorem 1 and Theorem 2
Now we prove (1.11) and (1.12). Let U be given by (4.5). Let \(a\in (0,\infty )\) be arbitrarily given. Using Lemma 19, we have
for every \(t\in {\mathbb {R}}\). By combining Lemma 7 and [34, Theorem 5.8], there exists \(T_{\varepsilon }\in (0,\infty )\) for any given \(\varepsilon \in (0,1)\) such that we have
for every \(t\in {\mathbb {R}}\). Thus we obtain
for every \(t\in {\mathbb {R}}\). Since \(\varepsilon \in (0,1)\) can be taken arbitrarily, we obtain (1.11) and (1.12). The other assertions in these two theorems follow from Proposition 13, Lemmas 15, 18 and 19. \(\square \)
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Acknowledgements
The author expresses his sincere gratitude to Professor Hirokazu Ninomiya of Meiji University and Professor Sigurd Angenent of University of Wisconsin, Madison for stimulating discussions. This work is supported by JSPS Grant-in-Aid for Scientific Research (C) Grant number 20K03702, JSPS Grant-in-Aid for Scientific Research (B) Grant number 20H01816 and JSPS Grant-in-Aid for Scientific Research (C) Grant number 22K03288.
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Taniguchi, M. Entire solutions with and without radial symmetry in balanced bistable reaction–diffusion equations. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02844-6
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DOI: https://doi.org/10.1007/s00208-024-02844-6