- Research
- Open Access
Complicated asymptotic behavior exponents for solutions of the evolution p-Laplacian equation with absorption
- Liangwei Wang1Email author,
- Jngxue Yin2 and
- Yuqiu Wu1
- Received: 26 December 2016
- Accepted: 8 May 2017
- Published: 19 May 2017
Abstract
In this paper, we investigate how the initial value belonging to spaces \(W_{\sigma}(\mathbb{R}^{N})\) (\(0<\sigma<N\)) affects the complicated asymptotic behavior of solutions for the Cauchy problem of the evolution p-Laplacian equation with absorption. In fact, we reveal the fact that \(\sigma=\frac{p}{q-p+1}\) is the critical exponent for the complicated asymptotic behavior of the solutions.
Keywords
- complicated asymptotic behavior
- evolution p-Laplacian equation
- critical exponent
- absorption
MSC
- 35B40
- 35K65
1 Introduction
In this paper, we study the complicated asymptotic behavior of solutions for the Cauchy problem of the evolution p-Laplacian equation with absorption
where \(p>2\), \(q>p-1+\frac{p}{N}\), \(N\ge1\), \(\lambda>0\) and \(\varphi (x)\in W_{\sigma}^{+}(\mathbb{R}^{N})\), i.e., \(\varphi(x)\geq0\) and \(\varphi \in W_{\sigma}(\mathbb{R}^{N})\equiv\{\phi\in L^{1}_{\mathrm{loc}}(\mathbb {R}^{N}); \vert x \vert ^{\sigma}\phi(x)\in L^{\infty}(\mathbb{R}^{N})\}\) with the norm \(\Vert \varphi \Vert _{W_{\sigma}(\mathbb{R}^{N})}= \Vert \vert \cdot \vert ^{\sigma}\varphi(\cdot) \Vert _{L^{\infty}(\mathbb{R}^{N})}\).
$$\begin{aligned} &\frac{\partial u}{\partial t}-\operatorname {div}\bigl( \vert \nabla u \vert ^{p-2}\nabla u \bigr)+ \lambda u^{q}=0 \quad \text{in }(0,\infty)\times \mathbb{R}^{N}, \end{aligned}$$
(1.1)
$$\begin{aligned} &u(x,0)=u_{0}(x)\quad \text{in }\mathbb{R}^{N}, \end{aligned}$$
(1.2)
For solutions of some evolution equations, different initial values may cause different asymptotic behaviors, see [1–5]. Consider Problem (1.1)-(1.2). If \(\lambda=0\) and the nonnegative initial value \(u_{0}\in L^{1}(\mathbb{R}^{N})\), it is well known that the solutions \(u(x,t)\) converge to the Barenblatt solution \(U_{M}\) in \(L^{1}(\mathbb{R}^{N})\) as \(t\to\infty\) [6]. If \(q>p-1+\frac{p}{N}\), \(0<\sigma<N\) and \(\lambda=1\), the initial value \(u_{0}(x)\in W^{+}_{\sigma}(\mathbb{R}^{N})\) and \(\lim_{ \vert x \vert \rightarrow\infty} \vert x \vert ^{\sigma}u_{0}(x)= A\), then the solutions satisfy
uniformly on the cone \(\{x\in\mathbb{R}^{N}; \vert x \vert \leq Ct^{\beta}\} \) as \(t\rightarrow+\infty\), where \(\beta=\frac{q-p+1}{p(q-1)}\) and \(w(x,t)=(\frac{1}{q-1})^{\frac {1}{q-1}}\) if \(0<\sigma<\frac{p}{q-p+1}\), or \(\beta=\frac{1}{\sigma (p-2)+p}\) and \(w(x,t)\) is the solution of the Cauchy problem of the evolution p-Laplacian equation without absorption
if \(\sigma=\frac{p}{q-p+1}\), or \(\beta=\frac{1}{\sigma(p-2)+p}\) and \(w(x,t)\) is the solution of Problem (1.1) with the initial value \(w(x,0)=A \vert x \vert ^{-\sigma}\) if \(\frac{p}{q-p+1}<\sigma <N\), see details in [7]. If \(\lambda=0\) and the initial value belongs to the bounded function space, it was first founded by Vázquez and Zuazua [8] that there exists an initial value \(u_{0}\in L^{\infty}(\mathbb{R}^{N})\) such that the rescaled solutions \(u(t_{n}^{\frac {1}{p}}x,t_{n})\) converge to different functions in the weak-star topology of \(L^{\infty}(\mathbb{R}^{N})\) along different sequences \(t_{n}\rightarrow \infty\). This result means that the bounded function space \(L^{\infty }(\mathbb{R}^{N})\) provides the work space where complicated asymptotic behavior of solutions takes place.
$$t^{\frac{\sigma}{\sigma(p-2)+p}} \bigl\vert u(x,t) -w(x,t) \bigr\vert \rightarrow0 $$
$$\begin{aligned} &\frac{\partial w}{\partial t}-\operatorname {div}\bigl( \vert \nabla w \vert ^{p-2}\nabla w \bigr)=0 \quad \text{in }(0,\infty)\times\mathbb{R}^{N}, \end{aligned}$$
(1.3)
$$\begin{aligned} &w(x,0)=w_{0}(x)=A \vert x \vert ^{-\sigma}\quad \text{in } \mathbb{R}^{N} \end{aligned}$$
(1.4)
Since then, much attention has been paid to studying the complicated asymptotic behavior of solutions for evolution equations [9–11]. For example, Cazenave et al. considered the Cauchy problem of the heat equation and got a series of important results about the complicated asymptotic behavior of the rescaled solutions \(t^{\mu}(t^{\beta}x, t)\) (\(\mu, \beta>0\)) in papers [12–16]. In our previous papers [17–19], we investigated the complicated asymptotic behavior of solutions for the porous medium equation. One can find some other interesting results of the partial differential equations in [20–23].
Inspired by the above papers, in this paper, we try to find out how the initial value belonging to \(W_{\sigma}(\mathbb{R}^{N})\) with different σ affects the complicated asymptotic behavior for the solutions of Problem (1.1)-(1.2) with \(\lambda=1\). In fact, we find that if \(0<\sigma<\frac{p}{q-p+1}\), the complicated asymptotic behavior for the solutions of Problem (1.1)-(1.2) with the initial value \(u_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})\) cannot happen. While if \(\frac{p}{q-p+1}\leq\sigma< N\), then the complicated asymptotic behavior for the solutions of Problem (1.1)-(1.2) with the initial value \(u_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})\) may happen. In fact, if \(\sigma=\frac{p}{q-p+1}\), there exists an initial value \(u_{0}\in B^{\sigma,+}_{M}\equiv\{f\in W_{\sigma}^{+}(\mathbb{R}^{N}); \Vert f \Vert _{W_{\sigma}^{+}(\mathbb{R}^{N})}\leq M\}\) such that for every \(\phi\in B^{\sigma,+}_{M}\), there exists a sequence \(\{t_{n}\}\) such that
uniformly on \(\mathbb{R}^{N}\), where \(v(x,t)\) is the solution of Problem (1.1) with the initial value \(v(x,0)=\phi\); or if \(\frac{p}{q-p+1}<\sigma<N\), then there exists an initial value \(u_{0}\in B^{\sigma,+}_{M}\) such that for every \(\varphi\in B^{\sigma,+}_{M}\), there exists a sequence \(\{t_{n}\}\) such that
uniformly on \(\mathbb{R}^{N}\), where \(w(x,t)\) is the solution of Problem (1.3)-(1.4) with the initial value \(w(x,0)=\varphi\). So, the complexity of asymptotic behavior of the solutions for \(\frac{p}{q-p+1}\leq\sigma< N\) occurs, according to Vázquez and Zuazua [8]. Therefore, we get that \(\sigma=\frac{p}{q-p+1}\) is the critical exponent for the complexity of asymptotic behavior of solutions. For convenience, in the rest of this paper, we define \(\gamma=\frac {1}{\sigma(p-2)+p}\) and put \(\lambda=1\) in (1.2).
$$\lim_{t_{n}\rightarrow\infty}t^{\frac{\sigma}{\sigma(p-2)+p}}_{n} u \bigl(t^{\frac{1}{\sigma(p-2)+p}}_{n}x,t_{n}\bigr)=v(x,1) $$
$$\lim_{t_{n}\rightarrow\infty}t^{\frac{\sigma}{\sigma(p-2)+p}}_{n} u \bigl(t^{\frac{1}{\sigma(p-2)+p}}_{n}x,t_{n}\bigr)=w(x,1) $$
The rest of this paper is organized as follows. In the next section, we give some concepts and lemmas. Section 3 is devoted to the study of the nonexistence of complexity for the asymptotic behavior of solutions. The complexity of asymptotic behavior for the solutions is considered for \(\sigma=\frac{p}{q-p+1}\) in Section 4 and for \(\frac{p}{q-p+1}<\sigma<N\) in Section 5, respectively.
2 Preliminaries
In this section, we first give some concepts as [24–26]. For \(f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\) and \(r>0\), we define
The space \(X_{0}\) is given by
with the norm \(\Vert \!\vert \cdot \vert \!\Vert _{1}\). The existence and uniqueness of global weak solution for Problem (1.1)-(1.2) with the initial value \(\varphi(x)\in X_{0}\) has been shown in [24, 25] and this solution satisfies
for some \(\alpha>0\). Note that for \(0<\sigma<N\),
So we can define an operator \(T(t): W_{\sigma}(\mathbb{R}^{N})\to C(\mathbb {R}^{N})\) as
where \(u(x,t)\) is the solution of Problem (1.1)-(1.2) with the initial value \(u_{0}(x)\).
$$\Vert \!\vert f \vert \!\Vert _{r}=\sup _{R\geq r}R^{-\frac {N(p-2)+p}{p-2}} \int_{\{ \vert x \vert \leq R\}} \bigl\vert f(x) \bigr\vert \,dx. $$
$$X_{0}\equiv \Bigl\{ \varphi\in L^{1}_{\mathrm{loc}} \bigl( \mathbb{R}^{N} \bigr); \Vert \!\vert \varphi \vert \!\Vert _{1}< \infty \text{ and } \lim_{r\rightarrow+\infty} \Vert \! \vert \varphi \vert \!\Vert _{r}=0 \Bigr\} $$
$$\begin{aligned} u(x,t)\in C^{\frac{\alpha}{2},\alpha} \bigl((0,\infty)\times \mathbb{R}^{N} \bigr) \end{aligned}$$
(2.1)
$$W_{\sigma}\bigl(\mathbb{R}^{N}\bigr)\subset X_{0}. $$
$$\begin{aligned} T(t)u_{0}(x)=u(x,t), \end{aligned}$$
(2.2)
Lemma 2.1
For
\(w_{0}\in X_{0}\), there exists a unique global weak solution
\(w(x,t)\)
of Problem (1.3)-(1.4). Moreover, the evolution
p-Laplacian equation generates a bounded semigroup in
\(X_{0}\)
given by
If
\(1\leq q\leq\infty\)
and
\(w_{0}\in L^{q}(\mathbb{R}^{N})\subset X_{0}\), then
\(S(t)\)
is a contraction bounded semigroup in
\(L^{q}(\mathbb{R}^{N})\).
$$S(t): w_{0}\rightarrow w(x,t). $$
The following two lemmas appeared in [27] to study the chaotic dynamic systems in the evolution p-Laplacian equation. Let us write \(\Omega(t)=\{x\in\mathbb{R}^{N}; w(x,t)>0\}\), and let \(d(x,\Omega(t))\) be the distance from x to \(\Omega(t)\).
Lemma 2.2
Propagation speed estimate [27]
Suppose
\(0<\sigma<N\). If
\(w(x,t)\)
is the weak solution of Problem (1.3)-(1.4) with the initial value
\(w_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})\), then for
\(0\leq t_{1}< t_{2}<\infty\), we have
where
\(\Omega_{\rho(t_{2}-t_{1})}(t_{1})\equiv\{x\in\mathbb{R}^{N}; d(x,\Omega (t_{1}))< \rho(t_{2}-t_{1})\}\)
and
\(\rho(t_{2}-t_{1})=C(t_{2}-t_{1})^{\frac{1}{\sigma(p-2)+p}} \Vert u_{0} \Vert _{W_{\sigma}(\mathbb{R}^{N})}^{\frac{p-2}{\gamma}}\).
$$\Omega(t_{2})\subset\Omega_{\rho(t_{2}-t_{1})}(t_{1}), $$
The following lemma concerns the decay estimate of the solutions.
Lemma 2.3
Space-time decay estimate [27]
Let
\(0<\sigma<N\). If
\(u_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})\), then for
\(t>0\)
and
\(x\in\mathbb{R}^{N}\),
$$\begin{aligned} S(t)u_{0}(x)\leq C \bigl(t^{\frac{2}{\gamma}}+ \vert x \vert ^{2} \bigr)^{-\sigma}. \end{aligned}$$
(2.3)
3 Nonexistence of complexity: \(0<\sigma<\frac{p}{q-p+1}\)
In this section, we consider the nonexistence of complicated asymptotic behavior for the solutions of Problem (1.1)-(1.2) with the initial value \(u_{0}\in W^{+}_{\sigma}(\mathbb{R}^{N})\). The ideas of the proof of the following lemma come from [1, 2, 7], we give it here for completeness.
Lemma 3.1
Let
and let
If
\(u(x,t)\)
is the solution of Problem (1.1)-(1.2) with the initial value
\(u_{0}(x)=\varphi(x)\), then
$$q>p-1+\frac{p}{N}, \quad\quad 0< \sigma< \frac{p}{q-p+1}, $$
$$\varphi\in W_{\sigma}^{+}\bigl(\mathbb{R}^{N}\bigr). $$
$$\begin{aligned} \lim_{t\to\infty}\sup_{ \vert x \vert \leq Ct^{\frac{q-p+1}{p(q-1)}}}t^{\frac {1}{q-1}}u(x,t) \leq \biggl( \frac{1}{q-1} \biggr)^{\frac{1}{q-1}}. \end{aligned}$$
(3.1)
Proof
Let
So, for every \(k >0\), \(u_{k}(x,t)\) is a solution of Problem (1.1)-(1.2) with the initial value
Since \(\overline{u(x,t)}=(\frac{1}{q-1})^{\frac{1}{q-1}}t^{-\frac {1}{q-1}}\) is a solution of equation (1.1), it follows from the comparison principle that for every \((x,t)\in(0,+\infty)\times\mathbb{R}^{N}\),
This uniform upper bound implies that the sequence \(\{u_{k}\}\) is equicontinuous on compact subsets of \((0,+\infty)\times\mathbb{R}^{N}\). So we can extract a convergent subsequence \(\{u_{k'}\}\) such that
on compact subsets of \((0,+\infty)\times\mathbb{R}^{N}\). Therefore, for every \(C>0\), putting \(t=1\), \(kx = x'\) and \(k^{\frac {p(q-1)}{q-p+1}} = t'\), we obtain, omitting the primes,
The proof of this lemma is complete. □
$$u_{k}(x,t)=k^{\frac{p}{q-p+1}}u\bigl(kx,k^{\frac{p(q-1)}{q-p+1}}t\bigr), \quad k>0. $$
$$u_{k}(x,0)=\varphi_{k}(x)=k^{\frac{p}{q-p+1}}\varphi(kx), \quad k>0. $$
$$u_{k}(x,t)\leq\overline{u(x,t)}. $$
$$u_{k'}(x,t)\stackrel{ k'\rightarrow\infty}{ \longrightarrow} U(x,t)\leq\overline{u(x,t)} $$
$$\lim_{t\to\infty}t^{\frac{1}{q-1}}\sup_{\{ \vert x \vert \leq Ct^{\frac{q-p+1}{p(q-1)}}\}}u(x,t) \leq \biggl( \frac{1}{q-1} \biggr)^{\frac{1}{q-1}}. $$
Theorem 3.2
Suppose
Let
\(u(x,t)\)
be the solutions of Problem (1.1)-(1.2) with the initial value
\(u_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})\). Then the complexity cannot occur in the asymptotic behavior of the rescaled solutions
\(t^{\frac{\sigma}{\gamma}}u(t^{\frac{1}{\gamma}} x,t)\)
in
\(L^{\infty }(\mathbb{R}^{N})\). In other words, if there exists a function
\(\phi\in W_{\sigma}(\mathbb{R}^{N})\)
and a sequence
\(t_{n}\to\infty\)
such that
uniformly on
\(\mathbb{R}^{N}\), then
$$q>p-1+\frac{p}{N} \quad\textit{and}\quad 0< \sigma< \frac{p}{q-p+1}. $$
$$\begin{aligned} & \lim_{t_{n}\to\infty}t_{n}^{\frac{\sigma}{\gamma}}u \bigl(t_{n}^{\frac{1}{\gamma}} x,t_{n} \bigr)=\varphi(x) \end{aligned}$$
(3.2)
$$\varphi(x)\equiv0. $$
Proof
Suppose that (3.2) holds for some \(\varphi(x)\not\equiv0\). And, without loss of generality, we assume that for some \(x_{0}\in\mathbb{R}^{N}\),
It follows from (3.2) that there exists an integer \(n_{1}\) such that if \(n\ge n_{1}\), then
Note that \(u_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})\). By using Lemma 3.1, we obtain that
uniformly on the sets \(\{x\in\mathbb{R}^{N}; \vert x \vert \leq Ct^{\frac{q-p+1}{p(q-1)}}\}\) for \(C>0\). It follows from (3.3) and the fact \({\frac{\sigma}{\gamma}}-\frac {1}{q-1}<0\) that there exists an integer \(n_{2}\) such that if \(n\ge n_{2}\), then
So, from (3.5), we have
Taking \(n=\max(n_{1}, n_{2})\), and then letting \(C=2 \vert x_{0} \vert t_{n}^{\frac{1}{\gamma}-{\frac{q-p+1}{p(q-1)}}}\) and \(x=t_{n}^{\frac {1}{\gamma}}x_{0}\), we have
Thus we deduce from (3.4) and (3.6) that
So we get a contradiction. Therefore, (3.2) cannot hold for \(\varphi(x)\not\equiv0\). This means that if (3.2) holds with \(0<\sigma<\frac{p}{q-p+1}\), then
and the proof is complete. □
$$\begin{aligned} \varphi(x_{0})>0. \end{aligned}$$
(3.3)
$$\begin{aligned} t_{n}^{{\frac{\sigma}{\gamma}}}u \bigl(t_{n}^{\frac{1}{\gamma }}x_{0},t_{n} \bigr)\ge\frac{1}{2}\varphi(x_{0}). \end{aligned}$$
(3.4)
$$\begin{aligned} \lim_{t\to\infty}t^{\frac{1}{q-1}}u(x,t) \leq \biggl( \frac{1}{q-1} \biggr)^{\frac{1}{q-1}} \end{aligned}$$
(3.5)
$$t_{n}^{{\frac{\sigma}{\gamma}}-\frac{1}{q-1}} {\biggl(\frac{1}{q-1}\biggr)^{\frac{1}{q-1}}}< \frac{1}{2}\varphi(x_{0}). $$
$$\begin{aligned} t_{n}^{{\frac{\sigma}{\gamma}}}u(x,t_{n})=t_{n}^{{\frac{\sigma }{\gamma}}-\frac{1}{q-1}} t^{\frac{1}{q-1}}u(x,t_{n})< \frac{1}{2} \varphi(x_{0}) \quad \text{for } x\in \bigl\{ y; \vert y \vert \leq Ct_{n}^{\frac{q-p+1}{p(q-1)}} \bigr\} . \end{aligned}$$
(3.6)
$$x\in\bigl\{ y; \vert y \vert \leq Ct_{n}^{\frac{q-p+1}{p(q-1)}}\bigr\} . $$
$$\frac{1}{2} \varphi(x_{0})\leq t_{n}^{{\frac{\sigma}{\gamma}}}u \bigl(t_{n}^{\frac {1}{\gamma}}x_{0},t_{n}\bigr)< \frac{1}{2} \varphi(x_{0}). $$
$$\varphi(x)\equiv0\quad\text{for } x\in\mathbb{R}^{N}, $$
4 Complexity: \(\sigma=\frac{p}{q-p+1}\)
To give the result about the complicated asymptotic behavior of solutions, we need introduce some concepts. For \(0<\sigma<N\) and \(M>0\), the convex closed set \(B^{\sigma,+}_{M}\) is defined as
For \(\lambda>0\), \(0<\sigma<N\), \(\varphi(x)\in L^{1}(\mathbb{R}^{N})\), we define
For \(\sigma=\frac{p}{q-p+1}\), it follows from this definition and (2.2) that the following commutative relation holds [28]:
Now we give the result that for \(\sigma=\frac{p}{q-p+1}\), the complicated asymptotic behavior for the solutions of Problem (1.1)-(1.2) with the initial value \(u_{0}\in W_{\sigma}^{+}(\mathbb {R}^{N})\) can happen.
$$B^{\sigma,+}_{M}\equiv\bigl\{ \varphi\in W^{+}_{\sigma}\bigl( \mathbb{R}^{N}\bigr)\cap C\bigl(\mathbb {R}^{N}\bigr); \Vert \varphi \Vert _{W^{+}_{\sigma}(\mathbb{R}^{N})}\leq M\bigr\} . $$
$$D^{\sigma}_{\lambda}\varphi(x)=\lambda^{\frac{2\sigma}{\gamma}}\varphi \bigl( \lambda^{\frac{2}{\gamma}}x\bigr). $$
$$\begin{aligned} D^{\sigma}_{\lambda} \bigl[T \bigl( \lambda^{2} t \bigr)u_{0} \bigr]=T(t) \bigl[D^{\sigma}_{\lambda}u_{0} \bigr]. \end{aligned}$$
(4.1)
Theorem 4.1
Suppose
\(q>p+\frac{p}{N}\)
and
\(M>0\). Let
Then there exists a function
\(u_{0}\in B^{\sigma,+}_{M}\)
such that for every
\(\varphi\in B^{\sigma,+}_{M}\), there exists a sequence
\(t_{n}\to \infty\)
satisfying
uniformly on
\(\mathbb{R}^{N}\), where
\(u(x,t)\)
is the solution of Problem (1.1)-(1.2) with the initial value
\(u_{0}(x)\).
$$\sigma=\frac{p}{q-p+1}. $$
$$\begin{aligned} \lim_{n\to\infty}t^{\frac{\sigma}{\gamma}}_{n}u \bigl(t^{\frac{1}{\gamma}}_{n}x, t_{n} \bigr)=T(1)\varphi(x) \end{aligned}$$
(4.2)
Proof
By the definition of \(B^{\sigma,+}_{M}\), there exists a countable set \(F=\{\phi_{i}; \phi_{i}\in B^{\sigma ,+}_{M}\bigcap L^{1}(\mathbb{R}^{N}), i=1,2,\ldots\}\) such that for every \(\epsilon>0\) and \(\varphi\in B^{\sigma,+}_{M}\), there exists a function \(\phi_{\epsilon}\in F\) satisfying
Therefore, there exists a sequence \(\{\varphi_{j}\}_{j\geq1}\subset F\) such that Let
Now we can follow the methods given in [9, 10, 12] to construct an initial value by
where
and \(\chi_{j}(x)\) is the cut-off function defined on \(\{x\in\mathbb {R}^{N}; 2^{-j}< \vert x \vert <2^{j}\}\) relatively to \(\{x\in\mathbb {R}^{N}; 2^{-j+1}< \vert x \vert <2^{j-1}\}\). Note first that if \(\varphi\in B^{\sigma,+}_{M}\), then
and
By (4.6) and (4.7), we have
so
It follows from (4.1) that
We thus conclude from the definition of \(\lambda_{j}\), comparison principle [29] and Lemma 2.2 that
and
so
hence
The same result holds for \(0< t<1\),
From the comparison principle [25, 29], Lemma 2.1, (4.5) and (4.6), we have
as \(n\to\infty\). For every \(\phi\in F\), it follows from (4.4) and (4.8) that there exists a sequence \(\{\varphi_{n_{k}}\}_{k\geq1}\) such that if
then
By the \(L^{1}\)-contraction principle [25, 26], we conclude from (4.6) and (4.10) that
as \(k\to\infty\). Note that
Thus we deduce from (4.11) and (4.12) that there exists a subsequence, which we still denote as \(T(\frac{1}{2})D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}]\), which satisfies
Therefore, the regularity of the solutions (see (2.1)) indicates that
Note that
So, for every \(\varepsilon>0\), we obtain from Lemma 2.2 and the comparison principle that there exists \(k_{1}>0\) such that if \(\vert x \vert \geq2^{n_{k1}}\), then
and
Therefore, from (4.13), (4.14) and (4.15), we get
uniformly on \(\mathbb{R}^{N}\). Combining this with (4.9), we get that for every \(\phi\in F\), there exists a sequence \(n_{k}\to\infty\) as \(k\to\infty\) such that
uniformly on \(\mathbb{R}^{N}\). Taking \(t_{k}=\lambda_{n_{k}}^{\frac{1}{2}}\) in (4.16), we can conclude from (4.3) that (4.2) holds, and the proof is complete. □
$$\begin{aligned} \Vert \phi_{\epsilon}-\varphi \Vert _{L^{\infty}(\mathbb{R}^{N})}< \epsilon. \end{aligned}$$
(4.3)
- I.For every \(\phi_{i}\in F\), there exists a subsequence \(\{\varphi_{i_{k}}\} _{k\geq1}\) of the sequence \(\{\varphi_{j}\}_{j\geq1}\) such that$$\begin{aligned} \varphi_{j_{k}}(x)=\phi_{j}\quad\text{for all } k\geq1; \end{aligned}$$(4.4)
- II.There exists a constant \(C>0\) such that$$\begin{aligned} \max{ \bigl( \Vert \varphi_{j} \Vert _{L^{\infty}(\mathbb{R}^{N})}, \Vert \varphi_{j} \Vert _{L^{1}(\mathbb {R}^{N})} \bigr)} \leq Cj \quad\text{for } j\geq1. \end{aligned}$$(4.5)
$$N\gamma-\sigma{\bigl(N(p-2)+p\bigr)}>0. $$
$$\begin{aligned} \lambda_{j}= \textstyle\begin{cases} 2,& j=1,\\ \max(j^{\frac{3\gamma}{N\gamma-\sigma[N(p-2)+p]}}\lambda_{j-1}, (2^{j}\lambda_{j-1}^{\frac{2}{\gamma}}+C2^{j}M^{\frac{p-2}{\gamma}})^{\frac {\gamma}{2}} ), &j>1. \end{cases}\displaystyle \end{aligned}$$
(4.6)
$$\begin{aligned} u_{0}(x)= =\sum_{j=1}^{\infty}D^{\sigma}_{\lambda_{j}^{-1}} \bigl[\chi_{j}(x)\varphi_{j}(x) \bigr]=u_{n}+v_{n}+w_{n}, \end{aligned}$$
(4.7)
$$\begin{aligned} \begin{gathered} u_{n} =\sum _{j=1}^{n-1} D^{\sigma}_{\lambda_{j}^{-1}} \bigl[ \chi_{j}(x)\varphi_{j}(x) \bigr],\quad \quad v_{n}=D^{\sigma}_{\lambda_{n}^{-1}} \bigl[ \chi_{n}(x)\varphi_{n}(x) \bigr], \\ w_{n} =\sum_{j=n+1}^{\infty} D^{\sigma}_{\lambda_{j}^{-1}} \bigl[\chi_{j}(x)\varphi_{j}(x) \bigr], \end{gathered} \end{aligned}$$
(4.8)
$$\Vert \varphi \Vert _{W_{\sigma}(\mathbb{R}^{N})}\leq M $$
$$0\leq\varphi\in C\bigl(\mathbb{R}^{N}\bigr). $$
$$\Vert u_{0} \Vert _{L^{\infty}(\mathbb{R}^{N})}\leq \Vert u_{0} \Vert _{W_{\sigma}(\mathbb{R}^{N})}\leq\sup_{j\geq1} \bigl\Vert { \lambda_{j}}^{-{\frac{2\sigma}{\gamma}}} \chi_{j} \bigl(x/ \lambda_{j}^{\frac{2}{\gamma}} \bigr) \varphi_{j} \bigl(x/ \lambda_{j}^{\frac{2}{\gamma}} \bigr) \bigr\Vert _{{W_{\sigma}(\mathbb {R}^{N})}}\leq M, $$
$$u_{0}\in B^{\sigma,+}_{M}. $$
$$D^{\sigma}_{\lambda_{n}} \bigl[T \bigl( \lambda_{n}^{2}t \bigr)u_{0} \bigr]= T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{0} \bigr] = T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+ D^{\sigma}_{\lambda_{n}}v_{n}+D^{\sigma}_{\lambda_{n}}w_{n} \bigr]. $$
$$\operatorname {supp}\bigl(T(1) \bigl[D^{\sigma}_{n}(w_{n}) \bigr] \bigr)\subset \bigl\{ x\in\mathbb{R}^{N}; \vert x \vert >2^{n}+CM^{\frac{p-2}{\gamma}} \bigr\} $$
$$\operatorname {supp}\bigl({T(1)} \bigl[D^{\sigma}_{n}(v_{n}+u_{n}) \bigr] \bigr)\subset \bigl\{ x\in\mathbb{R}^{N}; \vert x \vert < 2^{n}+CM^{\frac{p-2}{\gamma}} \bigr\} , $$
$$\operatorname {supp}\bigl(T(1)\bigl[D^{\sigma}_{n}(w_{n})\bigr] \bigr)\cap \operatorname {supp}\bigl({T(1)}\bigl[D^{\sigma}_{n}(v_{n}+u_{n}) \bigr]\bigr)=\emptyset, $$
$$T(1) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+ D^{\sigma}_{\lambda_{n}}v_{n}+D^{\sigma}_{\lambda_{n}}w_{n} \bigr] = T(1) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+D^{\sigma}_{\lambda_{n}}v_{n} \bigr] +T(1) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr]. $$
$$T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+ D^{\sigma}_{\lambda_{n}}v_{n}+D^{\sigma }_{\lambda_{n}}w_{n} \bigr] = T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+D^{\sigma}_{\lambda_{n}}v_{n} \bigr] +T(t) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr]. $$
$$ \begin{aligned}[b] \bigl\Vert T(1) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr] \bigr\Vert _{L^{\infty}(\mathbb{R}^{N})} &\leq \bigl\Vert S(1) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr] \bigr\Vert _{L^{\infty}(\mathbb{R}^{N})} \leq C \bigl\Vert D^{\sigma}_{\lambda_{n}}w_{n} \bigr\Vert _{L^{\infty}(\mathbb{R}^{N})} \\ & \leq C\lambda_{n}^{{\frac{2\sigma}{\gamma}}}\lambda_{n+1}^{-{\frac {2\sigma}{\gamma}}} \sum_{i=n+1}^{\infty}2^{-i}i \leq C2^{-n}\to0 \end{aligned} $$
(4.9)
$$x\in E_{k}\equiv\bigl\{ y\in\mathbb{R}^{N}; 2^{-n_{k}+1}< \vert y \vert < 2^{n_{k}-1}\bigr\} , $$
$$\begin{aligned} D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}(x)=D^{\sigma}_{\lambda_{n_{k}}} \bigl[D^{\sigma}_{\lambda_{n_{k}}^{-1}}\chi_{n_{k}}\varphi_{n_{k}} \bigr](x)=\chi_{n_{k}}(x)\varphi_{n_{k}}(x) =\phi(x). \end{aligned}$$
(4.10)
$$ \begin{aligned}[b] & \biggl\Vert T(1/2) \bigl[D^{\sigma}_{\lambda _{n}}(u_{n_{k}}+v_{n_{k}}) \bigr]- T \biggl(\frac{1}{2} \biggr)\phi \biggr\Vert _{L^{1}(\mathbb{R}^{N})} \\ &\quad \leq \bigl\Vert \bigl[D^{\sigma}_{\lambda_{n_{k}}}(u_{n_{k}}+v_{n_{k}}) \bigr]- \phi \bigr\Vert _{L^{1}(\mathbb{R}^{N})} \\ &\quad = C \bigl\Vert D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}} \bigr\Vert _{L^{1}(\mathbb{R}^{N})} +C \Vert \phi \Vert _{L^{1}(\mathbb {R}^{N}\setminus E_{n_{k}})} \\ &\quad \leq C {n_{k}} \biggl(\frac{\lambda_{{n_{k}}-1}}{\lambda_{n_{k}}} \biggr)^{{\frac{2\sigma}{\gamma}}(N(q-p+1)+p)} +C \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N}\setminus E_{n_{k}})} \\ &\quad \leq C{n_{k}}^{-2}+C \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N}\setminus E_{n_{k}})} \to0 \end{aligned} $$
(4.11)
$$\begin{aligned} \begin{aligned}[b] & \biggl\Vert T \biggl( \frac{1}{2} \biggr) \bigl[D^{\sigma}_{\lambda_{n}}(u_{n_{k}}+v_{n_{k}}) \bigr] \biggr\Vert _{L^{\infty}(\mathbb{R}^{N})} \\ &\quad \leq \biggl\Vert S \biggl(\frac{1}{2} \biggr)D^{{\frac{2\sigma}{\gamma }},\beta}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \biggr\Vert _{L^{\infty}} \\ &\quad \leq C \bigl\Vert D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \bigr\Vert _{L^{1}(\mathbb{R}^{N})}^{\frac{2}{N(m-1)+2}}\leq C \bigl( \Vert \phi \Vert _{L^{1}} \bigr). \end{aligned} \end{aligned}$$
(4.12)
$$T \biggl(\frac{1}{2} \biggr)D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \stackrel{w*}{\longrightarrow} T \biggl(\frac{1}{2} \biggr)\phi \quad \text{in } L^{\infty} \bigl(\mathbb{R}^{N} \bigr) \text{ as } k\rightarrow \infty. $$
$$\begin{aligned} T(1)D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \xrightarrow{k\rightarrow\infty} T(1)\phi\quad\text{in } L^{\infty }_{\mathrm{loc}} \bigl(\mathbb{R}^{N} \bigr). \end{aligned}$$
(4.13)
$$D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}}+v_{n_{k}}], \quad \phi\in B^{\sigma,+}_{M}. $$
$$\begin{aligned} T(1) \bigl[D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}+v_{n_{k}} \bigr](x)\leq S(1) \bigl[D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}+v_{n_{k}} \bigr](x)< \frac{\varepsilon}{3} \end{aligned}$$
(4.14)
$$\begin{aligned} T(1)\phi(x)\leq S(1)\phi(x)< \frac{\varepsilon}{3}. \end{aligned}$$
(4.15)
$$T(1) \bigl[D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}+v_{n_{k}} \bigr](x)\xrightarrow{k\to\infty} T(1)\phi(x) $$
$$\begin{aligned} D^{\sigma}_{\lambda_{n_{k}}} \bigl[T \bigl( \lambda_{n_{k}}^{2} \bigr)u_{0} \bigr]\xrightarrow{k \to\infty}T(1) \phi \end{aligned}$$
(4.16)
5 Complexity: \(\frac{p}{q-p+1}<\sigma<N\)
In this section, we investigate the complicated asymptotic behavior of solutions for Problem (1.1)-(1.2) with the initial value \(u_{0}\in W_{\sigma}(\mathbb{R}^{N})\) (\(\frac{p}{q-p+1}<\sigma<N\)).
Theorem 5.1
Suppose
\(q>p+\frac{p}{N}\)
and
\(M>0\). Let
Then there exists a function
\(u_{0}\in B^{\sigma,+}_{M}\)
such that for every
\(\phi\in B^{\sigma,+}_{M}\), there exists a sequence
\(t_{n}\to\infty \)
satisfying
uniformly on
\(\mathbb{R}^{N}\), where
\(u(x,t)\)
is the solution of Problem (1.1)-(1.2) with the initial value
\(u_{0}\).
$$\frac{p}{q-p+1}< \sigma< N. $$
$$\begin{aligned} \lim_{n\to\infty}t^{\frac{\sigma}{\gamma}}_{n}u \bigl(t^{\frac{1}{\gamma}}_{n}x, t_{n} \bigr)=S(1)\phi(x) \end{aligned}$$
(5.1)
Proof
In our previous paper [27], we have obtained the result that there exists a function \(u_{0}\in B^{\sigma,+}_{M}\) such that for every \(\phi\in B^{\sigma,+}_{M}\), there exists a sequence \(t_{n}\to\infty\) satisfying
uniformly on \(\mathbb{R}^{N}\), where \(w(x,t)\) is the solution of Problem (1.3)-(1.4) with the initial value \(w_{0}(x)=u_{0}(x)\). To get Theorem 5.1, we only need to prove that if \(u_{0}(x)=\varphi(x)\in W^{+}_{\sigma}(\mathbb{R}^{N})\), then for every sequence \(t_{n}\rightarrow\infty\), the following limit holds:
uniformly on \(\mathbb{R}^{N}\). The ideas of the following proof come from [1, 2, 7].
$$\begin{aligned} \lim_{n\to\infty}t^{\frac{\sigma}{\gamma}}_{n}w \bigl(t^{\frac{1}{\gamma}}_{n}x, t_{n} \bigr)=S(1)\phi(x) \end{aligned}$$
(5.2)
$$\begin{aligned} \lim_{{t_{n}}\rightarrow\infty} {t_{n}}^{\frac{\sigma}{p+\sigma(p-2)}} \bigl\vert u \bigl({t_{n}}^{\frac{1}{p+\sigma(p-2)}}x,{t_{n}} \bigr) -w \bigl({t_{n}}^{\frac{1}{p+\sigma(p-2)}}x,{t_{n}} \bigr) \bigr\vert =0 \end{aligned}$$
(5.3)
Without loss of generality, assuming that \(\Vert \varphi \Vert _{W_{\sigma}(\mathbb{R}^{N})}\leq M\), we consider the following problem:
Then we define the functions
and
It follows from the comparison principle that
Therefore
It is well known that
where \(f(x)\) is the positive solution of the equation
As in [7], there exists a constant \(C>0\) such that if \(k>0\), \(x\in \mathbb{R}^{N}\), \(t\geq\tau>0\), then
and
From these, we can get that
and
where \(k^{\gamma}\tau\geq1\). Let \(\xi\in C^{\infty}({Q_{T}})\) which vanishes at large x and at \(t=T\), then \(u_{k}\) and \(w_{k}\) satisfy the integral identity
where
and
Note that \(\{w_{k}\}\), \(\{u_{k}\}\) are uniformly bounded on any compact subsets of \(Q_{T}\setminus\{(0, 0)\}\), and that \(\{\nabla w_{k}\}\), \(\{ \nabla u_{k}\}\) are Hölder continuous on any compact subsets of \(Q_{T}\), see [25]. Then there exist subsequences \(\{v_{k_{\ell}}\}\) of \(\{w_{k}\}\) and \(\{u_{k_{\ell}}\}\) of \(\{u_{k}\}\), and two functions \(w'(x,t), u'(x,t)\in C(Q_{T})\cap L^{1}_{\mathrm{loc}}(0,T; W^{1}_{\mathrm{loc}}(\mathbb{R}^{N}))\) such that
in \(C(\mathbb{K})\) as \(k_{\ell}\rightarrow\infty\), where \(\mathbb{K}\) is a compact subset of \({S}_{T}\). So, letting \(k=k_{\ell}\rightarrow+\infty\) in (5.6) and applying (5.4), (5.5), we have
where
Applying the existence and uniqueness theorem [25, 26] to (5.7), we obtain that
hence the entire sequence
uniformly on any compact subset of \(\mathbb{R}^{N}\) as \(k\rightarrow\infty \). Put \(t=1\) and \(k=t_{n}^{\frac{1}{\gamma}}\) in (5.8), then
uniformly on any compact subset of \(\mathbb{R}^{N}\) as \(t_{n}\rightarrow \infty\). Note that \(0<\frac{p}{q-p+1}<\sigma<N\). It now follows from Lemma 3.1 that
for all \(t_{n}>0\) and all \(x\in\mathbb{R}^{N}\). Then, for every \(\epsilon>0\), there exists \(R>0\) such that
Using the comparison principle, we obtain that
Therefore, (5.9) and (5.10) indicate that (5.3) holds. Combining this with (5.2), we can get (5.1), and the proof is complete. □
$$\begin{aligned}& \frac{\partial V}{\partial t}-\operatorname {div}\bigl( \vert \nabla V \vert ^{p-2}\nabla V \bigr)=0\quad\text{in }\mathbb{R}^{N}\times(0,T), \\& V(x,0)=M \vert x \vert ^{-\sigma}\quad\text{in } \mathbb{R}^{N}\setminus\{0\}. \end{aligned}$$
$$w_{k}(x,t)=k^{\sigma}w\bigl(kx,k^{\gamma}t\bigr),\quad\quad u_{k}(x,t)=k^{\sigma}u\bigl(kx,k^{\gamma}t\bigr) $$
$$V_{k}(x,t)=k^{\sigma}V\bigl(kx,k^{\gamma}t\bigr). $$
$$V(x,t)=V_{k}(x,t)\geq w_{k}(x,t)\geq u_{k}(x,t). $$
$$u_{k}(x,t)\leq w_{k}(x,t)\leq CV_{k} \biggl(x,t+ \frac{1}{k^{\gamma}} \biggr),\quad k>0. $$
$$V(x,t)=t^{-\frac{\sigma}{\gamma}}f\biggl(\frac{ \vert x \vert }{t^{\frac {1}{\gamma}}}\biggr), $$
$$f''(\eta)+\biggl(\frac{n-1}{\eta}+ \frac{\eta}{\gamma}\biggr)f'(\eta) +\frac{\sigma}{\gamma}f(\eta)=0. $$
$$V_{k}(x,t)\leq C\tau^{-\frac{\sigma}{\gamma}}, $$
$$\lim_{\eta\rightarrow\infty}\eta^{\frac{\sigma}{\gamma}}f(\eta)=M. $$
$$\begin{aligned} \int _{0}^{\tau} \int_{B_{1}}V(x,t)\,dx\,dt\leq C\tau \end{aligned}$$
(5.4)
$$\begin{aligned} \int _{0}^{\tau} \int_{B_{1}}V^{q}(x,t)\,dx\,dt\leq C\tau+ C \textstyle\begin{cases}\tau^{\frac{N-\sigma q+\gamma}{\gamma}}&\text{if } N-\sigma q+\gamma>0, N\neq \sigma q,\\ \tau\log{\frac{1}{\tau}}&\text{if } N=\sigma q,\\ \log{(1+k^{\gamma}\tau)}&\text{if } N-\sigma q+\gamma=0,\\ k^{-N+\sigma q-\gamma}&\text{if } N-\sigma q+\gamma< 0, \end{cases}\displaystyle \end{aligned}$$
(5.5)
$$\begin{aligned} & \iint _{Q_{T}} \biggl[\xi_{t} (w_{k}-u_{k})- \frac{1}{\gamma}k^{-\alpha}\xi u_{k}^{q} \biggr]\,dx\,dt+ \iint _{Q_{T}}a^{ij}\frac{\partial(w_{k}-u_{k})}{\partial x_{i}} \frac{\partial\xi }{\partial x_{j}}\,dx\,dt =0, \end{aligned}$$
(5.6)
$$\alpha=\sigma(q-p+1)-p>\frac{p}{q-p+1}(q-p+1)-p=0 $$
$$\begin{aligned} a^{i,j}_{k}(x,t)&=\delta_{ij}\cdot \int^{1}_{0} \bigl\vert s\nabla u_{k}+(1-s)\nabla w_{k} \bigr\vert ^{p-2}\,ds \\ &\quad{} +(p-2) \int^{1}_{0} \bigl\vert s\nabla u_{k}+(1-s)\nabla w_{k} \bigr\vert ^{p-4} \bigl(su_{k}+(1-s)w_{k} \bigr)_{x_{i}} \bigl(su_{k}+(1-s)w_{k} \bigr)_{x_{j}}\,ds. \end{aligned}$$
$$\begin{aligned} &w_{k_{\ell}}(x,t)\rightarrow w'(x,t), \qquad u_{k_{\ell}}(x,t)\rightarrow u'(x,t), \\ &\nabla w_{k_{\ell}}(x,t)\rightarrow\nabla w'(x,t),\quad\quad \nabla u_{k_{\ell}}(x,t)\rightarrow\nabla u'(x,t), \end{aligned} $$
$$\begin{aligned} & \iint _{Q_{T}}\xi_{t} \bigl(w'-u' \bigr)\,dx\,dt+ \iint _{Q_{T}}a^{ij}\frac{\partial(w'-u')}{\partial x_{i}} \frac{\partial\xi }{\partial x_{j}}\,dx\,dt =0, \end{aligned}$$
(5.7)
$$\begin{aligned} a^{i,j}(x,t)& =\delta_{ij}\cdot \int^{1}_{0} \bigl\vert s\nabla u'+(1-s)\nabla w' \bigr\vert ^{p-2}\,ds \\ &= (p-2) \int^{1}_{0} \bigl\vert s\nabla u'+(1-s)\nabla w' \bigr\vert ^{p-4} \bigl(su_{k}+(1-s)w' \bigr)_{x_{i}} \bigl(su'+(1-s)w' \bigr)_{x_{j}}\,ds. \end{aligned}$$
$$u'(x,t)-w'(x,t)=0 \quad \text{a.e. on }Q_{T}, $$
$$\begin{aligned} u_{k}(\cdot,t)-w_{k}(\cdot,t)\rightarrow0 \end{aligned}$$
(5.8)
$$\begin{aligned} t_{n}^{\frac{\sigma}{\gamma}} \bigl\vert u \bigl(t_{n}^{\frac {1}{\gamma}} \cdot,t_{n} \bigr)- w \bigl(t_{n}^{\frac{1}{\gamma}} \cdot,t_{n} \bigr) \bigr\vert \rightarrow0 \end{aligned}$$
(5.9)
$$t_{n}^{\frac{\sigma}{\gamma}}V\bigl(t_{n}^{\frac{1}{\gamma}}x,t_{n} \bigr)\leq C\bigl(1+ \vert x \vert ^{2}\bigr)^{-\frac{\sigma}{2}} $$
$$\bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}V \bigl(t_{n}^{\frac{1}{\gamma}} \cdot,t \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}^{N}\setminus B_{R})}< \epsilon. $$
$$\begin{aligned} \begin{aligned}[b] \bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}u \bigl(t_{n}^{\frac{1}{\gamma}}\cdot,t \bigr) \bigr\Vert _{L^{\infty }(\mathbb{R}^{N}\setminus B_{R})} &\leq \bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}w \bigl(t_{n}^{\frac{1}{\gamma}} \cdot,t \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}^{N}\setminus B_{R})} \\ & \leq \bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}V \bigl(t_{n}^{\frac {1}{\gamma}} \cdot,t \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}^{N}\setminus B_{R})}< \epsilon. \end{aligned} \end{aligned}$$
(5.10)
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (11071099 and 11371153), Chongqing Fundamental and Frontier Research Project (cstc2016jcyjA0596) and the Chongqing Municipal Commission of Education (KJ1401003, KJ1601009), Innovation Team Building at Institutions of Higher Education in Chongqing (CXTDX201601035).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics and Statistics, Chongqing Three Gorges University, No. 666, Tian Xing Road, Wanzhou District, Chongqing, 404100, China
(2)
School of Mathematical Sciences, South China Normal University, No. 55, West of Zhong Shan Road, Tianhe District, Guangzhou, 510631, China
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