Complicated asymptotic behavior exponents for solutions of the evolution p-Laplacian equation with absorption

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.

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

$$\begin{aligned} &\frac{\partial u}{\partial t}-\operatorname {div}\bigl( \vert \nabla u \vert ^{p-2}\nabla u \bigr)+ \lambda u^{q}=0 \quad \text{in }(0,\infty)\times \mathbb{R}^{N}, \end{aligned}$$

(1.1)

$$\begin{aligned} &u(x,0)=u_{0}(x)\quad \text{in }\mathbb{R}^{N}, \end{aligned}$$

(1.2)

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 $\Vert \varphi \Vert _{W_{\sigma}(\mathbb{R}^{N})}= \Vert \vert \cdot \vert ^{\sigma}\varphi(\cdot) \Vert _{L^{\infty}(\mathbb{R}^{N})}$.

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

$$t^{\frac{\sigma}{\sigma(p-2)+p}} \bigl\vert u(x,t) -w(x,t) \bigr\vert \rightarrow0 $$

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

$$\begin{aligned} &\frac{\partial w}{\partial t}-\operatorname {div}\bigl( \vert \nabla w \vert ^{p-2}\nabla w \bigr)=0 \quad \text{in }(0,\infty)\times\mathbb{R}^{N}, \end{aligned}$$

(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)

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.

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

$$\lim_{t_{n}\rightarrow\infty}t^{\frac{\sigma}{\sigma(p-2)+p}}_{n} u \bigl(t^{\frac{1}{\sigma(p-2)+p}}_{n}x,t_{n}\bigr)=v(x,1) $$

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

$$\lim_{t_{n}\rightarrow\infty}t^{\frac{\sigma}{\sigma(p-2)+p}}_{n} u \bigl(t^{\frac{1}{\sigma(p-2)+p}}_{n}x,t_{n}\bigr)=w(x,1) $$

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).

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

$$\Vert \!\vert f \vert \!\Vert _{r}=\sup _{R\geq r}R^{-\frac {N(p-2)+p}{p-2}} \int_{\{ \vert x \vert \leq R\}} \bigl\vert f(x) \bigr\vert \,dx. $$

The space $X_{0}$ is given by

$$X_{0}\equiv \Bigl\{ \varphi\in L^{1}_{\mathrm{loc}} \bigl( \mathbb{R}^{N} \bigr); \Vert \!\vert \varphi \vert \!\Vert _{1}< \infty \text{ and } \lim_{r\rightarrow+\infty} \Vert \! \vert \varphi \vert \!\Vert _{r}=0 \Bigr\} $$

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

$$\begin{aligned} u(x,t)\in C^{\frac{\alpha}{2},\alpha} \bigl((0,\infty)\times \mathbb{R}^{N} \bigr) \end{aligned}$$

(2.1)

for some $\alpha>0$. Note that for $0<\sigma<N$,

$$W_{\sigma}\bigl(\mathbb{R}^{N}\bigr)\subset X_{0}. $$

So we can define an operator $T(t): W_{\sigma}(\mathbb{R}^{N})\to C(\mathbb {R}^{N})$ as

$$\begin{aligned} T(t)u_{0}(x)=u(x,t), \end{aligned}$$

(2.2)

where $u(x,t)$ is the solution of Problem (1.1)-(1.2) with the initial value $u_{0}(x)$.

Lemma 2.1

[24, 26]

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

$$S(t): w_{0}\rightarrow w(x,t). $$

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 $L^{q}(\mathbb{R}^{N})$.

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

$$\Omega(t_{2})\subset\Omega_{\rho(t_{2}-t_{1})}(t_{1}), $$

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 $\rho(t_{2}-t_{1})=C(t_{2}-t_{1})^{\frac{1}{\sigma(p-2)+p}} \Vert u_{0} \Vert _{W_{\sigma}(\mathbb{R}^{N})}^{\frac{p-2}{\gamma}}$.

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

$$q>p-1+\frac{p}{N}, \quad\quad 0< \sigma< \frac{p}{q-p+1}, $$

and let

$$\varphi\in W_{\sigma}^{+}\bigl(\mathbb{R}^{N}\bigr). $$

If $u(x,t)$ is the solution of Problem (1.1)-(1.2) with the initial value $u_{0}(x)=\varphi(x)$, then

$$\begin{aligned} \lim_{t\to\infty}\sup_{ \vert x \vert \leq Ct^{\frac{q-p+1}{p(q-1)}}}t^{\frac {1}{q-1}}u(x,t) \leq \biggl( \frac{1}{q-1} \biggr)^{\frac{1}{q-1}}. \end{aligned}$$

(3.1)

Proof

Let

$$u_{k}(x,t)=k^{\frac{p}{q-p+1}}u\bigl(kx,k^{\frac{p(q-1)}{q-p+1}}t\bigr), \quad k>0. $$

So, for every $k >0$, $u_{k}(x,t)$ is a solution of Problem (1.1)-(1.2) with the initial value

$$u_{k}(x,0)=\varphi_{k}(x)=k^{\frac{p}{q-p+1}}\varphi(kx), \quad k>0. $$

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 $(x,t)\in(0,+\infty)\times\mathbb{R}^{N}$,

$$u_{k}(x,t)\leq\overline{u(x,t)}. $$

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

$$u_{k'}(x,t)\stackrel{ k'\rightarrow\infty}{ \longrightarrow} U(x,t)\leq\overline{u(x,t)} $$

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,

$$\lim_{t\to\infty}t^{\frac{1}{q-1}}\sup_{\{ \vert x \vert \leq Ct^{\frac{q-p+1}{p(q-1)}}\}}u(x,t) \leq \biggl( \frac{1}{q-1} \biggr)^{\frac{1}{q-1}}. $$

The proof of this lemma is complete. □

Theorem 3.2

Suppose

$$q>p-1+\frac{p}{N} \quad\textit{and}\quad 0< \sigma< \frac{p}{q-p+1}. $$

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

$$\begin{aligned} & \lim_{t_{n}\to\infty}t_{n}^{\frac{\sigma}{\gamma}}u \bigl(t_{n}^{\frac{1}{\gamma}} x,t_{n} \bigr)=\varphi(x) \end{aligned}$$

(3.2)

uniformly on $\mathbb{R}^{N}$, then

$$\varphi(x)\equiv0. $$

Proof

Suppose that (3.2) holds for some $\varphi(x)\not\equiv0$. And, without loss of generality, we assume that for some $x_{0}\in\mathbb{R}^{N}$,

$$\begin{aligned} \varphi(x_{0})>0. \end{aligned}$$

(3.3)

It follows from (3.2) that there exists an integer $n_{1}$ such that if $n\ge n_{1}$, then

$$\begin{aligned} t_{n}^{{\frac{\sigma}{\gamma}}}u \bigl(t_{n}^{\frac{1}{\gamma }}x_{0},t_{n} \bigr)\ge\frac{1}{2}\varphi(x_{0}). \end{aligned}$$

(3.4)

Note that $u_{0}\in W_{\sigma}^{+}(\mathbb{R}^{N})$. By using Lemma 3.1, we obtain that

$$\begin{aligned} \lim_{t\to\infty}t^{\frac{1}{q-1}}u(x,t) \leq \biggl( \frac{1}{q-1} \biggr)^{\frac{1}{q-1}} \end{aligned}$$

(3.5)

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

$$t_{n}^{{\frac{\sigma}{\gamma}}-\frac{1}{q-1}} {\biggl(\frac{1}{q-1}\biggr)^{\frac{1}{q-1}}}< \frac{1}{2}\varphi(x_{0}). $$

So, from (3.5), we have

$$\begin{aligned} t_{n}^{{\frac{\sigma}{\gamma}}}u(x,t_{n})=t_{n}^{{\frac{\sigma }{\gamma}}-\frac{1}{q-1}} t^{\frac{1}{q-1}}u(x,t_{n})< \frac{1}{2} \varphi(x_{0}) \quad \text{for } x\in \bigl\{ y; \vert y \vert \leq Ct_{n}^{\frac{q-p+1}{p(q-1)}} \bigr\} . \end{aligned}$$

(3.6)

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

$$x\in\bigl\{ y; \vert y \vert \leq Ct_{n}^{\frac{q-p+1}{p(q-1)}}\bigr\} . $$

Thus we deduce from (3.4) and (3.6) that

$$\frac{1}{2} \varphi(x_{0})\leq t_{n}^{{\frac{\sigma}{\gamma}}}u \bigl(t_{n}^{\frac {1}{\gamma}}x_{0},t_{n}\bigr)< \frac{1}{2} \varphi(x_{0}). $$

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

$$\varphi(x)\equiv0\quad\text{for } x\in\mathbb{R}^{N}, $$

and the proof is complete. □

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

$$B^{\sigma,+}_{M}\equiv\bigl\{ \varphi\in W^{+}_{\sigma}\bigl( \mathbb{R}^{N}\bigr)\cap C\bigl(\mathbb {R}^{N}\bigr); \Vert \varphi \Vert _{W^{+}_{\sigma}(\mathbb{R}^{N})}\leq M\bigr\} . $$

For $\lambda>0$, $0<\sigma<N$, $\varphi(x)\in L^{1}(\mathbb{R}^{N})$, we define

$$D^{\sigma}_{\lambda}\varphi(x)=\lambda^{\frac{2\sigma}{\gamma}}\varphi \bigl( \lambda^{\frac{2}{\gamma}}x\bigr). $$

For $\sigma=\frac{p}{q-p+1}$, it follows from this definition and (2.2) that the following commutative relation holds [28]:

$$\begin{aligned} D^{\sigma}_{\lambda} \bigl[T \bigl( \lambda^{2} t \bigr)u_{0} \bigr]=T(t) \bigl[D^{\sigma}_{\lambda}u_{0} \bigr]. \end{aligned}$$

(4.1)

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.

Theorem 4.1

Suppose $q>p+\frac{p}{N}$ and $M>0$. Let

$$\sigma=\frac{p}{q-p+1}. $$

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

$$\begin{aligned} \lim_{n\to\infty}t^{\frac{\sigma}{\gamma}}_{n}u \bigl(t^{\frac{1}{\gamma}}_{n}x, t_{n} \bigr)=T(1)\varphi(x) \end{aligned}$$

(4.2)

uniformly on $\mathbb{R}^{N}$, where $u(x,t)$ is the solution of Problem (1.1)-(1.2) with the initial value $u_{0}(x)$.

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

$$\begin{aligned} \Vert \phi_{\epsilon}-\varphi \Vert _{L^{\infty}(\mathbb{R}^{N})}< \epsilon. \end{aligned}$$

(4.3)

Therefore, there exists a sequence $\{\varphi_{j}\}_{j\geq1}\subset F$ such that

• 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)

Note that $\frac{p}{q-p+1}=\sigma< N$. So the following inequality holds:

$$N\gamma-\sigma{\bigl(N(p-2)+p\bigr)}>0. $$

Let

$$\begin{aligned} \lambda_{j}= \textstyle\begin{cases} 2,& j=1,\\ \max(j^{\frac{3\gamma}{N\gamma-\sigma[N(p-2)+p]}}\lambda_{j-1}, (2^{j}\lambda_{j-1}^{\frac{2}{\gamma}}+C2^{j}M^{\frac{p-2}{\gamma}})^{\frac {\gamma}{2}} ), &j>1. \end{cases}\displaystyle \end{aligned}$$

(4.6)

Now we can follow the methods given in [9, 10, 12] to construct an initial value by

$$\begin{aligned} u_{0}(x)= =\sum_{j=1}^{\infty}D^{\sigma}_{\lambda_{j}^{-1}} \bigl[\chi_{j}(x)\varphi_{j}(x) \bigr]=u_{n}+v_{n}+w_{n}, \end{aligned}$$

(4.7)

where

$$\begin{aligned} \begin{gathered} u_{n} =\sum _{j=1}^{n-1} D^{\sigma}_{\lambda_{j}^{-1}} \bigl[ \chi_{j}(x)\varphi_{j}(x) \bigr],\quad \quad v_{n}=D^{\sigma}_{\lambda_{n}^{-1}} \bigl[ \chi_{n}(x)\varphi_{n}(x) \bigr], \\ w_{n} =\sum_{j=n+1}^{\infty} D^{\sigma}_{\lambda_{j}^{-1}} \bigl[\chi_{j}(x)\varphi_{j}(x) \bigr], \end{gathered} \end{aligned}$$

(4.8)

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

$$\Vert \varphi \Vert _{W_{\sigma}(\mathbb{R}^{N})}\leq M $$

and

$$0\leq\varphi\in C\bigl(\mathbb{R}^{N}\bigr). $$

By (4.6) and (4.7), we have

$$\Vert u_{0} \Vert _{L^{\infty}(\mathbb{R}^{N})}\leq \Vert u_{0} \Vert _{W_{\sigma}(\mathbb{R}^{N})}\leq\sup_{j\geq1} \bigl\Vert { \lambda_{j}}^{-{\frac{2\sigma}{\gamma}}} \chi_{j} \bigl(x/ \lambda_{j}^{\frac{2}{\gamma}} \bigr) \varphi_{j} \bigl(x/ \lambda_{j}^{\frac{2}{\gamma}} \bigr) \bigr\Vert _{{W_{\sigma}(\mathbb {R}^{N})}}\leq M, $$

so

$$u_{0}\in B^{\sigma,+}_{M}. $$

It follows from (4.1) that

$$D^{\sigma}_{\lambda_{n}} \bigl[T \bigl( \lambda_{n}^{2}t \bigr)u_{0} \bigr]= T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{0} \bigr] = T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+ D^{\sigma}_{\lambda_{n}}v_{n}+D^{\sigma}_{\lambda_{n}}w_{n} \bigr]. $$

We thus conclude from the definition of $\lambda_{j}$, comparison principle [29] and Lemma 2.2 that

$$\operatorname {supp}\bigl(T(1) \bigl[D^{\sigma}_{n}(w_{n}) \bigr] \bigr)\subset \bigl\{ x\in\mathbb{R}^{N}; \vert x \vert >2^{n}+CM^{\frac{p-2}{\gamma}} \bigr\} $$

and

$$\operatorname {supp}\bigl({T(1)} \bigl[D^{\sigma}_{n}(v_{n}+u_{n}) \bigr] \bigr)\subset \bigl\{ x\in\mathbb{R}^{N}; \vert x \vert < 2^{n}+CM^{\frac{p-2}{\gamma}} \bigr\} , $$

so

$$\operatorname {supp}\bigl(T(1)\bigl[D^{\sigma}_{n}(w_{n})\bigr] \bigr)\cap \operatorname {supp}\bigl({T(1)}\bigl[D^{\sigma}_{n}(v_{n}+u_{n}) \bigr]\bigr)=\emptyset, $$

hence

$$T(1) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+ D^{\sigma}_{\lambda_{n}}v_{n}+D^{\sigma}_{\lambda_{n}}w_{n} \bigr] = T(1) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+D^{\sigma}_{\lambda_{n}}v_{n} \bigr] +T(1) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr]. $$

The same result holds for $0< t<1$,

$$T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+ D^{\sigma}_{\lambda_{n}}v_{n}+D^{\sigma }_{\lambda_{n}}w_{n} \bigr] = T(t) \bigl[D^{\sigma}_{\lambda_{n}}u_{n}+D^{\sigma}_{\lambda_{n}}v_{n} \bigr] +T(t) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr]. $$

From the comparison principle [25, 29], Lemma 2.1, (4.5) and (4.6), we have

$$ \begin{aligned}[b] \bigl\Vert T(1) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr] \bigr\Vert _{L^{\infty}(\mathbb{R}^{N})} &\leq \bigl\Vert S(1) \bigl[D^{\sigma}_{\lambda_{n}}w_{n} \bigr] \bigr\Vert _{L^{\infty}(\mathbb{R}^{N})} \leq C \bigl\Vert D^{\sigma}_{\lambda_{n}}w_{n} \bigr\Vert _{L^{\infty}(\mathbb{R}^{N})} \\ & \leq C\lambda_{n}^{{\frac{2\sigma}{\gamma}}}\lambda_{n+1}^{-{\frac {2\sigma}{\gamma}}} \sum_{i=n+1}^{\infty}2^{-i}i \leq C2^{-n}\to0 \end{aligned} $$

(4.9)

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

$$x\in E_{k}\equiv\bigl\{ y\in\mathbb{R}^{N}; 2^{-n_{k}+1}< \vert y \vert < 2^{n_{k}-1}\bigr\} , $$

then

$$\begin{aligned} D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}(x)=D^{\sigma}_{\lambda_{n_{k}}} \bigl[D^{\sigma}_{\lambda_{n_{k}}^{-1}}\chi_{n_{k}}\varphi_{n_{k}} \bigr](x)=\chi_{n_{k}}(x)\varphi_{n_{k}}(x) =\phi(x). \end{aligned}$$

(4.10)

By the $L^{1}$-contraction principle [25, 26], we conclude from (4.6) and (4.10) that

$$ \begin{aligned}[b] & \biggl\Vert T(1/2) \bigl[D^{\sigma}_{\lambda _{n}}(u_{n_{k}}+v_{n_{k}}) \bigr]- T \biggl(\frac{1}{2} \biggr)\phi \biggr\Vert _{L^{1}(\mathbb{R}^{N})} \\ &\quad \leq \bigl\Vert \bigl[D^{\sigma}_{\lambda_{n_{k}}}(u_{n_{k}}+v_{n_{k}}) \bigr]- \phi \bigr\Vert _{L^{1}(\mathbb{R}^{N})} \\ &\quad = C \bigl\Vert D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}} \bigr\Vert _{L^{1}(\mathbb{R}^{N})} +C \Vert \phi \Vert _{L^{1}(\mathbb {R}^{N}\setminus E_{n_{k}})} \\ &\quad \leq C {n_{k}} \biggl(\frac{\lambda_{{n_{k}}-1}}{\lambda_{n_{k}}} \biggr)^{{\frac{2\sigma}{\gamma}}(N(q-p+1)+p)} +C \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N}\setminus E_{n_{k}})} \\ &\quad \leq C{n_{k}}^{-2}+C \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N}\setminus E_{n_{k}})} \to0 \end{aligned} $$

(4.11)

as $k\to\infty$. Note that

$$\begin{aligned} \begin{aligned}[b] & \biggl\Vert T \biggl( \frac{1}{2} \biggr) \bigl[D^{\sigma}_{\lambda_{n}}(u_{n_{k}}+v_{n_{k}}) \bigr] \biggr\Vert _{L^{\infty}(\mathbb{R}^{N})} \\ &\quad \leq \biggl\Vert S \biggl(\frac{1}{2} \biggr)D^{{\frac{2\sigma}{\gamma }},\beta}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \biggr\Vert _{L^{\infty}} \\ &\quad \leq C \bigl\Vert D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \bigr\Vert _{L^{1}(\mathbb{R}^{N})}^{\frac{2}{N(m-1)+2}}\leq C \bigl( \Vert \phi \Vert _{L^{1}} \bigr). \end{aligned} \end{aligned}$$

(4.12)

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

$$T \biggl(\frac{1}{2} \biggr)D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \stackrel{w*}{\longrightarrow} T \biggl(\frac{1}{2} \biggr)\phi \quad \text{in } L^{\infty} \bigl(\mathbb{R}^{N} \bigr) \text{ as } k\rightarrow \infty. $$

Therefore, the regularity of the solutions (see (2.1)) indicates that

$$\begin{aligned} T(1)D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}-1}+v_{n_{k}}] \xrightarrow{k\rightarrow\infty} T(1)\phi\quad\text{in } L^{\infty }_{\mathrm{loc}} \bigl(\mathbb{R}^{N} \bigr). \end{aligned}$$

(4.13)

Note that

$$D^{\sigma}_{\lambda_{n_{k}}}[u_{n_{k}}+v_{n_{k}}], \quad \phi\in B^{\sigma,+}_{M}. $$

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

$$\begin{aligned} T(1) \bigl[D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}+v_{n_{k}} \bigr](x)\leq S(1) \bigl[D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}+v_{n_{k}} \bigr](x)< \frac{\varepsilon}{3} \end{aligned}$$

(4.14)

and

$$\begin{aligned} T(1)\phi(x)\leq S(1)\phi(x)< \frac{\varepsilon}{3}. \end{aligned}$$

(4.15)

Therefore, from (4.13), (4.14) and (4.15), we get

$$T(1) \bigl[D^{\sigma}_{\lambda_{n_{k}}}u_{n_{k}}+v_{n_{k}} \bigr](x)\xrightarrow{k\to\infty} T(1)\phi(x) $$

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

$$\begin{aligned} D^{\sigma}_{\lambda_{n_{k}}} \bigl[T \bigl( \lambda_{n_{k}}^{2} \bigr)u_{0} \bigr]\xrightarrow{k \to\infty}T(1) \phi \end{aligned}$$

(4.16)

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. □

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

$$\frac{p}{q-p+1}< \sigma< N. $$

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

$$\begin{aligned} \lim_{n\to\infty}t^{\frac{\sigma}{\gamma}}_{n}u \bigl(t^{\frac{1}{\gamma}}_{n}x, t_{n} \bigr)=S(1)\phi(x) \end{aligned}$$

(5.1)

uniformly on $\mathbb{R}^{N}$, where $u(x,t)$ is the solution of Problem (1.1)-(1.2) with the initial value $u_{0}$.

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

$$\begin{aligned} \lim_{n\to\infty}t^{\frac{\sigma}{\gamma}}_{n}w \bigl(t^{\frac{1}{\gamma}}_{n}x, t_{n} \bigr)=S(1)\phi(x) \end{aligned}$$

(5.2)

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:

$$\begin{aligned} \lim_{{t_{n}}\rightarrow\infty} {t_{n}}^{\frac{\sigma}{p+\sigma(p-2)}} \bigl\vert u \bigl({t_{n}}^{\frac{1}{p+\sigma(p-2)}}x,{t_{n}} \bigr) -w \bigl({t_{n}}^{\frac{1}{p+\sigma(p-2)}}x,{t_{n}} \bigr) \bigr\vert =0 \end{aligned}$$

(5.3)

uniformly on $\mathbb{R}^{N}$. The ideas of the following proof come from [1, 2, 7].

Without loss of generality, assuming that $\Vert \varphi \Vert _{W_{\sigma}(\mathbb{R}^{N})}\leq M$, we consider the following problem:

$$\begin{aligned}& \frac{\partial V}{\partial t}-\operatorname {div}\bigl( \vert \nabla V \vert ^{p-2}\nabla V \bigr)=0\quad\text{in }\mathbb{R}^{N}\times(0,T), \\& V(x,0)=M \vert x \vert ^{-\sigma}\quad\text{in } \mathbb{R}^{N}\setminus\{0\}. \end{aligned}$$

Then we define the functions

$$w_{k}(x,t)=k^{\sigma}w\bigl(kx,k^{\gamma}t\bigr),\quad\quad u_{k}(x,t)=k^{\sigma}u\bigl(kx,k^{\gamma}t\bigr) $$

and

$$V_{k}(x,t)=k^{\sigma}V\bigl(kx,k^{\gamma}t\bigr). $$

It follows from the comparison principle that

$$V(x,t)=V_{k}(x,t)\geq w_{k}(x,t)\geq u_{k}(x,t). $$

Therefore

$$u_{k}(x,t)\leq w_{k}(x,t)\leq CV_{k} \biggl(x,t+ \frac{1}{k^{\gamma}} \biggr),\quad k>0. $$

It is well known that

$$V(x,t)=t^{-\frac{\sigma}{\gamma}}f\biggl(\frac{ \vert x \vert }{t^{\frac {1}{\gamma}}}\biggr), $$

where $f(x)$ is the positive solution of the equation

$$f''(\eta)+\biggl(\frac{n-1}{\eta}+ \frac{\eta}{\gamma}\biggr)f'(\eta) +\frac{\sigma}{\gamma}f(\eta)=0. $$

As in [7], there exists a constant $C>0$ such that if $k>0$, $x\in \mathbb{R}^{N}$, $t\geq\tau>0$, then

$$V_{k}(x,t)\leq C\tau^{-\frac{\sigma}{\gamma}}, $$

and

$$\lim_{\eta\rightarrow\infty}\eta^{\frac{\sigma}{\gamma}}f(\eta)=M. $$

From these, we can get that

$$\begin{aligned} \int _{0}^{\tau} \int_{B_{1}}V(x,t)\,dx\,dt\leq C\tau \end{aligned}$$

(5.4)

and

$$\begin{aligned} \int _{0}^{\tau} \int_{B_{1}}V^{q}(x,t)\,dx\,dt\leq C\tau+ C \textstyle\begin{cases}\tau^{\frac{N-\sigma q+\gamma}{\gamma}}&\text{if } N-\sigma q+\gamma>0, N\neq \sigma q,\\ \tau\log{\frac{1}{\tau}}&\text{if } N=\sigma q,\\ \log{(1+k^{\gamma}\tau)}&\text{if } N-\sigma q+\gamma=0,\\ k^{-N+\sigma q-\gamma}&\text{if } N-\sigma q+\gamma< 0, \end{cases}\displaystyle \end{aligned}$$

(5.5)

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

$$\begin{aligned} & \iint _{Q_{T}} \biggl[\xi_{t} (w_{k}-u_{k})- \frac{1}{\gamma}k^{-\alpha}\xi u_{k}^{q} \biggr]\,dx\,dt+ \iint _{Q_{T}}a^{ij}\frac{\partial(w_{k}-u_{k})}{\partial x_{i}} \frac{\partial\xi }{\partial x_{j}}\,dx\,dt =0, \end{aligned}$$

(5.6)

where

$$\alpha=\sigma(q-p+1)-p>\frac{p}{q-p+1}(q-p+1)-p=0 $$

and

$$\begin{aligned} a^{i,j}_{k}(x,t)&=\delta_{ij}\cdot \int^{1}_{0} \bigl\vert s\nabla u_{k}+(1-s)\nabla w_{k} \bigr\vert ^{p-2}\,ds \\ &\quad{} +(p-2) \int^{1}_{0} \bigl\vert s\nabla u_{k}+(1-s)\nabla w_{k} \bigr\vert ^{p-4} \bigl(su_{k}+(1-s)w_{k} \bigr)_{x_{i}} \bigl(su_{k}+(1-s)w_{k} \bigr)_{x_{j}}\,ds. \end{aligned}$$

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

$$\begin{aligned} &w_{k_{\ell}}(x,t)\rightarrow w'(x,t), \qquad u_{k_{\ell}}(x,t)\rightarrow u'(x,t), \\ &\nabla w_{k_{\ell}}(x,t)\rightarrow\nabla w'(x,t),\quad\quad \nabla u_{k_{\ell}}(x,t)\rightarrow\nabla u'(x,t), \end{aligned} $$

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

$$\begin{aligned} & \iint _{Q_{T}}\xi_{t} \bigl(w'-u' \bigr)\,dx\,dt+ \iint _{Q_{T}}a^{ij}\frac{\partial(w'-u')}{\partial x_{i}} \frac{\partial\xi }{\partial x_{j}}\,dx\,dt =0, \end{aligned}$$

(5.7)

where

$$\begin{aligned} a^{i,j}(x,t)& =\delta_{ij}\cdot \int^{1}_{0} \bigl\vert s\nabla u'+(1-s)\nabla w' \bigr\vert ^{p-2}\,ds \\ &= (p-2) \int^{1}_{0} \bigl\vert s\nabla u'+(1-s)\nabla w' \bigr\vert ^{p-4} \bigl(su_{k}+(1-s)w' \bigr)_{x_{i}} \bigl(su'+(1-s)w' \bigr)_{x_{j}}\,ds. \end{aligned}$$

Applying the existence and uniqueness theorem [25, 26] to (5.7), we obtain that

$$u'(x,t)-w'(x,t)=0 \quad \text{a.e. on }Q_{T}, $$

hence the entire sequence

$$\begin{aligned} u_{k}(\cdot,t)-w_{k}(\cdot,t)\rightarrow0 \end{aligned}$$

(5.8)

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

$$\begin{aligned} t_{n}^{\frac{\sigma}{\gamma}} \bigl\vert u \bigl(t_{n}^{\frac {1}{\gamma}} \cdot,t_{n} \bigr)- w \bigl(t_{n}^{\frac{1}{\gamma}} \cdot,t_{n} \bigr) \bigr\vert \rightarrow0 \end{aligned}$$

(5.9)

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

$$t_{n}^{\frac{\sigma}{\gamma}}V\bigl(t_{n}^{\frac{1}{\gamma}}x,t_{n} \bigr)\leq C\bigl(1+ \vert x \vert ^{2}\bigr)^{-\frac{\sigma}{2}} $$

for all $t_{n}>0$ and all $x\in\mathbb{R}^{N}$. Then, for every $\epsilon>0$, there exists $R>0$ such that

$$\bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}V \bigl(t_{n}^{\frac{1}{\gamma}} \cdot,t \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}^{N}\setminus B_{R})}< \epsilon. $$

Using the comparison principle, we obtain that

$$\begin{aligned} \begin{aligned}[b] \bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}u \bigl(t_{n}^{\frac{1}{\gamma}}\cdot,t \bigr) \bigr\Vert _{L^{\infty }(\mathbb{R}^{N}\setminus B_{R})} &\leq \bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}w \bigl(t_{n}^{\frac{1}{\gamma}} \cdot,t \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}^{N}\setminus B_{R})} \\ & \leq \bigl\Vert t_{n}^{\frac{\sigma}{\gamma}}V \bigl(t_{n}^{\frac {1}{\gamma}} \cdot,t \bigr) \bigr\Vert _{L^{\infty}(\mathbb{R}^{N}\setminus B_{R})}< \epsilon. \end{aligned} \end{aligned}$$

(5.10)

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. □