Optimal and Sharp Convergence Rate of Solutions for a Semilinear Heat Equation with a Critical Exponent and Exponentially Approaching Initial Data

Abstract

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion.

1 Introduction and Results

In this paper, we investigate the behavior of solutions of the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} u_t = \varDelta u + |u|^{ p-1 }u, &{} \quad x \in \mathbb {R}^N ,\quad t > 0, \\ u(x,0) = u_0 (x), &{} \quad x \in \mathbb {R}^N , \\ \end{array} \right. \end{aligned}$$

(1.1)

where $u=u(x,t)$, $\varDelta$ is the Laplace operator with respect to x, $p>1$, and $u_0 \not \equiv 0$ is a given continuous function on $\mathbb {R}^N$ that decays to zero as $|x| \rightarrow \infty$. The problem (1.1) has been studied in many papers, since Fujita studied the blow-up problem [6]. Among them, the stability problem of stationary solutions is one of the most important problems and we study the problem (1.1) along this line.

It is known that there exist critical exponents p that govern the structure of solutions. The exponent

$$\begin{aligned} p_S = {\left\{ \begin{array}{ll} \frac{N+2}{N-2} &{}\quad N > 2,\\ \infty &{}\quad N \le 2, \end{array}\right. } \end{aligned}$$

is well known as the Sobolev exponent that is critical for the existence of positive stationary solution of (1.1). Namely, there exists a classical positive radial solution $\varphi$ of

$$\begin{aligned} \varDelta \varphi + \varphi ^p =0, \quad \quad x \in \mathbb {R}^N, \end{aligned}$$

if and only if $p \ge p_S$ [1, 2, 8]. We denote the solution by $\varphi =\varphi _\alpha (r), r=|x|, \alpha >0$, where $\varphi _\alpha (0) = \alpha$. Then $\varphi _\alpha (r)$ satisfies the initial value problem

$$\begin{aligned} \left\{ \begin{array}{l} \varphi _{\alpha ,rr} + \frac{N-1}{r} \varphi _{\alpha ,r} +\varphi _\alpha ^p = 0, \\ \varphi _\alpha (0) = \alpha , \quad \varphi _{\alpha ,r}(0) = 0. \end{array} \right. \end{aligned}$$

(1.2)

For each $\alpha >0$, the solution $\varphi _\alpha$ is strictly decreasing in |x| and satisfies

$$\begin{aligned} \varphi _\alpha \rightarrow 0 \quad \text {as} \quad |x| \rightarrow \infty . \end{aligned}$$

We extend the solution by setting $\varphi _{\alpha } = -\varphi _{-\alpha }$ for $\alpha <0$ and $\varphi _0 = 0$. Then the set $\{ \varphi _\alpha ; \alpha \in \mathbb {R} \}$ forms a one-parameter family of radial stationary solutions.

The exponent

$$\begin{aligned} p_c = {\left\{ \begin{array}{ll} \frac{(N-2)^2 -4N + 8\sqrt{N-1}}{(N-2)(N-10)} &{}\quad N > 10,\\ \infty &{}\quad N \le 10, \end{array}\right. } \end{aligned}$$

is another important exponent which appeared first in [13]. It is known that for $p_S \le p < p_c$, any pair of positive stationary solutions intersects each other. For $p \ge p_c$, Wang [18] showed that the family of stationary solutions forms a simply ordered set, that is, $\varphi _\alpha$ is strictly increasing in $\alpha$ for each x. We call it the ordering property of $\{\varphi _\alpha \}$. Moreover, $\varphi _\alpha$ satisfies

$$\begin{aligned} \lim _{\alpha \rightarrow 0} \varphi _\alpha (|x|) =0 ,\quad \lim _{\alpha \rightarrow \infty } \varphi _\alpha (|x|) = \varphi _\infty (|x|), \end{aligned}$$

for each x, where $\varphi _\infty (|x|)$ is a singular stationary solution given by

$$\begin{aligned} \varphi _\infty (|x|) = L|x|^{-m}, \quad x \in \mathbb {R}^N {\setminus } \{0\}, \end{aligned}$$

with

$$\begin{aligned} m = \frac{2}{p-1}, \quad L=\{ m (N-2-m) \}^{1/(p-1)}. \end{aligned}$$

(1.3)

It was also shown in [9] that each positive stationary solution has the expansion

$$\begin{aligned} \varphi _\alpha (|x|) = {\left\{ \begin{array}{ll} L|x|^{-m} - a_\alpha |x|^{-m - \lambda _1} + \mathrm{h.o.t.} &{}\quad p > p_c , \\ L|x|^{-m} - a_\alpha |x|^{-m - \lambda } \log |x| + \mathrm{h.o.t.} &{}\quad p = p_c , \end{array}\right. } \end{aligned}$$

(1.4)

as $|x| \rightarrow \infty$, where for $p>p_c$, $\lambda _1, \lambda$ is a positive constant. $\lambda _1$ is given by

$$\begin{aligned} \lambda _1 = \lambda _1( N , p ) := \frac{N-2-2m-\sqrt{(N-2-2m)^2-8(N-2-m)}}{2}, \end{aligned}$$

$\lambda$ is given later and $a_\alpha = a(\alpha )$ is a positive number that is monotone decreasing in $\alpha$. Note that $\lambda _1$ is a smaller root of the quadratic equation

$$\begin{aligned} h(\lambda ) := \lambda ^2 - (N-2-2m) \lambda +2(N-2-m) =0. \end{aligned}$$

(1.5)

We define for $p>p_c$ by

$$\begin{aligned} \lambda _2 = \lambda _2( N , p ) := \frac{N-2-2m + \sqrt{(N-2-2m)^2-8(N-2-m)}}{2}, \end{aligned}$$

a larger root of the quadratic equation (1.5).

Concerning the stability problem, Gui et al. [9, 10] showed that any regular positive radial stationary solution is unstable in any reasonable sense if $p_S< p < p_c$ and “weakly asymptotically stable" in a weighted $L^\infty$ norm if $p \ge p_c$. For $p > p_c$, Poláčik and Yanagida [15, 16] improved the above results and proved that the solutions approach a set of stationary solutions for a wide class of the initial data. As a by-product, they also showed the existence of global unbounded solutions. We note that the study of global unbounded solutions of (1.1) [3, 4] is closely related to our problem on bounded solutions mentioned later.

Fila et al. [5] carried out the further investigation about the convergence of solutions of (1.1). They studied the following more general problem: Let u and $\tilde{u}$ denote solutions of (1.1) with initial data $u_0$, $\tilde{u}_0$ respectively. They considered how fast these two solutions approach each other as $t \rightarrow \infty$. In particular, in the case of $\tilde{u}_0 = \varphi _\alpha (|x|)$, then the rate of approach corresponds to the convergence rate to the stationary solution. More precisely, they showed that if $p > p_c$ , $m+\lambda _1< l < m+ \lambda _2$ and $u_0$, $\tilde{u}_0$ satisfy

$$\begin{aligned} |u_0|,\, | \tilde{u}_0| \le \varphi _\alpha (|x|), \quad x \in \mathbb {R}^N \end{aligned}$$

(H1)

and

$$\begin{aligned} |u_0 - \tilde{u}_0| \le c_1(1+|x|)^{-l}, \quad x \in \mathbb {R}^N\ \end{aligned}$$

(H2)

with some constants $\alpha > 0$ and $c_1 >0$, then $\Vert u( \cdot ,t) - \tilde{u}(\cdot ,t )\Vert _{L^\infty }$ decays faster in time than the rate $t^{-(l-m-\lambda _1)/2}$.

The above result is no longer valid for large l and in fact they found a universal lower bound for the rate of approach which applies to any initial data. More precisely, they showed that if $p \ge p_c$ and $0 \le \tilde{u}_0(x) < u_0(x) \le \varphi _\infty (|x|)$ then $\Vert u( \cdot ,t) - \tilde{u}(\cdot ,t )\Vert _{L^\infty }$ decays more slowly in time than the rate $t^{-(N- m- \lambda _1)/2}$. We note that there exists a gap of the convergence rate between the rate $t^{-(\lambda _2-\lambda _1)/2}$ which is obtained for the case $l=m+\lambda _2$ and a universal lower bound of the rate $t^{-(N- m -\lambda _1)/2}$.

On the other hand, for the grow-up problem which can be regarded as a stability problem of singular stationary solution, a sharp universal upper bound of the grow-up rate was found by Mizoguchi [14], and optimal lower bound of the grow-up rate was found by Fila et al. [4]. The results on the grow-up problem strongly suggest that the above result of the convergence rate is not optimal.

For $p > p_c$, We obtain a sharp bound of the convergence rate in the case of $m+ \lambda _1< l < m+ \lambda _2 +2$ which leads to its optimal convergence rate in [11]. In fact, we improve the results in [5] and if two solutions are initially close exponentially near the spatial infinity, then we obtain optimal estimate of the convergence rate of approaching two solutions for the super critical exponent in [11, Theorem1.2] corresponding to $l=m+\lambda _2+2$. Our result in [11, Theorem1.2] implies that the convergence rate can be extended in the case of $l=m+\lambda +2$ for the critical exponent $p=p_c$.

Stinner studied a similar problem in [17] with the critical exponent $p=p_c$ in the case of $m+ \lambda< l < m+ \lambda +2$ and shows the sharp convergence rate of the solution that approaches a stationary solution $\varphi _\alpha (|x|)$. In this case, the equation (1.5) has the double root

$$\begin{aligned} \lambda := \frac{N-2-2m}{2}. \end{aligned}$$

Actually, the sharp convergence rate $t^{-(l- m- \lambda )/2} (\log t)^{-1}$ is obtained in [17]. The characteristic point is that the convergence rate contains a logarithmic term.

Our purpose of this paper is to show there exists a optimal and sharp estimate of the convergence rate for $l=m+\lambda +2$ with the approaching solutions which applies an exponentially approaching two initial data. In fact, with the critical exponent $p=p_c$ and in the case of $l= m+ \lambda +2$, we can extend the results as follows. Our results also show that a logarithmic term appear in the case.

Theorem 1.1

Let $p = p_c$. Suppose that $|u_0|, |\tilde{u}_0| \le \varphi _{\alpha }(|x|)$ with some $\alpha$ and satisfy

$$\begin{aligned} | u_0(x) - \tilde{u}_0(x)| \le c_1 \exp { ( -\nu _1 |x|^2 ) } \end{aligned}$$

with some $c_1, \nu _1 > 0$. Then there exists constant $C_1 > 0$ such that

$$\begin{aligned} \Vert u( \cdot ,t) - \tilde{u}(\cdot ,t )\Vert _{L^\infty } \le C_1 ( t +3 )^{-1} \big (\log (t+3)^{1/2}\big )^{-1} \end{aligned}$$

for all $t>0$.

Theorem 1.2

Let $p = p_c$. Suppose that $\varphi _{\alpha }(|x|) \le \tilde{u}_0 \le u_0 \le \varphi _{\infty }(|x|)$ with some $\alpha$ and satisfy

$$\begin{aligned} | u_0(x) - \tilde{u}_0(x)| \ge c_2 \exp {\big ( -\nu _2 |x|^2 \big ) } \end{aligned}$$

with some $c_2, \nu _2 > 0$. Then there exists constant $C_2 > 0$ such that

$$\begin{aligned} \Vert u( \cdot ,t) - \tilde{u}(\cdot ,t )\Vert _{L^\infty } \ge C_2 ( t + 3 )^{-1} \big (\log (t+3)^{1/2}\big )^{-1} \end{aligned}$$

for all $t>0$.

Proofs of the above theorems are obtained by a comparison technique that is based on matched asymptotic expansion. This expansion consists of two parts which are called the inner expansion and the outer expansion. The inner expansion is used to approximate the behavior of solutions near the origin and the outer expansion is used to approximate the behavior of solutions near the spatial infinity. The inner expansion is the same as in [17] and the key of our proof is the careful description of the outer expansion. In fact, we will find a solution which behaves in a self-similar way near the spatial infinity. Then we construct super and sub-solutions by matching these inner and outer solutions.

This paper is organized as follows. In Sect. 2, we recall preliminary results in [4, 5]. We note that the results of this section imply the reason why logarithmic term appear. The formal analysis in this section will give the idea of constructing super and sub-solutions, and a matching condition of these expansion implies the convergence rate. In Sect. 3, we derive the upper bound of the convergence rate, and in Sect. 4, we derive the lower bound of the convergence rate with the initial data satisfying the above conditions. These results show that the sharp and the optimal convergence rate are obtained in the case.

2 Preliminary Results on the Linearized Equation

In this section, we summarize previous results on the linear equation that are needed in subsequent sections. For proofs of the results, see [4, 5].

We consider radial solutions $U=U(r,t)$, $r=|x|$, of the linearized equation of (1.1) at $\varphi _\alpha$. Namely, let $\mathcal{P}_\alpha$ be the linearized operator defined by

$$\begin{aligned} \mathcal{P}_\alpha U := U_{rr} + \frac{N-1}{r} U_r + p{\varphi _\alpha }^{p-1}U \end{aligned}$$

and let U(r, t) be a solution of

$$\begin{aligned} \left\{ \begin{array}{ll} U_t = \mathcal{P}_\alpha U, &{}\quad r> 0, \;\, t> 0 ,\\ U_r(0,t) = 0,&{} \quad t>0, \\ U(r,0) = U_0(r), &{} \quad r \ge 0, \end{array} \right. \end{aligned}$$

(2.1)

where $U_0$ is a continuous function that decays to zero as $r \rightarrow \infty$. From the maximum principle, we see that $U(\cdot ,t) > 0$ for all $t>0$ if $U_0 \ge 0$ and $U_0 \not \equiv 0$. We will describe some fundamental properties for the solution of (2.1).

2.1 Comparison Principle

Let u and $\tilde{u}$ be solutions of (1.1) with initial data $u_0$ and $\tilde{u}_0$ respectively. We recall some comparison results for $u - \tilde{u}$ and the solution U of (2.1), which comes from the ordering property and the convexity of nonlinearity.

Lemma 2.1

([5, Lemma 2.1]) Let $p \ge p_c$. Suppose that $u_0$ and $\tilde{u}_0$ satisfy (H1). If

$$\begin{aligned} |u_0(x) - \tilde{u}_0(x)| \le U_0(|x|), \quad x \in \mathbb {R}^N, \end{aligned}$$

then

$$\begin{aligned} |u(x,t) - \tilde{u}(x,t)| \le U(|x|,t), \quad x \in \mathbb {R}^N \end{aligned}$$

for all $t > 0$.

Lemma 2.2

([5, Lemma 2.2]) Let $p \ge p_c$. Suppose that $u_0$ and $\tilde{u}_0$ satisfy

$$\begin{aligned} \varphi _\alpha (|x|) \le \tilde{u}_0(x) \le u_0(x) \le \varphi _\infty (|x|) ,\quad x \in \mathbb {R}^N {\setminus } \{0\} \end{aligned}$$

with some $\alpha > 0$. If

$$\begin{aligned} 0 \le U_0(|x|) \le u_0(x) - \tilde{u}_0(x) , \quad x \in \mathbb {R}^N, \end{aligned}$$

then

$$\begin{aligned} 0 \le U(|x|,t) \le u(x,t) - \tilde{u}(x,t), \quad x \in \mathbb {R}^N \end{aligned}$$

for all $t > 0$.

2.2 Formal Matched Asymptotics

By the above comparison results, we may only consider the convergence of radial solution of the linearized equation (2.1). In the following, we recall the asymptotic analysis, which is only formal but will be useful in the rigorous analysis in subsequent sections.

First, following Fila et al. [5], the formal expansion of a solution of (2.1) near the origin is given by

$$\begin{aligned} U(r,t)=\sigma (t)\psi (r) + \sigma _t(t)\varPsi (r) + \mathrm{h.o.t.}, \end{aligned}$$

(2.2)

where, $\sigma (t)=U(0,t)$, $\psi$ and $\varPsi$ satisfy

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal{P}_\alpha \psi = 0, &{}\quad r>0, \\ \psi (0)=1, &{} \quad \psi _r(0)=0 \end{array} \right. \end{aligned}$$

(2.3)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal{P}_\alpha \varPsi =\psi , &{}\quad r>0, \\ \varPsi (0)=0, &{} \quad \varPsi _r(0)=0, \end{array} \right. \end{aligned}$$

(2.4)

respectively (see also [5, 7] for details). We recall some results in [5] on the above linear differential equations (2.3) and (2.4) in the following.

Lemma 2.3

For all $\alpha >0$ and $r \ge 0$, $\alpha \mapsto \varphi _\alpha (r)$ is differentiable and

$$\begin{aligned} \psi (r) := \frac{\partial }{\partial \alpha } \varphi _\alpha \end{aligned}$$

satisfies (2.3). Moreover, if $p = p_c$, then $\psi (r)$ is positive and satisfies

$$\begin{aligned} \psi (r) = c_\alpha r^{-m-\lambda }\log r + o(r^{-m-\lambda }\log r) \quad {\mathrm{as }} \, r \rightarrow \infty , \end{aligned}$$

where $c_\alpha$ is a constant given by $c_\alpha =\frac{ a_1 \lambda }{m} \alpha ^{-\frac{ m + \lambda }{m}}$ and $a_1=a(1)$ is a constant independent of $\alpha$.

Proof

The proof is same manner as in [5, Lemma 2.3]. That is to say we use a scaling invariance property. In fact, we find

$$\begin{aligned} \varphi _\alpha (r) \equiv \alpha \varphi _1\left( \alpha ^{1/m}r \right) \end{aligned}$$

(2.5)

for all $\alpha >0$ and differentiating (1.2) yields

$$\begin{aligned} \psi (r) = \frac{\partial }{\partial \alpha } \varphi _\alpha (r) \end{aligned}$$

satisfies (2.3). Furthermore, $\varphi _\alpha (r)$ is monotone increasing with respect to $\alpha$, it follows $\psi \ge 0$ for all $r \ge 0$. Then, the uniqueness of the solution and $\psi (0) = 1$ shows $\varphi _\alpha (r)$ is positive.

Next, we differentiate (2.5) and using (1.4), we obtain

$$\begin{aligned} \psi (r)&= \frac{\partial }{\partial \alpha } \varphi _\alpha = \frac{\partial }{\partial \alpha } \alpha \varphi _1\left( \alpha ^{1/m}r \right) \\&=\varphi _1\left( \alpha ^{1/m}r \right) + \alpha \frac{\partial }{\partial \alpha } \varphi _1\left( \alpha ^{1/m}r \right) \\&=L\left( \alpha ^{1/m}r \right) ^{-m} - a_1 \left( \alpha ^{1/m} r \right) ^{-m- \lambda } \log \left( \alpha ^{1/m}r \right) + \mathrm{h.o.t.} \\&\quad + \alpha \left( \frac{\partial }{\partial \alpha } \left( \left( L\alpha ^{1/m}r \right) ^{-m} - a_1 \left( \alpha ^{1/m}r \right) ^{-m- \lambda } \log \left( \alpha ^{1/m}r \right) + \mathrm{h.o.t.} \right) \right) \\&=L \alpha ^{-1} r^{-m} - a_1 \alpha ^{(-m- \lambda )/m} r^{-m- \lambda } \left( \log \alpha ^{1/m}+ \log r \right) + \mathrm{h.o.t.} \\&\quad + \alpha \left( \frac{\partial }{\partial \alpha } \left( L \alpha ^{-1} r^{-m} - a_1 \alpha ^{(-m- \lambda )/m} r^{-m- \lambda } \left( \log \alpha ^{1/m}+ \log r \right) + \mathrm{h.o.t.} \right) \right) \\&=L \alpha ^{-1} r^{-m} - a_1 \alpha ^{(-m- \lambda )/m} r^{-m- \lambda } \left( \frac{1}{m}\log \alpha + \log r \right) + \mathrm{h.o.t.} \\&\quad + \alpha \Big ( - L \alpha ^{-2} r^{-m} - a_1 \frac{ m + \lambda }{m} \alpha ^{((- m- \lambda )/m -1)} r^{-m- \lambda } \left( \frac{1}{m} \log \alpha + \log r \right) \\&\quad - a_1 \alpha ^{(- m- \lambda )/m} \frac{1}{m \alpha } r^{-m- \lambda }+ \mathrm{h.o.t.} \Big )\\&=L \alpha ^{-1} r^{-m} - a_1 \alpha ^{(-m- \lambda )/m} r^{-m- \lambda } \left( \frac{1}{m}\log \alpha + \log r \right) \\&\quad - L \alpha ^{-1} r^{-m} +a_1 \frac{ m + \lambda }{m} \alpha ^{ (- m- \lambda )/m } r^{-m- \lambda } \left( \frac{1}{m} \log \alpha + \log r \right) \\&\quad - a_1 \alpha ^{(- m- \lambda )/m} \frac{1}{m} r^{-m- \lambda }+ \mathrm{h.o.t.}\\&=- a_1 \alpha ^{(-m- \lambda )/m} r^{-m- \lambda } \left( \frac{1}{m}\log \alpha + \log r \right) \\&\quad +a_1 \left( 1 + \frac{\lambda }{m} \right) \alpha ^{ (- m- \lambda )/m } r^{-m- \lambda } \left( \frac{1}{m} \log \alpha + \log r \right) \\&\quad - \frac{a_1}{m}\alpha ^{(- m- \lambda )/m} r^{-m- \lambda }+ \mathrm{h.o.t.}\\&=\frac{a_1 \lambda }{m} \alpha ^{(-m- \lambda )/m} r^{-m- \lambda } \log r +\frac{a_1}{m} \alpha ^{(- m- \lambda )/m} \left( \frac{\lambda }{m} \log \alpha -1 \right) r^{-m- \lambda }+ \mathrm{h.o.t.}\\ \end{aligned}$$

Thus we finish the proof. $\square$

Remark 2.1

The function $\psi$ defined in Lemma 2.3 satisfies $\psi _r < 0$ for all $r>0$. Indeed, we see from (2.3) that $\psi$ does not attain a positive local minimum by the positivity of $\varphi _\alpha$ and $\psi$.

Next, we need to prepare an elementary assertion on the asymptotic behavior of solutions of certain perturbed Euler-type ODE as in [3, Lemma 2.4] and [12]. Namely, we need to know a behavior of $\varPsi$, for that purpose, we prove the behavior of solutions an Euler-type ODE as follows.

Lemma 2.4

Let $\mu =(1-a_1)/2$ and suppose $y=y(s)$ is a solution of

$$\begin{aligned} y''+\frac{a_1}{s}y'+\left( \frac{a_2}{s^2}+g_1(s) \right) y = a_3 s^\gamma \log s +g_2(s) , \quad s \in [1, \infty ). \end{aligned}$$

where $a_1, a_2, a_3$ and $\gamma$ are real numbers satisfying $a_3 \ne 0$, $(a_1-1)^2-4a_2=0$, $\gamma >\mu -2$ and $g_1, g_2$ are continuous functions satisfying

$$\begin{aligned} |g_1(s)|< \frac{\epsilon _1(s)}{s^2} \quad \text {and} \quad |g_2(s)| < \epsilon _2(s) s^\gamma \log s \end{aligned}$$

for all $s \ge 1$ with $\epsilon _1(s) {\searrow }0$ and $\epsilon _2(s) {\searrow }0$ as $s \rightarrow \infty$. Then the solution y(s) satisfies

$$\begin{aligned} y(s)= & {} \frac{a_3}{(\gamma +2 - \mu )^2}s^{\gamma + 2}\log s + o(s^{\gamma + 2}\log s) , \\ y'(s)= & {} \frac{a_3(\gamma + 2)}{(\gamma +2 - \mu )^2}s^{\gamma + 1}\log s + o(s^{\gamma + 1}\log s) \end{aligned}$$

as $s \rightarrow \infty$.

Proof

The proof is same manner as in [3, 12]. That is to say, we use the contraction mapping theorem.

Setting $z = (z_1, z_2)^t$ , we rewrite the equation as a system of the form

$$\begin{aligned} \left\{ \begin{array}{l} z_1' = z_2, \\ z_2' = -\frac{a_2}{s^2}z_1 -\frac{a_1}{s}z_2 +a_3 s^\gamma \log s +g_2(s) - g_1(s) z_1\\ \end{array} \right. \end{aligned}$$

(2.6)

From the variation of constants formula, it follows that for arbitrary $s_0 \ge 1$

$$\begin{aligned} z(s)&= \left( {\begin{array}{cc} {s^\mu } &{}\quad {s^\mu \log s} \\ {\mu s^{\mu - 1} } &{}\quad {\mu s^{\mu - 1} \log s + s^{\mu - 1} } \\ \end{array} } \right) \left\{ v + \int _{s_0}^s \tau ^{-\mu +1} f(\tau ) \left( \begin{array}{c} - \log \tau \\ 1 \\ \end{array} \right) d\tau \right\} \nonumber \\&=: (Fz)(s) \end{aligned}$$

(2.7)

holds with the constant vector

$$\begin{aligned} v= \left( \begin{array}{c} v_1 \\ v_2 \\ \end{array} \right) =\left( {\begin{array}{cc} {s_0^{-\mu }(\mu \log s_0 +1)} &{}\quad {-s_0^{-\mu +1} \log s_0} \\ { -\mu s_0^{-\mu } } &{}\quad {s_0^{-\mu + 1} } \\ \end{array} } \right) \left( \begin{array}{c} z_1(s_0) \\ z_2(s_0) \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} f(\tau ) := a_3\tau ^{\gamma }\log \tau + g_2(\tau ) - g_1(\tau ) w_1(\tau ). \end{aligned}$$

We claim that if $s_0$ is sufficiently large, then (2.6) has a unique solution z in the closed subset

$$\begin{aligned} M:=\left\{ w\in X \left| w(s_0) = \left( \begin{array}{c} z_1(s_0) \\ z_2(s_0) \\ \end{array} \right) \right\} \right. \end{aligned}$$

of the Banach space

$$\begin{aligned} X:= \{ w \in C^0([s_0, \infty ) ; \mathbb {R}^2)| \, \Vert w\Vert _X< \infty \} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert w\Vert _X:= \Vert s^{-\gamma -2}(\log s)^{-1}w_1 \Vert _{L^\infty }+ \Vert s^{-\gamma -1}(\log s)^{-1}w_2 \Vert _{L^\infty }. \end{aligned}$$

We denote first component of Fw $(Fw)_1$ and second component of Fw $(Fw)_2$ in the follwing. For $w \in M$ by using integration by parts, we have

$$\begin{aligned} |(Fw)_1 |\le & {} s^\mu |v_1| + s^\mu \log s |v_2| \\&+ s^\mu \int _{s_0}^s \tau ^{-\mu +1} \Big ( a_3 \tau ^\gamma \log \tau + |g_2(\tau )| + |g_1(\tau )| |w_1| \Big ) \log \frac{s}{\tau } d\tau \\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| \\&+ s^\mu \int _{s_0}^s \tau ^{-\mu +1} \Big ( a_3 \tau ^\gamma \log \tau + \epsilon _2(\tau ) \tau ^\gamma \log \tau + \tau ^{-2} \epsilon _1(\tau ) |w_1| \Big ) \log \frac{s}{\tau } d\tau \\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| \\&+ s^\mu \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 + \epsilon _2(\tau ) + \epsilon _1(\tau ) \Vert w \Vert _X \Big ) \log \tau \log \frac{s}{\tau } d\tau \\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| \\&+ s^\mu \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert _X \Big ) \log \tau \log \frac{s}{\tau } d\tau \\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| + ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert _X ) \\&s^\mu \Big ( \left[ \frac{\tau ^{\gamma -\mu +2}}{\gamma -\mu +2} \log \tau \log \frac{s}{\tau } \right] _{s_0 }^s - \int _{s_0}^s \frac{\tau ^{\gamma -\mu +2}}{\gamma -\mu +2} \frac{1}{\tau } \log \frac{s}{\tau ^2} d\tau \Big )\\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert _X ) \\&s^\mu \int _{s_0}^s \frac{\tau ^{\gamma -\mu +1}}{\gamma -\mu +2} \left( - \log \frac{s}{\tau ^2} \right) d\tau \\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| + ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert _X )\\&s^\mu \int _{s_0}^s \frac{\tau ^{\gamma -\mu +1}}{\gamma -\mu +2} \left( - \log \frac{1}{s} \right) d\tau \\\le & {} s^\mu |v_1| + s^\mu \log s |v_2| + c s^{\gamma +2} \log s \end{aligned}$$

For the second component, we similarly obtain

$$\begin{aligned} |(Fw)_2|&\le s^{\mu -1} (\mu |v_1|+|v_2|) + \mu s^{\mu -1} \log s |v_2| \\&\quad + s^{\mu -1} \int _{s_0}^s \tau ^{-\mu +1} \Big ( a_3 \tau ^\gamma \log \tau + |g_2(\tau ) | \tau ^\gamma \log \tau + |g_1(\tau )| |w_1| \log \tau \Big )\\&\quad \left( \mu \log \frac{s}{\tau }+1\right) d\tau \\&\le s^{\mu -1} (\mu |v_1|+|v_2|) + \mu s^{\mu -1} \log s |v_2| \\&\quad + s^{\mu -1} \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 \log \tau + \epsilon _2(\tau ) \log \tau + \epsilon _1(\tau ) \log \tau \Vert w \Vert \Big )\\&\quad \left( \mu \log \frac{s}{\tau }+1 \right) d\tau \\&\le s^{\mu -1} (\mu |v_1|+|v_2|) + \mu s^{\mu -1} \log s |v_2| \\&\quad + s^{\mu -1} \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert \Big )\log \tau \left( \mu \log \frac{s}{\tau }+1\right) d\tau \\&= s^{\mu -1} (\mu |v_1|+|v_2|) + \mu s^{\mu -1} \log s |v_2| \\&\quad + \mu s^{\mu -1} \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert \Big )\log \tau \log \frac{s}{\tau } d\tau \\&\quad + s^{\mu -1} \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert \Big ) \log \tau d\tau \\&\le s^{\mu -1} (\mu |v_1|+|v_2|) + \mu s^{\mu -1} \log s |v_2| + \mu c s^{\gamma +1} \log s\\&\quad + s^{\mu -1} \int _{s_0}^s \tau ^{\gamma -\mu +1} \Big ( a_3 + \epsilon _2(1) + \epsilon _1(1) \Vert w \Vert \Big )\log s d\tau \\&\le s^{\mu -1} (\mu |v_1|+|v_2|) + \mu s^{\mu -1} \log s |v_2| + d s^{\gamma +1} \log s \end{aligned}$$

with some $d>0$. Taking norms and checking the initial value, we infer that $Fw \in M$ Moreover, if $w, \tilde{w} \in M$ then

$$\begin{aligned} |(Fw - F\tilde{w})_1 |&\le s^\mu \int _{s_0}^s \tau ^{-\mu +1} |g_1(\tau )| |w_1 - \tilde{w}_1| \log \frac{s}{\tau } d\tau \\&\le s^\mu \int _{s_0}^s \tau ^{-\mu +1} |g_1(\tau )| \tau ^{\gamma +2} \Vert w_1- \tilde{w}_1\Vert _X \log \tau \log \frac{s}{\tau } d\tau \\&\le s^\mu \int _{s_0}^s \tau ^{\gamma -\mu +1} \epsilon _1(\tau ) \Vert w_1- \tilde{w}_1\Vert _X \log \tau \log \frac{s}{\tau } d\tau \\&\le s^\mu \int _{s_0}^s \tau ^{\gamma -\mu +1} \epsilon _1(s_0) \Vert w_1- \tilde{w}_1\Vert _X \log \tau \log \frac{s}{\tau } d\tau \\&\le c' \epsilon _1(s_0) s^{\gamma +2} \log s \Vert w_1- \tilde{w}_1\Vert _X \end{aligned}$$

with some constant $c'>0$. Proceeding in the same way for the second component, we obtain

$$\begin{aligned} |(Fw - F\tilde{w})_2 |&\le s^{\mu -1} \int _{s_0}^s \tau ^{-\mu +1} |g_1(\tau )| |w_1 - \tilde{w}_1| \left( \mu \log \frac{s}{\tau } +1 \right) d\tau \\&\le s^{\mu -1} \int _{s_0}^s \tau ^{-\mu +1} |g_1(\tau )| \tau ^{\gamma +2} \Vert w_1- \tilde{w}_1\Vert _X \log \tau \left( \mu \log \frac{s}{\tau } +1 \right) d\tau \\&\le s^{\mu -1} \int _{s_0}^s \tau ^{\gamma -\mu +1} \epsilon _1(\tau ) \Vert w_1- \tilde{w}_1\Vert _X \log \tau \left( \mu \log \frac{s}{\tau } +1 \right) d\tau \\&\le s^{\mu -1} \Big ( \int _{s_0}^s \tau ^{\gamma -\mu +1} \epsilon _1(s_0) \Vert w_1- \tilde{w}_1\Vert _X \log \tau \log \frac{s}{\tau } d\tau \\&\quad + \int _{s_0}^s \tau ^{\gamma -\mu +1} \epsilon _1(s_0) \Vert w_1- \tilde{w}_1\Vert _X \log \tau d\tau \Big ) \\&\le d' \epsilon _1(s_0) s^{\gamma +1} \log s \Vert w_1- \tilde{w}_1\Vert _X \end{aligned}$$

with some constant $d'>0$. Hence we see

$$\begin{aligned} \Vert (Fw - F \tilde{w}) \Vert _X \le (c' + d') \epsilon _1(s_0) \Vert (Fw - F \tilde{w}) \Vert _X \end{aligned}$$

Then we finished to prove that F is a contraction of the closed set M for large $s_0>0$.

Consequently, the solution z of (2.6) lies in X, so that particularly

$$\begin{aligned} |g_2(s)-g_1(s)z_1| \le c (\epsilon _1(s)+ \epsilon _2(s)) s^\gamma \log s \end{aligned}$$

for large s. Using this in (2.7), we easily obtain the conclusion. Indeed, we can check

$$\begin{aligned} z_1(s)&= s^\mu \int _{s_0}^s a_3 \tau ^{\gamma -\mu +1} \log \frac{s}{\tau } d\tau \\&\quad + s^\mu v_1 + s^\mu \log s v_2 + s^\mu \int _{s_0}^s \tau ^{-\mu +1} ( g_2(\tau ) - g_1(\tau ) z_1(\tau ) ) \log \frac{s}{\tau } d\tau \\&=\frac{a_3}{(\gamma +2 - \mu )^2}s^{\gamma + 2}\log s + o(s^{\gamma + 2}\log s) \end{aligned}$$

as $s \rightarrow \infty$ by using dominated convergence theorem and integration by parts. We can use the same argument as the above for the second component, then complete the proof. $\square$

Lemma 2.5

If $p = p_c$, then the solution $\varPsi$ of (2.4) has the following properties : 

• (i) $\varPsi /\psi$ is strictly increasing in $r >0$. In particular, $\varPsi$ is positive for all $r > 0$.

• (ii) $\varPsi$ satisfies

  $$\begin{aligned} \varPsi (r) = C_\alpha r^{-m-\lambda +2}\log r + o(r^{-m-\lambda +2}\log r) \quad {\mathrm{as }} \quad r \rightarrow \infty , \end{aligned}$$

  where

  $$\begin{aligned} C_\alpha =\frac{c_\alpha }{g(m + \lambda -2 )},\quad g(\mu ) := h(\mu -m ). \end{aligned}$$

Proof

Proof $(\mathrm{i})$ is the same as in [5, Lemma 2.4]. In order to prove $(\mathrm{ii})$, we can apply Lemma 2.4 as follows.

$$\begin{aligned} \varPsi _{rr}+\frac{N-1}{r}\varPsi _r +\left( \frac{pL^{p-1}}{r^2} + \eta _1(r) \right) \varPsi =c_\alpha r^{-m-\lambda } \log r +\eta _2(r), \quad r > 0, \end{aligned}$$

where $\eta _1, \eta _2$ are functions satisfying $r^2\eta _1(r) {\searrow }0$ and $r^{m+\lambda }\eta _2(r) {\searrow }0$ as $r \rightarrow \infty$ for $r > 1$. Then we can apply Lemma 2.4 which asserts

$$\begin{aligned} r^{m+\lambda -2}(\log r)^{-1} \varPsi (r) \rightarrow \frac{c_\alpha }{g(m+\lambda -2)} >0 \quad \text {as} \quad r \rightarrow \infty , \end{aligned}$$

the proof of (ii) is finished. $\square$

Next, let us consider the expansion of a solution of (2.1) near $r=\infty$. By the expansion of $\varphi _\alpha (r)$ near $r=\infty$, U(r, t) satisfies approximately

$$\begin{aligned} U_t = U_{rr} + \frac{N-1}{r}U_r + \frac{pL^{p-1}}{r^2}U, \quad r \simeq \infty . \end{aligned}$$

(2.8)

Following [3, 4], we assume that U is of a self-similar form for $r \gg 1$

$$\begin{aligned} U(r,t)=t^{-l/2}F(\eta ), \quad \eta = t^{-1/2}r. \end{aligned}$$

(2.9)

so that the specific scaling for $r \gg 1$ corresponding to the outer region is in fact $r = O(t^{1/2})$ as $t \rightarrow \infty$. Substituting this in (2.8), we see that F satisfies

$$\begin{aligned} F_{\eta \eta }+\frac{N-1}{\eta } F_\eta +\frac{\eta }{2} F_\eta + \frac{l}{2} F+\frac{pL^{p-1}}{\eta ^2} F = 0. \end{aligned}$$

(2.10)

In order that the outer expansion matches with the inner solution (2.2), $F(\eta )$ must satisfy

$$\begin{aligned} \lim _{\eta \rightarrow 0} \eta ^{m+\lambda }F(\eta ) = a_0 >0 \end{aligned}$$

(2.11)

in view of the spatial order of Lemma 2.3, where $a_0$ is a constant depending on initial data.

We will know in the next section that (2.10) has a positive solution satisfying (2.11).

Matching the inner expansion (2.2) by using Lemmas 2.3 and 2.5,

$$\begin{aligned} U(r,t)&=\sigma (t)\psi (r) + \sigma _t(t)\varPsi (r) + \mathrm{h.o.t.}\\&\simeq \sigma (t) r^{-m-\lambda } \log r + \sigma _t(t)r^{-m-\lambda +2} \log r + \mathrm{h.o.t.}\\&\simeq \sigma (t) t^{-(m+\lambda )/2} \eta ^{-m-\lambda }\big ( \log \eta t^{1/2}\big )\\&\quad + \sigma _t(t) t^{-(m+\lambda )/2} \eta ^{-m-\lambda +2}\big (\log \eta t^{1/2}\big ) + \mathrm{h.o.t.} \end{aligned}$$

and the outer expansion (2.9),

$$\begin{aligned} U(r,t)= t^{-l/2}F(\eta ), \end{aligned}$$

we obtain

$$\begin{aligned} \sigma (t) t^{(-m-\lambda )/2}\log t^{1/2}&\simeq t^{-(l/2)}.\end{aligned}$$

This implies the convergence rate

$$\begin{aligned} \sigma (t) \simeq t^{-(l-m-\lambda )/2} \big (\log t^{1/2}\big )^{-1} \end{aligned}$$

which is the same convergence rate given in Theorems 1.1 and 1.2 for $l=m+\lambda +2$. We use theses results, and also obtain

$$\begin{aligned} \sigma _t \simeq - \frac{ l-m - \lambda }{2} t^{-(l-m-\lambda )/2 -1} (\log t^{1/2})^{-1} -\frac{1}{2} t^{-(l-m-\lambda )/2 -1} (\log t^{1/2})^{-2}. \end{aligned}$$

We substitute above results in (2.2), then we obtain a formal expansion near the origin as follows.

$$\begin{aligned} U(r,t)&=\sigma (t)\psi (r) + \sigma _t(t)\varPsi (r) + \mathrm{h.o.t.}\\&\simeq t^{-q} (\log t^{1/2})^{-1} \psi (r) - \left( qt^{-q-1} \big (\log t^{1/2}\right) ^{-1} \\&\quad +\frac{1}{2} t^{-q-1} \big (\log t^{1/2}\big )^{-2} \big ) \varPsi (r), \end{aligned}$$

where $q=( l - m - \lambda ) /2$. The above expansion suggests a construction of inner super and sub-solutions.

2.3 Properties of Self-similar Solutions

In this subsection, we recall the behavior of solutions of (2.10) satisfying

$$\begin{aligned} \lim _{\eta \rightarrow 0} \eta ^{m + \lambda } F(\eta ) = a_0 > 0, \end{aligned}$$

where $a_0 > 0$ is a constant. To this end, we set

$$\begin{aligned} f(\eta ) = \eta ^{m + \lambda } F(\eta ). \end{aligned}$$

Substituting this in (2.10), we see that $f(\eta )$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} f_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f_{\eta } + \frac{\eta }{2}f_{\eta } + \frac{ l - m - \lambda }{2} f=0 , &{}\quad \eta> 0, \\ f(0)=a_0 > 0,\quad f_\eta (0)=0. \end{array} \right. \end{aligned}$$

(2.12)

The following lemma characterizes the behavior of f as $\eta \rightarrow \infty$, and explains why $l=m + \lambda + 2$ is critical.

Lemma 2.6

([4, Lemma 3.1]) Let f be the solution of (2.12).

• (i) If $l \in (m + \lambda , m + \lambda +2 )$, then $f > 0$ and $f_\eta <0$ for all $\eta > 0$. Moreover, for each $\eta _0 >0$, there exist $d_-(\eta _0) > 0$ such that

  $$\begin{aligned} f(\eta ) \ge d_-(\eta _0) \eta ^{-(l-m-\lambda )} \quad \text { for } \quad \eta \ge \eta _0, \end{aligned}$$

  and $d_+ > 0$ such that

  $$\begin{aligned} f(\eta ) \le d_+ \eta ^{-(l-m-\lambda )} \quad \text {for all } \quad \eta > 0. \end{aligned}$$

• (ii) If $l=m + \lambda +2$, then $f(\eta )$ is given explicitly by $f(\eta ) = a_0 \exp ({-{\eta }^2/4})$.

• (iii) If $l > m + \lambda +2$, then $f(\eta )$ vanishes at some finite $\eta$.

Remark 2.2

By the proof of Lemma 3.1 in [4], we can also show $f_\eta <0$ as far as $f>0$ in the case of Lemma 2.6 (iii).

3 Upper Bound

In this section, we prove that there exists a optimal upper bound of the convergence rate which applies to an initial data exponentially close from above or below to a stationary solution in the case $\tilde{u}=\varphi _\alpha .$

First, we recall the initial value problem (2.12):

$$\begin{aligned} \left\{ \begin{array}{ll} f_{\eta \eta } + \frac{n-1}{\eta }f_{\eta } + \frac{\eta }{2}f_{\eta } + \frac{\beta }{2}f=0,&{}{}\quad \eta> 0,\\ f(0)=a_0 > 0,\quad f_\eta (0)=0, \end{array} \right. \end{aligned}$$

where $n=N -2(m+\lambda )$, $\beta = l-m-\lambda$, and throughout this section, l is fixed to $l=m+\lambda +2$. We note that f is given explicitly by $f(\eta )=a_0 \exp {(-\eta ^2 / 4) }$.

3.1 Outer Super-solution

In this subsection, we will construct a suitable super-solution of (2.1).

Lemma 3.1

We define

$$\begin{aligned} U^+_\mathrm{out}(r,t) :=(t+\tau )^{-\frac{l}{2}} F^+(\eta )=(t+\tau )^{-\frac{l}{2}} \eta ^{-m - \lambda } f^+(\eta ), \end{aligned}$$

where $f^+(\eta ) = a_0 \exp (-\eta ^2/4)$ with $\eta = (t+\tau )^{-1/2}r$ and $\tau$ is a positive constant determined later. Then $U^+_\mathrm{out}$ is a super-solution of (2.1) .

Proof

We note that $F^+(\eta )$ satisfies

$$\begin{aligned} F^+_{\eta \eta }+\frac{N-1}{\eta } F^+_\eta +\frac{\eta }{2} F^+_\eta + \frac{l}{2} F^++\frac{pL^{p-1}}{\eta ^2} F^+ = 0. \end{aligned}$$

Then we have

$$\begin{aligned}&U^+_{\mathrm {out},t}-\mathcal {P}_\alpha U^+_\mathrm {out}\\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \left( \frac{l}{2} F + \frac{\eta }{2} F^+_\eta + F^+_{\eta \eta }+ \frac{N-1}{\eta } F^+_\eta + p(t+\tau )\varphi _\alpha ^{p-1} F^+ \right) \\ {}&=-(t+\tau )^{-\frac{l}{2}-1} \left( - \frac{pL^{p-1}}{\eta ^2} F^+(\eta ) + p(t+\tau )\varphi _\alpha ^{p-1} F^+(\eta ) \right) \\ {}&=(t+\tau )^{-\frac{l}{2}-1} \left( p(t+\tau )\big (\varphi _\infty ^{p-1}(r)-\varphi _\alpha ^{p-1}(r)\big ) \right) F^+(\eta )\\ {}&=(t+\tau )^{-\frac{l}{2}-1} \left( p(t+\tau )\big (\varphi _\infty ^{p-1}(r)-\varphi _\alpha ^{p-1}(r)\big ) \right) \eta ^{-m-\lambda } f^+(\eta )\\ {}&= p(t+\tau )^{-\frac{l}{2}} \left( \big (\varphi _\infty ^{p-1}(r)-\varphi _\alpha ^{p-1}(r)\big ) \right) \eta ^{-m-\lambda } a_0 \exp {(-\eta ^2 / 4) }. \end{aligned}$$

Then from the ordering property of $\{\varphi _\alpha \}$ and the positivity of $a_0 \exp {(-\eta ^2 / 4)}$, we have

$$\begin{aligned} U^+_{\mathrm{out},t}-\mathcal{P}_\alpha U^+_\mathrm{out} > 0 \end{aligned}$$

for all $r,t > 0$. $\square$

Since the super-solution as above decays too fast as $r \rightarrow \infty$, we shall only use it in an outer region $r \ge r^*(t)$ with suitable positive function $r^*(t)$ (see Lemma 3.2).

In the inner region, we shall work with a different class of super-solutions defined in Lemma 3.2.

3.2 Inner Super-solution and Matching

We use the same inner super-solution in [17, Lemma3.2].

Lemma 3.2

For $q>0$. We define

$$\begin{aligned} U^+_\mathrm{in}(r,t)&:= ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \\&\qquad - (t+\tau )^{-q-1} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1} + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-2} \right) \varPsi (r), \end{aligned}$$

where $q=(l-m-\lambda )/2$, and we take $l=m+\lambda +2$ later. If $\tau > 1$ is sufficiently large, then there exist constants $B_1 > 0$ satisfies $B_1 \tau ^{1/2} > 3$ and $c >0$ such that the following inequalities hold.

• (i) $U^+_{\mathrm{in},t} \ge \mathcal{P}_\alpha U^+_\mathrm{in}$ for all $r > 0 \,$ and $t >0$.

• (ii) $U^+_\mathrm{in}(r,t) >0$ for all $t >0$ and $r \in [0,B_1(t+\tau )^{\frac{1}{2}}]$.

• (iii) $U^+_\mathrm{in}(r,t) > c U^+_\mathrm{out}(r,t)$ at $r=B_1(t+\tau )^{\frac{1}{2}}$ for all $t >0$.

Proof

Although proof is similar manner in [17], we prove the Lemma here for the reader’s convenience. We prove (i) for any $B_1> 0, \tau > 0$ determined later.

$$\begin{aligned} U^+_{\mathrm {in},t} - \mathcal {P}_\alpha U^+_\mathrm {in}&= -q(t+\tau )^{-q-1} \big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r)\\ {}&\quad +(q+1) (t+\tau )^{-q-2} \\ {}&\quad \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1}+\frac{1}{2}(\log B_1(t+\tau )^{1/2})^{-2} \right) \varPsi (r) \\ {}&\quad - (t+\tau )^{-q}\\ {}&\quad \left( \big (\log B_1(t+\tau )^{1/2}\big )^{-2} B_1^{-1}(t+\tau )^{-1/2} B_1/2(t+\tau )^{-1/2} \right) \psi (r)\\ {}&\quad + (t+\tau )^{-q-1} \\ {}&\quad \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-2} B_1^{-1} (t+\tau )^{-1/2} B_1/2 (t+\tau )^{-1/2} \right) \varPsi (r)\\ {}&\quad +(t+\tau )^{-q-1} \\ {}&\quad \left( \big (\log B_1(t+\tau )^{1/2}\big )^{-3} B_1^{-1}(t+\tau )^{-1/2} B_1/2 (t+\tau )^{-1/2} \right) \varPsi (r)\\ {}&\quad -( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1}\mathcal {P}_\alpha \psi (r) \\ {}&\quad + (t+\tau )^{-q-1}\\ {}&\quad \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1} + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-2} \right) \mathcal {P}_\alpha \varPsi (r)\\ {}&= -q(t+\tau )^{-q-1} \big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r)\\ {}&\quad +(q+1) (t+\tau )^{-q-2}\\ {}&\quad \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1}+\frac{1}{2}\big (\log B_1(t+\tau )^{1/2}\big )^{-2} \right) \varPsi (r) \\ {}&\quad -\frac{1}{2} (t+\tau )^{-q-1} \left( \big (\log B_1(t+\tau )^{1/2}\big )^{-2} \right) \psi (r)\\ {}&\quad +\frac{1}{2}(t+\tau )^{-q-2} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-2} \right) \varPsi (r)\\ {}&\quad +\frac{1}{2}(t+\tau )^{-q-2} \left( \big (\log B_1(t+\tau )^{1/2}\big )^{-3} \right) \varPsi (r)\\ {}&\quad + (t+\tau )^{-q-1} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1} + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-2} \right) \psi (r)\\ {}&=(q+1) (t+\tau )^{-q-2} \\ {}&\quad \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1}+\frac{1}{2}(\log B_1(t+\tau )^{1/2})^{-2} \right) \varPsi (r)\\ {}&\quad +\frac{1}{2}(t+\tau )^{-q-2} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-2} +\big (\log B_1(t+\tau )^{1/2}\big )^{-3} \right) \varPsi (r)\\ {}&\ge 0, \, \text { for } \text { all } \, r\ge 0, t > 0, \end{aligned}$$

by Lemma 2.5, (2.3) and (2.4). Hence $U^+_\mathrm{in}$ is a super-solution of (2.1).

Next, let us show (ii) and (iii). By Lemmas 2.3 and 2.4, we can choose positive constant $c_\alpha ^-$ and $C_\alpha ^+$ such that

$$\begin{aligned} \psi (r) \ge c_\alpha ^- r^{-m-\lambda } \log r \quad \text {for} \quad r \ge 3, \end{aligned}$$

(3.1)

and

$$\begin{aligned} \varPsi (r) \le C_\alpha ^+ r^{-m-\lambda +2} \log r \quad \text {for} \quad r \ge 3, \end{aligned}$$

(3.2)

respectively. Then we fix $B_1>0$ such that

$$\begin{aligned} c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2 >0. \end{aligned}$$

(3.3)

Next, we take $\tau > 1$ so large that

$$\begin{aligned} B_1 \tau ^{\frac{1}{2}} > 3 \end{aligned}$$

(3.4)

and satisfies

$$\begin{aligned} \tau - \left( q + \frac{1}{2} \right) \frac{\varPsi (3)}{\psi (3)} >0 \end{aligned}$$

(3.5)

and

$$\begin{aligned} \nu _1 - 1/(4\tau )>0. \end{aligned}$$

(3.6)

Finally $c > 0$ so small such that

$$\begin{aligned} c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2 > c f^+(B_1). \end{aligned}$$

(3.7)

Let us now verify (ii) and (iii). For $r \in [0,3]$, it follows from due to the monotonicity of $\varPsi / \psi$, positivity of $\psi (r)$ (see Lemma 2.5 and Remark 2.1), (3.4) and (3.5) that

$$\begin{aligned} U^+_\mathrm{in}(r,t)&= ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \\&\quad - (t+\tau )^{-q-1} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1} + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-2} \right) \varPsi (r) \\&= ( t+\tau )^{-q-1}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \\&\quad \left( (t+\tau ) - \left( q + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-1} \right) \frac{\varPsi (r)}{\psi (r)} \right) \\&\ge ( t+\tau )^{-q-1} \big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \left( \tau - \left( q + \frac{1}{2} \right) \frac{\varPsi (r)}{\psi (r)}\right) \\&\ge ( t+\tau )^{-q-1} \big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \left( \tau - \left( q + \frac{1}{2} \right) \frac{\varPsi (3)}{\psi (3)} \right) \\&>0 \quad \text {for all} \quad t>0. \end{aligned}$$

For $r \in [3, B_1(t+\tau )^{\frac{1}{2}}]$, (3.1), (3.2), (3.3) and (3.4) yield

$$\begin{aligned} U^+_\mathrm{in}(r,t)&= ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \\&\quad - (t+\tau )^{-q-1} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1} + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-2} \right) \varPsi (r)\\&=( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \\&\quad \left( \psi (r) - \left( q + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-1} \right) (t+\tau )^{-1} \varPsi (r) \right) \\&\ge ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1}\\&\quad \left( c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \big (\log B_1\tau ^{1/2}\big )^{-1} \right) r^2(t+\tau )^{-1} \right) r^{-(m+\lambda )} \log r \\&\ge ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1}\\&\quad \left( c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2 \right) r^{-(m+\lambda )} \log r \\&> 0 \quad \text {for all} \quad t>0, \end{aligned}$$

which proves (ii).

This also shows in view of (3.1), (3.2), (3.7) and the definition of q that

$$\begin{aligned} U^+_\mathrm{in}(r,t)&= ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \psi (r) \\&\quad -(t+\tau )^{-q-1} \left( q\big (\log B_1(t+\tau )^{1/2}\big )^{-1} + \frac{1}{2} \big ( \log B_1(t+\tau )^{1/2}\big )^{-2} \right) \varPsi (r)\\&\ge ( t+\tau )^{-q}\big (\log B_1(t+\tau )^{1/2}\big )^{-1} \left( c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2\right) r^{-(m+\lambda )} \log r \\&= ( t+\tau )^{-q}(\log r)^{-1} \left( c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2 \right) r^{-(m+\lambda )} \log r\\&= ( t+\tau )^{-(l-m-\lambda )/2}\left( c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2 \right) B_1^{-(m+\lambda )} (t+\tau )^{-(m+\lambda )/2}\\&=( t+\tau )^{-l/2}\left( c_\alpha ^- - C_\alpha ^+ \left( q +\frac{1}{2} \right) B_1^2 \right) B_1^{-(m+\lambda )}\\&> c ( t+\tau )^{-l/2}B_1^{-(m+\lambda )} f^+(B_1) \end{aligned}$$

at $r = B_1(t+\tau )^{\frac{1}{2}}$. On the other hand, we have at $r = B_1(t+\tau )^{\frac{1}{2}}$ that is to say at $\eta =B_1$,

$$\begin{aligned} cU^+_\mathrm{out}(r,t)&= c(t+\tau )^{-l/2}F^+(\eta )\\&= c(t+\tau )^{-l/2} B_1^{-(m+\lambda )} f^+(B_1). \end{aligned}$$

Hence we obtain by (3.7)

$$\begin{aligned} c U^+_\mathrm{out}(r,t) < U^+_\mathrm{in}(r,t)\quad \text {at} \quad r=B_1(t+\tau )^{\frac{1}{2}}, \quad t >0 . \end{aligned}$$

Thus (iii) is proved. $\square$

Proposition 3.3

Suppose that $l =m + \lambda +2$ and

$$\begin{aligned} 0< U_0(r) \le c_1 \exp {(-\nu _1 r^2)}, \quad r \ge 0 \end{aligned}$$

with some $c_1 >0$ and $\nu _1>0$. Then there exists a constant $C^+ > 0$ such that the solution of (2.1) satisfies

$$\begin{aligned} \Vert U(\cdot ,t )\Vert _{L^\infty } \le C^+ U^+(0,t) = C^+(t+\tau )^{-1} \big (\log (t+\tau )^{1/2}\big )^{-1} \quad \text {for all} \quad t > 0. \end{aligned}$$

Proof

Let $U^+_\mathrm{out}$ and $U^+_\mathrm{in}$ be as given in Lemmas 3.1 and 3.2 respectively, and define

$$\begin{aligned} U^+(r,t):={\left\{ \begin{array}{ll} U^+_\mathrm{in}(r,t) &{}\quad \text {for } \, r < r^*(t),\\ c U^+_\mathrm{out}(r,t) &{}\quad \text {for } \, r \ge r^*(t), \end{array}\right. } \end{aligned}$$

where $c>0$ is given in Lemma 3.2. Put

$$\begin{aligned} r^*(t):=\sup \{r>0\, | \,U^+_\mathrm{in}(\rho ,t) < c U^+_\mathrm{out}(\rho ,t) \text { for } \rho \in [0,r]\}. \end{aligned}$$

We note that $r^*(t) \in (0 , \infty ]$ is well defined for each $c>0$, in view of Lemma 3.2 (iii). It is clear that

$$\begin{aligned} U^+_r(0,t) = U^+_{\mathrm{in},r}(0,t) =0, \quad t>0. \end{aligned}$$

We will show that the initial data $U_0(r)$ lies below $C^+ U^+(r,0)$ if we take a constant $C^+ >0$ sufficiently large. In fact, we see from Lemma 2.6 that for $r \ge r^*(0)$,

$$\begin{aligned} U^+(r,0)&=cU^+_\mathrm{out}(r,0)\\&=c\tau ^{-\frac{l}{2}}F^+(\eta )\\&=c\tau ^{-\frac{l}{2}} \eta ^{-(m+\lambda )} f^+(\eta )\\&=c\tau ^{-\frac{l}{2}} \tau ^{(m+\lambda )/2} r^{-(m+\lambda )} f^+(\tau ^{-1/2}r)\\&= c a_0 \tau ^{-\frac{l}{2}} \tau ^{(m+\lambda )/2} r^{-(m+\lambda )} \exp {(-\tau ^{-1} r^2 / 4) }. \end{aligned}$$

Then, we show there exists $C^+>0$ so large that

$$\begin{aligned} U_0= c_1 \exp {(-\nu _1 r^2)} \le C^+ U^+(r,0) \quad \text {for} \quad r \ge r^*(0). \end{aligned}$$

Indeed, we can take $C^+>0$ satisfies

$$\begin{aligned} C^+ c a_0 \tau ^{-\frac{l-m-\lambda }{2}} m_1 - c_1 > 0, \end{aligned}$$

where $m_1>0$ is a constant given by

$$\begin{aligned} m_1:=\min _{r \ge r^*(0)}{ \left( r^{-(m+\lambda )} \exp \{(\nu _1 - 1/(4\tau ))r^2 \} \right) } \end{aligned}$$

using (3.6). Then we obtain

$$\begin{aligned}&~C^+ U^+(r,0) - U_0(r)\\&\ge C^+ c a_0 \tau ^{-\frac{l}{2}} \tau ^{(m+\lambda )/2} r^{-(m+\lambda )} \exp {(-r^2/4\tau ) } -c_1 \exp {(-\nu _1 r^2)}\\&= \left( C^+ c a_0 \tau ^{-\frac{l-m-\lambda }{2}} r^{-(m+\lambda )} \exp { \{(\nu _{1}-1/(4\tau ))r^{2} \}} -c_1\right) \exp {(-\nu _1 r^2)}\\&\ge \left( C^+ c a_0 \tau ^{-\frac{l-m-\lambda }{2}} m_1 - c_1 \right) \exp {(-\nu _1 r^2)}\\&\ge 0 \quad \text {for} \quad r \ge r^*(0). \end{aligned}$$

On the other hand, for $0 \le r \le r^*(0)$, we have

$$\begin{aligned} U^+(r,0)=U^+_\mathrm{in}(r,0)&= \tau ^{-q-1}\big (\log B_1\tau ^{1/2}\big )^{-1} \psi (r) \\&\left( \tau - \left( q + \frac{1}{2} \big ( \log B_1\tau )^{1/2}\big )^{-1} \right) \frac{\varPsi (r)}{\psi (r)} \right) . \end{aligned}$$

This shows that $U^+(r ,0)$ is monotone decreasing in $r \in [0,r^*(0)]$, and $U^+_\mathrm{in}$ attains its minimum at $r=r^*(0)$ (see Lemma 2.5 and Remark 2.1). Hence it is sufficient to choose $C^+$ so large that

$$\begin{aligned} C^+ U^+_\mathrm{in}(r^*(0),0) \ge c_1. \end{aligned}$$

By taking larger $C^+$ that satisfies the above conditions, we see that $U_0$ satisfies

$$\begin{aligned} 0 < U_0(r) \le C^+ U^+(r,0) , \quad r \ge 0. \end{aligned}$$

Then by the comparison principle, we obtain

$$\begin{aligned} 0 < U(r,t) \le C^+U^+(r,t) , \quad r \ge 0, \quad t >0. \end{aligned}$$

Since $U^+$ attains the exact decay rate at the origin by an argument similar to the above. we finish the proof. $\square$

Proof of Theorem 1.1

We take

$$\begin{aligned} U_0(r) := \max _{|x|=r}{|u_0(x) -\tilde{u}_0(x)|} >0 ,\quad r \ge 0. \end{aligned}$$

Then by Lemma 2.1, Proposition 3.3, and put $q=m+\lambda +2$, then U satisfies

$$\begin{aligned} \Vert U( \cdot , t)\Vert _{L^\infty }\le C^+U^+(0,t) \le C (t+3)^{-1} \big (\log (t+3)^{1/2}\big )^{-1} \quad \text {for all}\, t >0, \end{aligned}$$

with some constant $C > 0$. The proof is now complete. $\square$

4 Lower Bound

In this section, we prove that there exists a optimal lower bound of the convergence rate which applies to an initial data that does not exponentially close from above or below to a stationary solution in the case $\tilde{u}=\varphi _\alpha$.

4.1 Outer Sub-solution

In this subsection, we construct a suitable outer sub-solution of (2.1).

First, we recall that f satisfies

$$\begin{aligned} f_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f_{\eta } + \frac{\eta }{2}f_{\eta } + \frac{ l - m - \lambda }{2} f =0 \end{aligned}$$

(4.1)

and throughout sections, l is fixed to $l=m+\lambda +2$. In our case, (4.1) has an explicit solution $f(\eta ) = a_0 \exp ({-{\eta }^2/4})$ by Lemma 2.6. Although this solution is a super-solution of (2.1) used in the previous section, to make a sub-solution, we need to modify this solution to construct a sub-solution of (2.1) in an outer region.

Lemma 4.1

We define $F^-(\eta ):= \eta ^{-m - \lambda } f^-(\eta ), f^-(\eta ):=\exp (-\eta ^2) \log \eta$. There exists a $\eta =\eta _1>1$ satisfies

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \left( 6 ( \eta ^2 -1 ) \log \eta -7 \right) \\&\quad - mp(N-2-m) \left( 1 - \left( 1 - \frac{a_\alpha ^- }{L} \frac{1}{e \lambda } \right) ^{p-1} \right) \frac{1}{2e} \ge 0 \quad \text{ for } \quad \eta \ge \eta _1, \end{aligned} \end{aligned}$$

(4.2)

where $a^-_\alpha$ is a positive constant determined in the following inequality

$$\begin{aligned} \varphi _\alpha (r) \ge Lr^{-m} - a_\alpha ^- r^{-m - \lambda } \log r \quad \ \text {for} \quad r \ge 3 \end{aligned}$$

(4.3)

holds.

We define

$$\begin{aligned} U^-_\mathrm{out}(r,t) :={\left\{ \begin{array}{ll} (t+\tau )^{-l/2}F^-(\eta _1) =(t+\tau )^{-\frac{l}{2}} \eta _1^{-m - \lambda } f^-(\eta _1) &{}\quad \text {for} \, 0 \le \eta <\eta _1,\\ (t+\tau )^{-\frac{l}{2}} F^-(\eta ) =(t+\tau )^{-\frac{l}{2}} \eta ^{-m - \lambda } f^-(\eta ) &{}\quad \text {for} \, \eta \ge \eta _1, \end{array}\right. } \end{aligned}$$

with $\eta = (t+\tau )^{-1/2}r$, $\tau$ is a large positive constant such that $\eta _1 \tau ^{1/2} \ge 3$ determined later. Then $U^-_\mathrm{out}$ is a sub-solution of (2.1) .

Proof

It is trivial that $(t+\tau )^{-l/2}F^-(\eta _1)$ is a sub-solution. Then we only check the case where $\eta \ge \eta _1$. First, in general setting of our problem, the following differential inequality is computed.

$$\begin{aligned}&U^-_{\mathrm {out},t}-\mathcal {P}_\alpha U^-_\mathrm {out}\\ {}&=-(t+\tau )^{-\frac{l}{2}-1} \left( \frac{l}{2}F^- +\frac{\eta }{2}F^-_\eta + F^-_{\eta \eta }+ \frac{N-1}{\eta } F^-_\eta + p(t+\tau )\varphi _\alpha ^{p-1} F^- \right) \\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Big ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad +p(t+\tau ) \varphi _\alpha ^{p-1} f^- + \eta ^{-2} \left( (m+\lambda +1)(m+\lambda ) -(N-1)(m+\lambda ) \right) f^- \Big ) \\ {}&\le -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad +p(t+\tau ) \left( Lr^{-m} -a_\alpha ^- r^{-m-\lambda } \log r \right) ^{p-1} f^-\\ {}&\quad + \eta ^{-2} \left( (m+\lambda +1)(m+\lambda ) -(N-1)(m+\lambda ) \right) f^- \Bigg ) \\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad +p(t+\tau ) L^{p-1} r^{-m(p-1)} \left( 1 - \frac{a_\alpha ^-}{L} r^{-\lambda } \log r \right) ^{p-1} f^-\\ {}&\quad + \eta ^{-2} \left( (m+\lambda +1)(m+\lambda ) -(N-1)(m+\lambda ) \right) f^- \Bigg ) \\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad + \eta ^{-2} \Big ( (m+\lambda +1)(m+\lambda ) -(N-1)(m+\lambda ) \\ {}&\quad +p L^{p-1}\big ( 1 - \frac{a_\alpha ^- }{L}r^{-\lambda } \log r \big )^{p-1} \Big ) f^- \Bigg ) \\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad + \eta ^{-2} \Big ( (m+\lambda )^2 + 2(m+\lambda )- N(m+\lambda )\\ {}&\quad + (m+2)(N-2-m)\big ( 1-1+ \big ( 1 - \frac{a_\alpha ^- }{L}r^{-\lambda } \log r \big )^{p-1} \big ) \Big ) f^- \Bigg )\\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad + \eta ^{-2} \Big ( \lambda ^2 - (N-2-m)\lambda +m^2 +2m -Nm \\ {}&\quad + (m+2)(N-2-m)\big ( 1-1+ \big ( 1 - \frac{a_\alpha ^- }{L}r^{-\lambda } \log r \big )^{p-1} \big ) \Big ) f^- \Bigg )\\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( f^-_{\eta \eta } + \frac{N-1-2(m + \lambda )}{\eta }f^-_{\eta } + \frac{\eta }{2}f^-_{\eta } + \frac{ l - m - \lambda }{2} f^- \\ {}&\quad + \eta ^{-2} \Big ( \lambda ^2 - (N -2- 2 m)\lambda + 2(N-2-m) \\ {}&\quad -(m+2)(N-2-m) \big ( 1- \big ( 1 - \frac{a_\alpha ^- }{L}r^{-\lambda } \log r \big )^{p-1} \big ) \Big ) f^- \Bigg ). \end{aligned}$$

Here we use (1.3) and (4.3).

Next, we substitute $l=m+\lambda +2, f^-(\eta )=\exp (-\eta ^2) \log \eta$ with $r=(t+\tau )^{1/2} \eta$. We use $m + \lambda = (N-2)/2$ and (1.5), then we can simplify the above inequality as follows.

$$\begin{aligned}&U^-_{\mathrm {out},t}-\mathcal {P}_\alpha U^-_\mathrm {out}\\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda }\Bigg ( \Big ( \big ( 4\eta ^2 \log \eta - 2 \log \eta -4 - \eta ^{-2} \big ) + \frac{1}{\eta } \big ( -2\eta \log \eta + \eta ^{-1} \big )\\ {}&\quad + \frac{\eta }{2} \big ( -2 \eta \log \eta + \eta ^{-1} \big ) + \log \eta \Big ) \exp (-\eta ^2)\\ {}&\quad -\eta ^{-2} (m+2)(N-2-m) \left( 1 - \left( 1 - \frac{a_\alpha ^- }{L}r^{-\lambda } \log r \right) ^{p-1} \right) \exp (-\eta ^2) \log \eta \Bigg ) \\ {}&= -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( \frac{1}{2} \left( 6 ( \eta ^2 -1 ) \log \eta -7 \right) \\ {}&\quad - (m+2)(N-2-m) \left( 1 - \left( 1 - \frac{a_\alpha ^- }{L} r^{-\lambda } \log r \right) ^{p-1} \right) \eta ^{-2} \log \eta \Bigg )\exp (-\eta ^2) \\ {}&\le -(t+\tau )^{-\frac{l}{2}-1} \eta ^{-m-\lambda } \Bigg ( \frac{1}{2} \left( 6 ( \eta ^2 -1 ) \log \eta -7 \right) \\ {}&\quad - (m+2)(N-2-m) \left( 1 - \left( 1 - \frac{a_\alpha ^- }{L} \frac{1}{e \lambda } \right) ^{p-1} \right) \frac{ 1}{2 e} \Bigg )\exp (-\eta ^2) \\ {}&\le 0 \quad \text{ for } \quad \eta \ge \eta _1, \end{aligned}$$

from the assumption (4.2), we complete the proof. $\square$

Remark 4.1

We remark $\eta _1$ is a constant, it means $\eta _1$ does not depend on time and spatial variable and find $\eta _1 >1$ if the above inequality is satisfied.

4.2 Inner Sub-solution and Matching

We use a similar inner sub-solution as in [17, Lemma4.1].

Lemma 4.2

For any $q>0$, we define

$$\begin{aligned} U^-_\mathrm{in}(r,t) := (t+\tau )^{-q}(\log (t+\tau )^{1/2})^{-1} \psi (r). \end{aligned}$$

If $\tau$ is sufficiently large, then there exist constants $B_2 > 0$ satisfies $B_2 \tau ^{1/2} > 3$ and $c >0$ such that the following inequalities hold.

• (i) $U^-_{\mathrm{in},t} \le \mathcal{P}_\alpha U^-_\mathrm{in}$ for all $r > 0 \,$ and $t > 0$.

• (ii) $U^-_\mathrm{in}(r,t) >0$ for all $t >0$ and $r \in [0,B_2(t+\tau )^{\frac{1}{2}}]$.

• (iii) $cU^-_\mathrm{in}(r,t) < \, U^-_\mathrm{out}(r,t)$ at $r=B_2(t+\tau )^{\frac{1}{2}}$ for all $t >0$.

• (iv) $cU^-_\mathrm{in}(0,t) > \, U^-_\mathrm{out}(0,t)$ for all $t >0$.

Proof

We compute

$$\begin{aligned} U^-_{\mathrm{in},t} - \mathcal{P}_\alpha U^-_\mathrm{in}&= -q(t+\tau )^{-q-1}\big (\log (t+\tau )^{1/2}\big )^{-1}\psi (r)\\&\quad - \frac{1}{2}(t+\tau )^{-q-1/2}\big ((t+\tau )^{1/2}\big )^{-1}\big (\log (t+\tau )^{1/2}\big )^{-2} \psi (r)\\&\quad - (t+\tau )^{-q} \big (\log (t+\tau )^{1/2}\big )^{-1}\mathcal{P}_\alpha \psi (r) \\&= -q(t+\tau )^{-q-1}\big (\log (t+\tau )^{1/2}\big )^{-1}\psi (r)\\&\quad - \frac{1}{2}(t+\tau )^{-q-1/2}\big ((t+\tau )^{1/2}\big )^{-1}\big (\log (t+\tau )^{1/2}\big )^{-2} \psi (r)\\&< 0, \quad \text {for} \quad r>0, t>0, \end{aligned}$$

by (2.3) and Lemma 2.3. Hence $U^-_\mathrm{in}$ is a sub-solution of (2.1) which proves(i).

Next, let us shows (ii), (iii). We set

$$\begin{aligned} q := \frac{l- m - \lambda }{2} \end{aligned}$$

in Lemma 4.2. By Lemma 2.3, We can choose positive constants $c_\alpha ^+$ such that

$$\begin{aligned} \psi (r) \le c_\alpha ^+ r^{-m-\lambda }\log r \quad \text {for} \quad r \ge 3. \end{aligned}$$

(4.4)

First, we fix $B_2> \eta _1 >1$. Next, we take $c > 0$ satisfies

$$\begin{aligned} f^-(B_2) - c c_\alpha ^+ \left( \frac{\log B_2}{\log \tau ^{1/2}} +1\right) > 0. \end{aligned}$$

(4.5)

Finally, we take $\tau >0$ is sufficiently large such that

$$\begin{aligned} \tau ^{1/2}> 3, \, c \tau ^{\lambda /2} m_2 - \eta _1^{-m-\lambda } f^-(\eta _1)> 0, \, B_2 \tau ^{1/2}> 3,\, \tau > \nu _2, c_2 -\tau ^{-1}M \ge 0, \end{aligned}$$

(4.6)

where $m_2>0, M>0$ are given by

$$\begin{aligned} m_2 : = \min _{ t \ge 0} \{( t+3)^{\frac{m}{2}}(\log (t+3)^{1/2})^{-1} \} \end{aligned}$$

and

$$\begin{aligned} M:=\max _{ r \ge r^*(0) } \left\{ r^{-(m+\lambda )}\exp {(-(\tau - \nu _2)r^2)} \log (\tau ^{-1/2}r) \right\} . \end{aligned}$$

It is clear that

$$\begin{aligned} U^-_{\mathrm {in}}(r,t)&> 0 , \quad t>0, r \in \left[ 0,B_2(t+\tau )^{1/2}\right] . \end{aligned}$$

Indeed, we recall

$$\begin{aligned} U^-_\mathrm {in}(r,t)&:= (t+\tau )^{-q}\left( \log (t+\tau )^{1/2}\right) ^{-1} \psi (r) >0 \, \text{ for } \, t\ge 0, r \ge 0, \end{aligned}$$

which prove (ii) by using positivity of $\psi (r)$ and (4.6).

Next, let us show $U^-_\mathrm{out}(r_0, t)- cU^-_\mathrm{in}(r_0, t) >0$ at $r_0(t) := B_2(t+\tau )^{1/2}$ and $\eta _0 := (t+\tau )^{-1/2} r_0(t)$, namely at $\eta _0 = B_2$. We obtain

$$\begin{aligned}&U^-_\mathrm {out}(r_0(t),t)- c U^-_\mathrm {in}(r_0(t),t)\\ {}&\quad = (t+\tau )^{-\frac{l}{2}} F^-(\eta _0) - c(t+\tau )^{-q}\left( \log (t+\tau )^{1/2} \right) ^{-1} \psi (r_0)\\ {}&\quad \ge (t+\tau )^{-\frac{l}{2}} B_2^{-(m+\lambda )} f^-(B_2) \\ {}&\qquad - c (t+\tau )^{-\frac{l-m-\lambda }{2}} \left( \log (t+\tau )^{1/2}\right) ^{-1} c_\alpha ^+ r_0^{-(m+\lambda )}\log r_0 \\ {}&\quad = (t+\tau )^{-\frac{l}{2}} \Big ( B_2^{-( m+\lambda )} f^-(B_2) - c c_\alpha ^+ B_2^{-(m+\lambda )} \frac{\log B_2(t+\tau )^{1/2}}{\log (t+\tau )^{1/2}}\Big ) \\ {}&\quad = (t+\tau )^{-\frac{l}{2}} \left( f^-(B_2) - c c_\alpha ^+ \frac{\log B_2 + \log (t+\tau )^{1/2}}{\log ( t+\tau )^{1/2}} \right) B_2^{-(m+\lambda )} \\ {}&\quad = (t+\tau )^{-\frac{l}{2}} \left( f^-(B_2) - c c_\alpha ^+ \left( \frac{\log B_2}{\log (t+\tau )^{1/2}} +1 \right) \right) B_2^{-(m+\lambda )}\\ {}&\quad> (t+\tau )^{-\frac{l}{2}} \left( f^-(B_2) - c c_\alpha ^+ \left( \frac{\log B_2}{\log \tau ^{1/2}} +1 \right) \right) B_2^{-(m+\lambda )}\\ {}&\quad> 0,\quad \text{ for } \text{ all } \, t > 0, \end{aligned}$$

by (4.4), (4.5) and (4.6), thus (iii) is proved.

Finally, let us show $cU^-_\mathrm{in}(0,t)-U^-_\mathrm{out}(0,t)>0$, we obtain

$$\begin{aligned}&c U^-_\mathrm{in}(0,t) - U^-_\mathrm{out}(0,t)\\&= c(t+\tau )^{-\frac{l-m-\lambda }{2}}\left( \log (t+\tau )^{1/2}\right) ^{-1} \psi (0) - (t+\tau )^{-\frac{l}{2}}\eta _1^{-m-\lambda }f^-(\eta _1)\\&=(t+\tau )^{-\frac{l}{2}}\big ( c(t+\tau )^{\frac{m+\lambda }{2}}\left( \log (t+\tau )^{1/2}\right) ^{-1} - \eta _1^{-m-\lambda }f^-(\eta _1) \big )\\&= (t+\tau )^{-\frac{l}{2}}\big ( c (t+\tau )^{\frac{\lambda }{2}}(t+\tau )^{\frac{m}{2}}\left( \log (t+\tau )^{1/2}\right) ^{-1} - \eta _1^{-m-\lambda }f^-(\eta _1) \big )\\&\ge (t+\tau )^{-\frac{l}{2}}\big ( c (t+\tau )^{\frac{\lambda }{2}} m_2 - \eta _1^{-m-\lambda }f^-(\eta _1) \big )\\&\ge (t+\tau )^{-\frac{l}{2}}\big ( c \tau ^{\frac{\lambda }{2}} m_2- \eta _1^{-m-\lambda }f^-(\eta _1) \big )\\&>0 \end{aligned}$$

by using (4.6). This proves (iv). Then we complete the proof. $\square$

Proposition 4.3

Suppose that $l =m + \lambda +2$ and

$$\begin{aligned} U_0(r) \ge c_2 \exp {(-\nu _2 r^2)}, \quad r \ge 0 \end{aligned}$$

with some $c_2 >0$ and $\nu _2>0$. Then there exists a constant $C^- > 0$ such that the solution of (2.1) satisfies

$$\begin{aligned} \Vert U(\cdot ,t )\Vert _{L^\infty } \ge C^- U(0,t) = C^- (t+\tau )^{-1}(\log (t+\tau )^{1/2})^{-1} \quad \text { for all } \, t > 0. \end{aligned}$$

Proof

Let $U^-_\mathrm{out}(r,t)$ and $U^-_\mathrm{in}(r,t)$ be as in Lemmas 4.1 and 4.2 respectively, and define

$$\begin{aligned} U^-(r, t) :={\left\{ \begin{array}{ll} cU^-_\mathrm{in}(r, t) &{}\quad \text {for} \, r < r^*(t),\\ U^-_\mathrm{out}(r, t) &{}\quad \text {for} \, r \ge r^*(t), \end{array}\right. } \end{aligned}$$

where $r^*(t)$ is defined

$$\begin{aligned} r^*(t) : = \sup \{r>0 | cU^-_\mathrm{in}(\rho ,t) > U^-_\mathrm{out}(\rho ,t) \, \text {for} \, \rho \in [0, r)\}. \end{aligned}$$

From Lemma 4.2 (iii) and (iv), we obtain

$$\begin{aligned} 0< r^*(t)< r_0(t) < \infty \quad \text { for all } t > 0. \end{aligned}$$

We note that $r^*(t) \in (0 , \infty ]$ is well defined since $0< r^*(t) < \infty$. This fact implies, $c U^-_\mathrm{in}(r, t)$ intersects $U^-_\mathrm{out}(r, t)$ at $r^*(t)$.

It is clear that

From the construction of $U^{-}$, it attains the exact decay rate at the origin. Thus it is shown that $U^- (r,t)$ is a sub-solution of (2.1) which satisfies

$$\begin{aligned} U^-(0,t)&= cU^-_\mathrm{in}(0,t)\\&= c (t+\tau )^{-(l-m-\lambda )/2}\left( \log (t+\tau )^{1/2}\right) ^{-1}\psi (0)\\&= c (t+\tau )^{-1}\left( \log (t+\tau )^{1/2}\right) ^{-1} \end{aligned}$$

for all $t > 0$ with $l=m+\lambda +2$.

We will show that the $C^- U^-(r,0)$ lies below the initial data $U_0(r)$ if we take a constant $C^- >0$ sufficiently small. In fact, we can take a constant $C^->0$ small enough to hold that

$$\begin{aligned} C^- U^-(r,0) \le U_0(r), \quad r \ge 0. \end{aligned}$$

Indeed, if we take $0 < C^- \le 1$ so small that

$$\begin{aligned} C^- c U^-_\mathrm{in}(r,0)&= C^- c\tau ^{-(l-m-\lambda )/2}\left( \log \tau ^{1/2}\right) ^{-1}\psi (0)\\&= C^- c\tau ^{-1}\left( \log \tau ^{1/2}\right) ^{-1}\le c_2 \le U_0(r) \quad \text {for} \, 0 \le r \le r^*(0) \end{aligned}$$

and

$$\begin{aligned}&U_0(r) - C^- U^-_\mathrm{out}(r,0)\\&\ge c_2\exp {(-\nu _2 r^2)}- C^- U^-_\mathrm{out}(r,0)\\&=c_2\exp {(-\nu _2 r^2)}-C^- \tau ^{-\frac{l}{2}} F^-(\eta )\\&= c_2\exp {(-\nu _2 r^2)}-C^- \tau ^{-\frac{l}{2}} \eta ^{-m-\lambda }\exp {(-\tau r^2)} \log \tau ^{-1/2}r \\&\ge \Big ( c_2- C^- \tau ^{\frac{-(l - m - \lambda )}{2}} r^{-(m+\lambda )}\exp {(-( \tau - \nu _2 )r^2)} \log \tau ^{-1/2}r \Big ) \exp {(-\nu _2 r^2)}\\&\ge \left( c_2 - C^- \tau ^{\frac{-(l - m - \lambda )}{2}} M \right) \exp {(-\nu r^2)}\\&=\left( c_2 - C^- \tau ^{-1} M \right) \exp {(-\nu r^2)}\\&\ge 0 \, \text {for} \, r \ge r^*(0) \end{aligned}$$

by using (4.6). Then the initial condition is satisfied by the above argument, and by the comparison principle, we obtain

$$\begin{aligned} C^- U^-(r,t) \le U(r,t) \quad \text {for} \quad r>0, \, t>0 . \end{aligned}$$

Since $U^-$ attains the exact decay rate at the origin, we finish the proof. $\square$

Proof of Theorem 1.2

We take

$$\begin{aligned} U_0(r) := \min _{|x|=r}{|u_0(x) -\tilde{u}_0(x)|} >0 ,\quad r \ge 0. \end{aligned}$$

Then by Lemma 2.2, Proposition 4.3, and put $l=m+\lambda +2$ we have

$$\begin{aligned} \Vert u(\cdot ,t) - \tilde{u}(\cdot ,t)\Vert _{L^\infty } \ge C^- U^-(0,t) \ge C (t+3)^{-1}\left( \log (t+3)^{1/2}\right) ^{-1} \end{aligned}$$

for all $t > 0$ with some constant $C > 0$. The proof is now complete. $\square$