Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry

1.   Introduction

In this paper, we investigate the asymptotic symmetry and monotonicity of positive solutions to the following reaction-diffusion equation:

where $B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N\geqslant2$.

Reaction-diffusion equations are fundamental mathematical models that describe how the concentration of substances evolves over time under the influence of both chemical reactions and diffusion. They are widely applied in fields such as biology for pattern formation, chemistry for reaction waves, ecology for population dynamics, neuroscience for modeling brain activity, and physics for phase transitions and material science ^[1,2]. These equations provide critical insights into self-organization in complex systems and form a theoretical basis for understanding various spatial structures and dynamic behaviors in nature. In terms of the application to the asymptotic symmetry of solutions, they also have been used in the proofs of convergence results for some autonomous and time-periodic equations ^[3].

For elliptic equations, the method of moving planes, initially introduced by Alexandrov ^[4] and Serrin ^[5], and later developed by Berestycki and Nirenberg ^[6], Gidas, Ni, and Nirenberg ^[7], and Chen and Li ^[8], among others, is a powerful tool for investigating the symmetry and monotonicity of solutions. In ^[7], Gidas, Ni, and Nirenberg proved that if for each $u\in (0, \infty)$, the function $r\mapsto f(r, u):(0, 1)\to \mathbb{R}$ is non-increasing, then any positive $C^2(\bar B_R(0))$ solution of

is radially symmetric and decreasing in $r$. In recent years, several systematic approaches have emerged for studying symmetry and monotonicity in both local and nonlocal elliptic equations. These include the method of moving planes in integral form ^[9,10,11,12], the direct method of moving planes ^{[13,14,15,16,17,18]}, the method of moving spheres ^{[19,20,21,22,23]}, and sliding methods ^[6,24,25]. For further details on these methods, we refer to ^{[26,27,28,29]} and the references therein.

Notably, by combining hyperbolic geometry with the spirit of the moving plane method, a completely new monotonicity result can be achieved. Under the condition that $(1-r^2)^{(N+2)/2}f(r, (1-r^2)^{-(N-2)/2}u)$ is decreasing in $r\in(0, 1)$, for a fixed $u\in(0, \infty)$, Naito, Nishimoto, and Suzuki have sequentially achieved that each positive solution of (1.2) is radially symmetric and $(1-r^2)^{-(N-2)/2}u$ is decreasing in $r\in(0, 1)$ in the cases of $N = 2$ ^[30] and $N\geqslant2$ ^[31]. In ^[32], using not only hyperbolic geometry but also elliptic geometry, Shioji and Watanabe established the symmetry and monotonicity properties of a wide class of strong solutions of (1.2).

As to the parabolic equations, the situation becomes significantly more intricate. There have been some preliminary studies of symmetric solutions of the periodic-parabolic problem ^[33,34,35] and further results concerning entire solutions where the time variable $t\in\mathbb{R}$ ^[36,37]. In a different type of symmetry considering the Cauchy-Dirichlet problem of reaction-diffusion equations, asymptotic symmetry shows a tendency of positive solutions to improve their symmetry as time variable $t\in(0, \infty)$ increases, becoming "symmetric and monotone in the limit" as $t\to\infty$. In this context, Li ^[38] obtained symmetry for positive solutions in the case that the initial solutions are symmetric. Without the symmetric initial solutions, in bounded, symmetric, and strictly convex domain $\Omega$, Hess and Poláčik ^[39] showed the asymptotic symmetry and monotonicity of positive solutions to the following problem:

It was assumed in ^[39] that $f$ is uniformly Lipschitz-continuous in $u$ and Hölder-continuous of exponent $(\alpha, (\alpha/2))$ with respect to $(u, t)$. Subsequently, in both bounded and unbounded domains, Poláčik ^[40,41] extended the result to the following generalized fully nonlinear parabolic equation:

In an independent work, Babin and Sell ^[42] gave a similar result. With the symmetry conclusion of entire solutions of (1.4), Poláčik ^[37] gave a survey that summarized the following limitations of existing asymptotic symmetry results: the regularity requirements of the time-dependence of the nonlinearities and domain, the compactness requirements of spatial derivatives, and the strong positivity requirements on the nonlinearities in the case of nonsmooth domains ^[43]. Moreover, Saldaña and Weth ^[44] established the asymptotic foliated Schwarz symmetry which indicates that all positive solutions of (1.1) become axially symmetric with respect to a common axis passing through the origin as $t\to\infty$.

The motivation for this paper is to extend the results of ^[30,31,32] to the framework of reaction-diffusion equations. Our approach, which integrates elliptic geometry with the method of moving planes, is inspired by the work of ^[45] and ^[32].

In order to clarify the theorem, we first introduce the $\omega$-limit-set of $u$:

By the discussion of $u$ in the Appendix, the orbit $\{u(\cdot, t):t > 0\}$ is relatively compact in $C(\overline{B_1(0)})$ and $\omega(u)$ is a nonempty compact subset of $C(\overline{B_1(0)})$.

From now on, we will assume the nonlinearity $f:(0, 1)\times(0, \infty)\times[0, \infty)\to\mathbb{R}$ satisfies

(F1) for each $M > 0$, $f(r, u, t)$ is Lipschitz-continuous in $u$ uniformly with respect to $t$ and $r$ in the region $(0, 1)\times[-M, M]\times[0, \infty)$;

(F2) for each $r\in (0, 1)$ and $\tau > 0$ there exist a constant $H$ and a small constant $\varepsilon_0$ both independent of $\tau$ such that

for all $u\in(0, \infty)$ and $(r_1, t_1), \, (r_2, t_2)\in (0, 1)\times[\tau-\varepsilon_0, \tau+\varepsilon_0]$;

(F3) for all $t\in[0, \infty)$ and each fixed $u\in(0, \infty)$, $(1+r^2)^{\frac{N+2}{2}}f(r, (1+r^2)^{-\frac{N-2}{2}}u, t)$ is decreasing in $r\in (0, 1)$.

Theorem 1.1. Let $f$ satisfy conditions (F1), (F2), and (F3). Assume $u\in C^{2, 1}(B_1(0)\times(0, \infty))\cap C(\overline{B_1(0)}\times[0, \infty))$ is a positive bounded solution of (1.1) satisfies that $\partial u/\partial t$ is non-negative, bounded, and $\nabla u(x, t)$ is bounded for all $(x, t)\in B_1(0)\times(0, \infty)$. Then for each $\varphi(x)\in\omega(u)$, the following holds:

● Either $\varphi(x)\equiv 0$;

● Or $\varphi(x)$ is radially symmetric about the origin and satisfies $\partial_{r}\left((1+|r|^2)^{\frac{N-2}{2}}\varphi(r)\right) < 0$ for $r = |x|\in(0, 1)$, where $x\in B_1(0)$.

The structure of the paper is as follows: Section 2 presents the preliminary results that form the foundation for our main findings. In Section 3, we introduce the asymptotic narrow region principle, which plays a crucial role in the proof of our main theorem. The proof of Theorem 1.1 is provided in detail in Section 4. Finally, the Appendix includes additional properties of the solutions.

2.   Preliminaries

In this section, we will establish some essential preliminaries for applying the moving plane method in the space $(B_1(0), g)$, where the metric tensor $g$ is defined by

Here, $|\cdot|$ denotes the standard Euclidean norm, consistent with the notation used in other sections. For each $\lambda \in (0, 1)$, let $T_\lambda \subset B_1(0)$ be a totally geodesic plane intersecting the $x_1$-axis orthogonally at the point $(\lambda, 0, \dots, 0)$. It follows that

where

Define

For each $x \in \Sigma_\lambda$, let $x^\lambda$ denote the reflection of $x$ with respect to $T_\lambda$ in the space $(B_1(0), g)$. This reflection can be expressed as

We remark that

The proof is presented in Lemma 5.1.

The Laplace-Beltrami operator $\Delta_{(g, x)}$ in the space $(B_1(0), g)$ at $x\in B_1(0)$ is defined by

where $\Delta = \sum_{i = 1}^{N}\frac{\partial^2}{\partial x_i^2}$.

Let $u(x, t)$ be a solution to the parabolic problem given by equation (1.1). For each $\lambda\in(0, 1)$, we introduce new functions $v(x, t)$, $w_\lambda(x, t)$, and $z_\lambda(x, t)$ to compare the value of $u(x, t)$ with $u(x^\lambda, t)$ and to simplify the analysis of the gradient's impact. These functions are defined as follows:

From (2.8)–(2.10), for $(x, t)\in\Sigma_\lambda\times(0, \infty)$, we obtain

Clearly, $z_\lambda\in C^{2, 1}(\Sigma_\lambda\times(0, \infty))\cap C(\overline{\Sigma_\lambda}\times[0, \infty)])$ and $z_\lambda = 0$ on $T_\lambda$. For each $\varphi(x)\in\omega(u)$, denote

which is an $\omega$-limit of $z_\lambda(x, t)$.

By virtue of the definitions given in (2.7) and (2.8), we observe that for $x\in B_1(0)$, the function $v$ satisfies

In addition to the above notations, we now present the following lemmas to establish the properties of $z_\lambda(x, t)$.

Lemma 2.1. Let $u\in C^{2, 1}(B_1(0)\times(0, \infty))\cap C(\overline{B_1(0)}\times[0, \infty))$ be a positive bounded solution of (1.1) that satisfies $\frac{\partial u}{\partial t}(x, t)\geqslant 0$ for all $(x, t)\in B_1(0)\times(0, \infty)$. Assume that the function $(1+r^2)^{\frac{N+2}{2}}f(r, (1+r^2)^{-\frac{N-2}{2}}s, t)$ is nonincreasing in $r\in(0, 1)$ for each fixed $s\in(0, \infty)$ and $t\in(0, \infty)$. Under these conditions, $z_\lambda$ satisfies

in $\Sigma_\lambda\times(0, \infty)$, where

Proof. Let $\lambda\in(0, 1)$, $x\in\Sigma_\lambda$, and set $y = x^\lambda$. Since the Laplace-Beltrami operator is invariant under the isometry, as shown in Lemma 5.2 in the Appendix, we have

Using this equality, together with (2.6) and the monotonicity assumption (F3) of $(1+r^2)^{\frac{N+2}{2}}f(r, (1+r^2)^{-\frac{N-2}{2}}s, t)$, we deduce that

Thus we obtain

From (2.10), an elementary computation shows that

Therefore, $z_\lambda$, as defined in (2.10), satisfies the inequality given in (2.14). □

Lemma 2.2. Assume that for some $(x_0, t_0)\in T_\lambda\times(0, \infty)$, there holds

where $n$ denotes the unit outer normal to $\partial\Sigma_\lambda$. Then,

Proof. Since $z_\lambda(x, t) = 0$ on $T_\lambda\times(0, \infty)$, we have

We define $x_p$ as $x_p = x_0-pn(x_0)$, $p > 0$, and $n(x_0)$ is the unit outer normal to $\partial\Sigma_\lambda$ at point $x_0$. Specifically, $n(x_0) = -(x_0-e_\lambda)/|x_0-e_\lambda|$.

For $x_p\in \Sigma_\lambda$, using the definition in (2.5), we find that

where

This result follows from the property given in (2.5) that

Then, it follows that

From (2.19), we conclude that inequality (2.18) holds. □

3.   Asymptotic maximum principles

In this section, we present the following asymptotic narrow region principle, which will play a crucial role in establishing Theorem 1.1.

Lemma 3.1. (Asymptotic narrow region principle) Assume that $\Omega$ is a bounded narrow region with respect to $e_\lambda$ contained within the annulus defined by

for some small $\delta > 0$, where $e_\lambda$ is defined by (2.3).

For sufficiently large $\overline{t}$, assume that $z_\lambda(x, t)\in C^2(\Omega)\times C^1([\, \overline{t}, \infty])$ is bounded and lower semi-continuous in $x$ on $\overline{\Omega}$, and satisfies

where $c_\lambda(x, t)$ is bounded from above. Then for sufficiently small $\delta$ the following statement holds:

Proof. Let $m$ be a fixed positive constant that will be determined later. Define

Then $\tilde{z}_\lambda(x, t)$ satisfies

and

Thus, we find that

Let $\phi(x)$ be defined as

For each $x\in\Omega$, we have

Direct calculation shows

Thus,

We claim that for any $T > \overline{t}$,

If (3.6) is not true, by (3.2) and the lower semi-continuity of $z_\lambda$ on $\overline{\Omega}\times[\overline{t}, T]$, there exists $(x_0, t_0)$ in $\Omega\times(\overline{t}, T]$ such that

Since

and

where $T_\lambda$ is defined in (2.2), the minimum point $x_0$ is an interior point of $\Omega$. Therefore,

From (3.4), (3.5), and (3.8)–(3.10), we have

Since $c_\lambda(x, t)$ is bounded from above for all $(x, t)\in B_1(0)\times(0, \infty)$, we can choose $\delta$ small enough such that

Taking $m = 1/2\delta^2$, we derive that the right-hand side of the second inequality of (3.11) is strictly greater than $0$, since $\tilde{z}_\lambda(x_0, t_0) < 0$. This contradicts with (3.11).

Therefore, given the boundedness of $z_\lambda$, there exists a constant $C > 0$ such that

Thus, we have

Taking the limit $t\to\infty$, we obtain

This completes the proof of Lemma 3.1. □

In the proof of the main theorem, we will also use the following two classical maximum principles. For the convenience of the reader, we now show them in the version suitable for this article.

Lemma 3.2. (Strong parabolic maximum principle for a not necessarily non-negative coefficient) For each $\lambda\in(0, 1)$, assume $z(x, t)\in C^{2, 1}\left(\Sigma_\lambda\times(0, \infty)\right)\cap C\left(\overline{\Sigma_\lambda}\times[0, \infty)\right)$ and for $(x, t)\times\Sigma_\lambda\times(0, \infty)$, $z(x, t)\geqslant0$ satisfies that

where $c(x, t)$ is bounded in $\Sigma_\lambda\times(0, \infty)$. If $z(x, t)$ attains its minimum $0$ over $\overline{\Sigma_\lambda}\times [0, \infty)$ at a point $(x_0, t_0)\in\Sigma_\lambda\times(0, \infty)$, then $z(x, t)\equiv0$ in $\Sigma_\lambda\times(0, t_0]$.

Proof. For each $\lambda\in(0, 1)$, we set $c_0 = \mathop{sup}\limits_{\Sigma_\lambda\times(0, \infty)}c(x, t)$ and let

and then for $(x, t)\times\Sigma_\lambda\times(0, \infty)$, $\overline{z}(x, t) = e^{-c_0t}z(x, t)\geqslant 0$ and satisfies that

If $z(x, t)$ attains its minimum $0$ at $(x_0, t_0)$, then $\overline{z}(x, t)$ also gets its minimum $0$ at the point $(x_0, t_0)$ over $\overline{\Sigma_\lambda}\times[0, \infty)$. Considering $c_0-c(x, t)\geqslant 0$ in $\Sigma_\lambda\times(0, \infty)$, then from the strong parabolic maximum principle with $c(x, t)\geqslant 0$ (Theorem 12 in ^[46], Chapter 7), we can obtain that for $(x, t)\in\Sigma_\lambda\times(0, t_0]$, we have $\overline{z}(x, t)\equiv0$ and therefore $z(x, t) = e^{c_0t}\, \overline{z}(x, t)\equiv0$. □

Lemma 3.3 (Parabolic Hopf's lemma for a not necessarily non-negative coefficient). For each $\lambda\in(0, 1)$, we let $(x_0, t_0)$ be a point on the boundary of $\Sigma_\lambda\times(0, T)$ for $\forall T > 0$ such that $z(x_0, t_0) = 0$ is the minimum in $\overline{\Sigma_\lambda}\times[0, T]$. Assume that there exists a neighborhood $V: = |x-x_0|^2+|t-t_0|^2 < R^2_0$ of $(x_0, t_0)$ such that for $(x, t)\in V\cap\left(\Sigma_\lambda\times(0, T)\right)$, $z(x, t) > 0$ and satisfies

where $c(x, t)$ is bounded in $\Sigma_\lambda\times(0, T)$. If there exists a sphere $S: = |x-x'|^2+|t-t'|^2 < R$ passing through $(x_0, t_0)$ and contained in $\overline{\Sigma_\lambda}\times[0, T]$ and $(x_0, t_0)\not = (x', t')$, then under the assumptions made above, we have

where $n$ is the unit outer normal of $\partial \Sigma_\lambda\,$ for the fixed $t_0$.

Proof. For each $\lambda\in(0, 1)$, we set $c_0 = \mathop{sup}\limits_{\Sigma_\lambda\times(0, \infty)}c(x, t)$ and let

Then at the point $(x_0, t_0)$, $z(x, t)$ also gets its minimum $0$ and for $(x, t)\in V\cap\left(\Sigma_\lambda\times(0, T)\right)$, $\overline{z}(x, t) > 0$ and satisfies that

Considering $(c_0-c(x, t))\geqslant 0$ in $\sigma_\lambda\times(0, T)$, then from the parabolic Hopf's lemma (Theorem 2 in ^[47]), we can derive that every outer non-tangential derivative $\partial\overline{z}\backslash\partial\nu$ at $(x_0, t_0)$ is negative, where $\nu$ represents any outer non-tangential vector. Particularly, by the definition of $\overline{z}(x, t)$, for the fixed $t_0$ and the unit outer normal $n$ of $\partial \Sigma_\lambda$, we have $\partial z\backslash\partial n(x_0, t_0) = \partial \overline{z}\backslash\partial n(x_0, t_0) < 0$. □

4.   Proof of the main theorem

We will carry out the proof in two steps. For simplicity, choose any direction within the region to be the $x_1$ direction. The first step is to show that for $\lambda$ sufficiently close to the right end of the domain, the following holds for all $\varphi\in\omega(u)$:

This provides the initial position to move the plane. We then move the plane $T_\lambda$ to the left as long as the inequality (4.1) continues to hold, until it reaches its limiting position. Define

We will show that $\lambda_0 = 0$. Since the $x_1$ direction can be chosen arbitrarily, this implies that for any $\varphi\in\omega(u)$, $\varphi(x)$ is radially symmetric and $(1+|x|^2)^{\frac{N-2}{2}}\varphi(x)$ is monotone decreasing about the origin. We will now detail these two steps.

Proof of Theorem 1.1. For all $\varphi\in\omega(u)$, we assume $\varphi\not\equiv0$ in $B_1(0)$.

Step 1. We show that for $\lambda < 1$ and sufficiently close to $1$, the following holds:

The Lipschitz continuity assumption (F1) on $f$ implies that $c_\lambda(x, t)$ is bounded. Additionally, we have

since $u(x, t) = 0$, for $(x, t)\in \partial B_1(0)\times(0, \infty)$, and $u$ is positive in $B_1(0)$. Combining this with (2.14), we can apply Lemma 3.1 to conclude that (4.3) holds.

Step 2. We will demonstrate that the parameter $\lambda_0$, as defined in (4.2), is equal to zero, that is to say

Assume for contradiction that $\lambda_0 > 0$. We will demonstrate that $T_{\lambda_0}$ can be shifted slightly to the left, thereby contradicting the definition of $\lambda_0$.

To begin with, we intend to determine the sign of $\psi_{\lambda_0}$ for any $\varphi\in\omega(u)$ and all $x\in\Sigma_{\lambda_0}$ under the case $\lambda_0 > 0$. To achieve this, according to the definition (2.12), we now need to discuss the inequality that $z_{\lambda_0}(x, t)$ satisfies as $t\to\infty$. From the definition of $\omega(u)$, for each $\varphi\in \omega(u)$, there exists a sequence $\{t_k\}$ such that $u(x, t_k)\to \varphi(x)$ as $t_k\to \infty$. Define

and

Then we have $u_k(x, 1)\to\varphi(x)$ in the sense of $C(B_1(0))$ as $k\to\infty$. Let $Q_1: = B_1(0)\times[1-\varepsilon_0, 1+\varepsilon_0]$. We now have

Similarly as in (2.8)–(2.11), we have the following definitions of functions:

It follows from (4.7) and Lemma 2.1 that

where

Based on the relationship between the functions (4.8)–(4.10) and $u_k$ defined as (4.5), using Lemma 5.3, we deduce the existence of subsequences $v_k$, $w_{\lambda_0, k}$, and $z_{{\lambda_0}, k}$ which converge uniformly to the respective functions $v_\infty$, $w_{\lambda_0, \infty}$, and $z_{{\lambda_0}, \infty}$ all in the sense of $C^{2, 1}\left(\Sigma_{\lambda_0}\times[1-\varepsilon_0, 1+\varepsilon_0]\right)$ as $k \to \infty$ and they satisfy

Furthermore, both in the sense of $C\left(\Sigma_{\lambda_0}\times[1-\varepsilon_0, 1+\varepsilon_0]\right)$ as $k\to\infty$ we have

where $c_{{\lambda_0}, \infty}(x, t)$ is bounded in $\Sigma_{\lambda_0}\times[1-\varepsilon_0, 1+\varepsilon_0]$. This follows from the proof of Lemma 5.3, where it is established that the sequence $f_k$ within the definition of $c_{\lambda_0, k}$ uniformly converges to $f_\infty$ satisfying (F1) in the sense of $C\left(\Sigma_{\lambda_0}\times[1-\varepsilon_0, 1+\varepsilon_0]\right)$. For any $\varphi\in\omega(u)$, by the definition of the limit set $\omega(u)$, there exists $t_k$ such that $z_{\lambda_0}(x, t_k)\to\psi_{\lambda_0}(x)$ in the sense of $C(\Sigma_{\lambda_0})$ as $t_k\to\infty$. Particularly, according to the convergence of $z_{\lambda_0, k}$, in the sense of $C^2(\Sigma_{\lambda_0})$ we have

Combining the continuity of $z_{\lambda_0, \infty}$ with respect to $t$ and the definition of $\lambda_0$, we deduce that $z_{{\lambda_0}, \infty}(x, t)$ satisfies the following inequalities:

We apply Lemma 3.2 to (4.12) in $\Sigma_{{\lambda_0}}\times[1-\varepsilon_0, 1+\varepsilon_0]$ to get either

or

In case (4.13), since $z_{{\lambda_0}, \infty}(x, t)\equiv0$, for all $(x, t)\in\Sigma_{\lambda_0}\times[1-\varepsilon_0, 1+\varepsilon_0]$, we have

From (2.13), for $(x, t)\in\Sigma_{\lambda_0}\times(1-\varepsilon_0, 1+\varepsilon_0]$, we have

and

By (4.15) we can obtain that

for $(x, t)\in\Sigma_{\lambda_0}\times[1-\varepsilon_0, 1+\varepsilon_0]$. Combining (4.16) and (4.17), we finally arrive at

for $(x, t)\in\Sigma_{\lambda_0}\times(1-\varepsilon_0, 1+\varepsilon_0]$. Since Lemma 5.1 implies that $|x| > |x^{\lambda_0}|$, (4.18) contradicts the assumption (F3) of $f$. It follows that case (4.13) is invalid.

Next from case (4.14), we can derive that for all $\varphi\in\omega(u)$,

Since $\lambda_0 > 0$, we now attempt to slightly shift $\lambda_0$ to the left. If $\lambda_0$ still meets definition (4.2), we can construct a contradiction. For any small $l > 0$, we set $\overline{V_{\lambda_0+l}} = x\in\left\{x\in\overline{B_1(0)}\, \left||x-e_{\lambda_0}|\geqslant\frac{1+\lambda_0^2}{2\lambda_0}+l\right.\right\}$, and for $\psi_{\lambda_0}$, there exists a $C_\varphi > 0$, such that

We now show that, for all $\varphi\in\omega(u)$, there exists a universal constant $C_0$ such that

If not, there exists a sequence of functions $\{\psi_{\lambda_0}^k\}$ with respect to ${\varphi^k}\subset\omega(u)$ and a sequence of points $\{x^k\}\subset\overline{V_{\lambda_0+l}}$ such that for each $k$ we have

By the compactness of $\omega(u)$ in $C(\overline{B_1(0)})$, there exists $\psi_{\lambda_0}^0$ which corresponds to some $\varphi^0\in\omega(u)$ and $x^0\in\overline{V_{\lambda_0+l}}$ such that

as $k\to\infty$ in the sense of $C(B_1(0))$. Now by (4.22) and the definition of $\lambda_0$, we obtain

which contradicts (4.19), since $\varphi^0\in\omega(u)$. Thus (4.21) must be established.

From (4.21) and the continuity of $\psi_\lambda$ with respect to $\lambda$, for each $\psi_\lambda$, under a fixed $C_0$, there exists $\varepsilon_\varphi > 0$ such that

Similarly due to the compactness of $\omega(u)$ in $C(\overline{B_1(0)})$, for all $\psi_\lambda$, there exists a universal $\varepsilon > 0$ such that

Consequently, for $t$ sufficiently large, we have

Since $l > 0$ is small, by (4.22), we can choose $\varepsilon > 0$ small, such that $\Sigma_\lambda\backslash V_{\lambda_0+l}$ is a narrow region defined as (3.1) for $\lambda\in(\lambda_0-\varepsilon, \lambda_0)$, and then applying the asymptotic narrow region principle (Lemma 3.1), we arrive at

Combining (4.23) and (4.24), we derive that

This contradicts the definition of $\lambda_0$. Therefore, $\lambda_0 = 0$ must be true.

As a result, (4.4) implies that for all $\varphi\in\omega(u)$,

that is to say, for all $\varphi\in\omega(u)$ and $x\in\Sigma_0$,

Since $x^0 = (-x_1, x_2, \cdots, x_N)$ for $x\in\Sigma_0$ with $0 < x_1 < 1$, finally we can get that

Since the $x_1$ direction can be chosen arbitrarily, (4.25) implies that all $\varphi(x)$ are radially symmetric about the origin. Combining with (4.12) and the proof of (4.19), we can derive that for $0 < \lambda < 1$, $z_{\lambda, \infty}$ satisfies

where $c_{\lambda, \infty}$ is bounded. Then we can apply Lemma 3.3 to obtain that

where $n$ is the outer normal vector of $\partial\overline{\Sigma_\lambda}$. From Lemma 2.2, we can conclude that

Under the conclusion that all $\varphi(x)$ are radially symmetric about the origin, from (4.26) we can infer that for all $0 < r < 1$,

which shows asymptotic monotonicity. Now we complete the proof of Theorem 1.1. □

5.   Appendix

We note that $B_1(0) = \{x\in\mathbb{R}^N:|x| < 1\}$, $N\geqslant3$, we set $S^+ = \{X\in\mathbb{R}^{N+1}:|X| = 1, X_{N+1} > 0\}$, and define a mapping $P:(S^+, |dX|^2)\to(B_1(0), g)$ by

The overall idea of the proof of (2.2)–(2.5) is similar to that in ^[32] in the case $a = 1$, so we omit the proof process. Strongly inspired by Lemma A.1 and Lemma A.2 in ^[31], here we give the proof of (2.6) and (2.16).

Lemma 5.1. We have

Therefore, we can derive $|x| > |x^\lambda|$ for $x\in\Sigma_\lambda$.

Proof. We define $a = -e_\lambda /|e_\lambda|^2 = (2\lambda/(1-{\lambda}^2), 0, \cdots, 0)$. From this definition, (2.3), and (2.5), by some elementary computations, we note that

Then we have

where we use the property of distance operations that for $p, q\in\mathbb{R}^N\backslash\{0\}$,

Thus we obtain

which implies (5.1). Due to the definition of $\Sigma_\lambda$, for $x\in\Sigma_\lambda$, we have

and then we can find that $|x| > |x^\lambda|$. □

Lemma 5.2. Assume that $v(x)\in C^{2}(B_1(0))$. Let $y = x^\lambda$ and $v(y)$ is a function with $y$ as the independent variable. We define $v_\lambda(x) = v(x^\lambda)$ as a new form of the function with $x$ as the independent variable. Then

where $\Delta_x = \sum_{i = 1}^{N}\partial^2/\partial x_i^2$, $x\cdot\nabla_x = \sum_{i = 1}^{N}x_i\partial/\partial x_i$, and $\Delta_{(g, x)}$ are defined as in (2.7).

Proof. To compare the values of $\Delta_{(g, x)}v_\lambda(x)$ and $\Delta_{(g, y)}v(y)$ through direct computation, we define $u(y)$ and $u_\lambda(x)$ as

Then we can find that

For simplicity, we define

By (2.5), (5.1), and (5.5), it follows that

Then we can calculate that

Then we obtain

From (5.7) we also have

Through the above calculation process, we finally get that

By (5.6), for each $i$, we have

Then (5.9) implies

By (5.1), we have

From (5.3) and (5.4), we can conclude that (5.2) holds. □

For each fixed $\tau > 0$ and $0 < \alpha < 1$, we define the parabolic cylinder $Q_{\tau}: = B_1(0)\times[\tau-\varepsilon_0, \tau+\varepsilon_0]$, where $\varepsilon_0$ is a small constant. For any points $(x, t), (\tilde{x}, \tilde{t})\in Q_\tau$, since $u(x, t)\in C^{2, 1}(Q_\tau)$, there exist two constants $\xi_t\in(\min\{t, \tilde{t}\}, \max\{t, \tilde{t}\})$, $\xi_x\in(0, 1)$, such that

Again by $u(x, t)\in C^{2, 1}(Q_\tau)$ and the boundedness of $Q_\tau$, we have $\left|u_t(x, \xi_t)\right|$, $\left|\nabla u(x+\xi_x(\tilde{x}-x), t)\right|$, $\left|t-\tilde{t}\right|^{1-\frac{\alpha}{2}}$, and $\left|x-\tilde{x}\right|^{1-\alpha}$ are all bounded. Then for some $0 < \alpha < 1$, there exists a constant $C_0 > 0$ such that

Combining (5.10) with the boundedness of $u(x, t)$ in $Q_\tau$, for any $\tau > 0$, the orbit $\{u(\cdot, t), \, t\in[\tau-\varepsilon_0, \tau+\varepsilon_0]\}$ is relatively compact in $C(B_1(0))$. To directly address the properties of the functions in $\omega(u)$, we present the following lemma.

Lemma 5.3. Let $M: = sup\{\Arrowvert u(\cdot, t)\Arrowvert_{L^{\infty}}:t > 0\}$, and then, under definitions (4.5) and (4.6), for each $k$, $u_k$ and $f_k$ satisfy the problem (4.7).

Then there exist some functions $u_\infty\in C^{2, 1}(B_1(0)\times[1-\varepsilon_0, 1+\varepsilon_0])$ and $f_\infty\in C((0, 1)\times[-M, M]\times[1-\varepsilon_0, 1+\varepsilon_0])$ such that $u_k\to u_\infty$ in the sense of $C^{2, 1}(B_1(0)\times[1-\varepsilon_0, 1+\varepsilon_0])$ and $f_k\to f_\infty$ as $k\to\infty$ in the sense of $C((0, 1)\times[-M, M]\times[1-\varepsilon_0, 1+\varepsilon_0])$. With a constant $\varepsilon_0$, $u_\infty(x, t)$ satisfies

Proof. For each $K$ and any $(x, t), (u, \tilde{u}), (\tilde{x}, \tilde{t})$ in the bounded domain $Q_u: = (0, 1)\times[-M, M]\times[1-\varepsilon_0, 1+\varepsilon_0]$, by (5.13) and definition (4.6), we set $C'$ to be the maximum of $C_u$, $C_x$, and $C_t$ appearing in (5.13), with any $\varepsilon > 0$, if $\left||x|-|\tilde{x}|\right|^{\alpha}+\left|u-\tilde{u}\right|+\left|t-\tilde{t}\right|^{\frac{\alpha}{2}} < \delta(\varepsilon) = \varepsilon\backslash C'$, and then we have

which means the sequence $\left\{f_k\right\}_{k\in\mathbb{N}}$ is equicontinuous in $Q_u$. In addition, owing to the boundedness of $Q_u$ and the continuity of $f$ in $Q_u$, we deduce that $\left\{f_k\right\}_{k\in\mathbb{N}}$ is bounded in $Q_u$ for each $k$. Then the Ascoli-Azela theorem implies that there exists a function $f_\infty\in C(Q_u)$ such that $f_k\to f_\infty$ in the sense of $C(Q_u)$ as $k\to\infty$, and $f_\infty$ is Lipschitz continuous in $u$ by (F1).

Next we discuss the convergence of $\{u_k\}_{k\in\mathbb{N}}$. Set $Q_1: = B_1(0)\times[1-\varepsilon_0, 1+\varepsilon_0]$, and from (4.7), for each $k$, let $\tilde{f}_k(x, t) = f_k(|x|, u_k(x, t), t)$. Then we have

Additionally, for any $(x, t), (\tilde{x}, \tilde{t})\in Q_1$, by conditions (F1) and (F2), we have three constants $C_x$, $C_u$, and $C_t$ such that

By (5.10), we set $C = \max\{C_x, C_uC_0, C_t\}$, and then for any $(x, t), (\tilde{x}, \tilde{t})\in Q_1$, we have

which means for any two points $P(x, t)$ and $\tilde{P}(\tilde{x}, \tilde{t})$, we define $d(P, \tilde{P}) = (|x-\tilde{x}|+|t-\tilde{t}|^{\frac{1}{2}})$, and then we can get that

Given that $u_k$ is the solution of (4.7), then the existence and uniqueness theorem of classical solutions for heat equations (Theorem 3.3.7 in ^[48]) implies that

where $C$ is a constant independent of $k$, $Du_k = \left(\partial u_k/\partial x_1, \partial u_k/\partial x_2, \cdots, \partial u_k/\partial x_N\right)$, and $D^2u_k$ is the Hessian matrix of $u_k$. On the one hand (5.16) shows the equicontinuity of $u_k$, $Du_k$, $D^2u_k$, and $\partial u_k/\partial t$ in $C(Q_1)$, which means for a given $\varepsilon > 0$, we can choose $\delta > 0$ independent of $k$ such that

for all $P, \tilde{P}\in Q_1$ and $k$. On the other hand (5.16) also ensures that for each $k$, $u_k$, $Du_k$, $D^2u_k$, and $\partial u_k/\partial t$ are all bounded in $Q_1$. By the Ascoli-Azela theorem, there exist functions $u_\infty$, $u_{t\infty}$, $\left\{u_{i\infty}\right\}_{i = 1}^N$, $\left\{u_{ij\infty}\right\}_{i, j = 1}^N\in C(Q_1)$ such that as $k\to\infty$ for each $i$ and $j$ we have

which are all in the sense of $C(Q_1)$. Employing the fundamental theorem of caculus and (5.16), for $\varepsilon > 0$, $(x, t)\in Q_1$, and fixed $i$, we can choose $\overline{\delta} = \min(\delta, 1) > 0$ independent of $k$ such that $B_{\overline{\delta}}(x)\times[1-\varepsilon_0, 1+\varepsilon_0]\in Q_1$ and for all $|h| < \overline{\delta}$ and $k$ we have

Similarly we can get that

Letting $k\to\infty$, we can derive that

Then (5.17) means that for each $i$ and $j$,

as $h\to 0$. The above discussion shows that $u_\infty\in C^{2, 1}(Q_1)$ and as $k\to\infty$ we observe that $u_k \to u_\infty$ in the sense of $C^{2, 1}(Q_1)$ and

all in the sense of $C(Q_1)$. By (5.19) and the existence of $u_\infty$ and $f_\infty$, considering the problem (4.7) as $k\to\infty$, we can deduce that $u_\infty$ satisfies

Then we complete the proof of Lemma 5.3. □

Author contributions

Baiyu Liu: Methodology, writing–review and editing; Wenlong Yang: Writing–original draft preparation.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors wish to thank the handing editor and the referees for their valuable comments and suggestions. Baiyu Liu is supported by the National Natural Science Foundation of China (No. 12471089).

Conflict of interest

The authors declare there is no conflict of interest.