Relation Between Solutions and Initial Values for Double-Nonlinear Diffusion Equation

Abstract

In this paper, we consider the Cauchy problem of the double-nonlinear diffusion equation. We establish the propagation speed estimates and space-time decay estimates for the solutions and study the equivalent relation between the solutions and the initial values. As an application of this relationship, we prove two different asymptotic behaviors for the solutions in the last of this paper.

1 Introduction

In this paper, we consider the Cauchy problem of the double-nonlinear diffusion equation

$$\begin{aligned}&u_{t}=\mathrm {div}(|\nabla u^{m}|^{p-2}\nabla u^{m})&\mathrm{in}\quad \mathbb {R}^{N}\times (0,\infty ), \end{aligned}$$

(1.1)

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

(1.2)

where $p>1,m>0,m(p-1)-1>0$ and

$$\begin{aligned} u_{0}\in W_{\sigma }\equiv \{\varphi \in L^{1}_{loc}(\mathbb {R}^{N});|x|^{\sigma }\varphi (x)\in L^{\infty }(\mathbb {R}^{N})\}. \end{aligned}$$

The doubly nonlinear diffusion equation (DNLE) is derived from many diffusion phenomena, such as soil physics, reaction chemistry, combustion theory, fluid dynamics, one can see [1, 2, 5, 14] and the references. Both the evolution p-Laplacian equation (PLE) and the porous media equation (PME) are special cases of it. For the evolution p-Laplacian equation, Vázquez and Zuazua [17] proved the complicated asymptotic behavior of solutions by the relation between the $\omega$-limit set of solution and the $\omega$-limit set of initial values. For the heat equation, Cazenave, Dickstein and Weissler [3, 4, 12] used the relation between the rescaled solutions $t^{\frac{\mu }{2}}u(t^{\beta }x,t)$ and the initial values found the complicated asymptotic behavior for the solutions. We also get the complicated asymptotic behavior of solutions for the medium porous equation in [18,19,20, 22].

Inspired by the above papers, our interest here is to study the asymptotic behavior of solutions for the problem (1.1)–(1.2) by using the relation between the solutions and the initial values. It is worth noting that the double-nonlinear diffusion equation has the nonlinearity and degeneracy that the heat equation [3, 4] does not have. To overcome these difficulties, we establish the space-time decay estimate and propagation speed estimate to obtain the equivalent relation that if $0\le u_{0}\in W_{\sigma }(\mathbb {R}^{N})$ with $0<\sigma <N$, then

$$\begin{aligned} \omega ^{\sigma }(u_{0})=S(1)\varOmega ^{\sigma }(u_{0}), \end{aligned}$$

(1.3)

where

$$\begin{aligned}&\omega ^{\sigma }(u_{0})\equiv \{f\in C_{0}(\mathbb {R}^{N}); \exists t_{n}\rightarrow \infty \ s.t.\\&t_{n}^{\frac{\sigma }{\sigma [m(p-1)-1]+p}}u(t_{n}^{\frac{1}{\sigma [m(p-1)-1]+p}}x,t_{n})\rightarrow f(x) \ \mathrm{in} \ C_{0}(\mathbb {R}^{N})\} \end{aligned}$$

and

$$\begin{aligned}&\varOmega ^{\sigma }(u_{0})\equiv \{\varphi \in W_{\sigma }(\mathbb {R}^{N});\exists t_{n}\rightarrow \infty \ s.t. \,\\&\lambda _{n}^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}u_{0}(\lambda _{n}^{\frac{2}{\sigma [m(p-1)-1]+p}}x){\mathop {\rightharpoonup }\limits ^{*}}\varphi (x) \ \mathrm{in} \ W_{\sigma }(\mathbb {R}^{N})\}. \end{aligned}$$

We use the relation (1.3) to prove differential asymptotic behavior of solutions for the problem (1.1)–(1.2). We first use the relations (1.3) to show that the complicated asymptotic behavior of solutions for the problem (1.1)–(1.2) can happen, accord to Vázquez and Zuazua[17].

If A is a positive constant and the initial value $u_0(x)\in W_\sigma (\mathbb {R}^N)$ such that

$$\begin{aligned} \lim _{|x|\rightarrow \infty }|x|^{\sigma }u_0(x)=A, \end{aligned}$$

we use the relations (1.3) to get that the solutions u(x, t) of problem (1.1)–(1.2) satisfy

$$\begin{aligned} \lim _{|x|\rightarrow \infty }t^{\frac{\sigma }{\sigma [m(p-1)-1]+p}}|u(t^{\frac{1}{\sigma [m(p-1)-1]+p}}x,t)-W(t^{\frac{1}{\sigma [m(p-1)-1]+p}}x,t)|=0 \ \mathrm{in} \ L^\infty (\mathbb {R}^{N}), \end{aligned}$$

where W(x, t) is the solution of Cauchy problem of equation (1.1) with the initial value $W(x,0)=A|x|^{-\sigma }.$

Remark 1

For $m=1$, some similar results as above have been got in [23], and for $p=2$, in [11, 13], for $m=1$ and $p=2$, in [8,9,10].

The rest of this paper is organized as follows. In Sect. 2, we establish the space-time decay estimate and propagation speed estimate. In Sect. 3, we study the relationship between solutions and initial values. Section 4 is devote to using this relationship to study the different asymptotic behaviors of the solutions.

2 Some Estimates

In this paper, in order to get the relation (1.3), we give some definitions and prove some estimates in this section.

Definition 1

([16, 24]) For $f\in L^{1}_{loc}(\mathbb {R}^{N})$, let

$$\begin{aligned} |||f|||_{r}=\sup _{R\ge r}R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{x\le R}|f(x)|dx \ and \ \ell (f)=\lim _{r\rightarrow \infty }|||f|||_{r}. \end{aligned}$$

The space $X_{0}$ is given by

$$\begin{aligned} X_{0}\equiv \{{\varphi \in L^{1}_{loc}(\mathbb {R}^{N}); |||\varphi |||<\infty \ and \ \ell (f)=0}\}. \end{aligned}$$

Obviously if $0< \sigma < N$, then $W_{\sigma }(\mathbb {R}^{N})\subset X_{0}$.

Proposition 1

([15, 21]) For $u_{0}\in X_{0}$, there exists a unique global weak solution u(x, t) of Problem (1.1)–(1.2). Moreover, the doubly nonlinear diffusion equation generates a bounded semigroup in $X_{0}$ given by

$$\begin{aligned} S(t):u_{0}\rightarrow u(x,t). \end{aligned}$$

(2.1)

Then, S(t) is a contraction bounded semigroup in $L^{q}(\mathbb {R}^{N})$ for all $u_{0}\in L^{q}(\mathbb {R}^{N})$ with $q\ge 1$.

The space-time dilation $\Gamma ^{\sigma }_{\lambda }$ is defined by

$$\begin{aligned} \Gamma ^{\sigma }_{\lambda }[S(t)u_{0}](x)=D^{\sigma }_{\lambda }[S(\lambda ^{2}t)u_{0}](x)=\lambda ^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}u(\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}x,\lambda ^{2}t), \end{aligned}$$

where $\lambda ,\sigma >0$, $D^{\sigma }_{\lambda }\varphi (x)=\lambda ^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}\varphi (\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}x)$ and $u_{0}\in X_{0}$. We can get the following commutative relations

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

(2.2)

In fact, let

$$\begin{aligned} v(x,t)=\lambda ^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}u(\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}x,\lambda ^{2}t). \end{aligned}$$

(2.3)

So

$$\begin{aligned} \frac{\partial v}{\partial t}=\lambda ^{2-\frac{2p}{\sigma [m(p-1)-1]+p}-\frac{2\sigma }{\sigma [m(p-1)-1]+p}(mp-m-1)}div(|\nabla v^{m}|^{p-2}\nabla v^{m}). \end{aligned}$$

Assume that

$$\begin{aligned} w(x,t)=v(x,\lambda ^{-2+\frac{2p}{\sigma [m(p-1)-1]+p}+\frac{2\sigma }{\sigma [m(p-1)-1]+p}(mp-m-1)}t), \end{aligned}$$

(2.4)

then w(x, t) is weak solution of the following Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} w_{t}=\mathrm {div}(|\nabla w^{m}|^{p-2}\nabla w^{m}) &{}\mathrm{in} \ R^{N}\times (0,\infty ),\\ w(x,0)=\lambda ^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}u_{0}(\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}x)=D^{\sigma }_{\lambda }u_{0} &{}\mathrm{in} \ R^{N}. \end{array}\right. } \end{aligned}$$

Hence,

$$\begin{aligned} w(x,t)=S(t)(D^{\sigma }_{\lambda }u_{0})(x), \end{aligned}$$

in other words, via (2.3) and (2.4), we have

$$\begin{aligned} \Gamma ^{\sigma }_{\lambda }u_{0}&=v(x,t)=w(x,\lambda ^{2-\frac{2p}{\sigma [m(p-1)-1]+p}-\frac{2\sigma }{\sigma [m(p-1)-1]+p}(mp-m-1)}t)\\&=S(\lambda ^{2-\frac{2p}{\sigma [m(p-1)-1]+p}-\frac{2\sigma }{\sigma [m(p-1)-1]+p}(mp-m-1)}t)(D^{\sigma }_{\lambda }u_{0})(x). \end{aligned}$$

Thus, we get the following commutative relations between the semigroup operators S(t) and the dilation operators $D^{\sigma }_{\lambda }$

$$\begin{aligned} \Gamma ^{\sigma }_{\lambda }u_{0}=D^{\sigma }_{\lambda }[S(\lambda ^{2}t)u_{0}]=S(\lambda ^{2-\frac{2p}{\sigma [m(p-1)-1]+p}-\frac{2\sigma }{\sigma [m(p-1)-1]+p}(mp-m-1)}t)(D^{\sigma }_{\lambda }u_{0}). \end{aligned}$$

Lemma 1

([21]) Let $x\in \mathbb {R}^{N}$ and u(x, t) is the nonnegative solution of (1.1)–(1.2), then $u(x,t)>0$ for every $t>0$ if and only if

$$\begin{aligned} B(x)=\sup _{R>0}R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{B_{R}(x)}u_{0}(y)dy=\infty , \end{aligned}$$

where

$$\begin{aligned} B_{R}(x)=\{y;|x-y|<R\}. \end{aligned}$$

Moreover, let $B(x)<\infty$, we have $u(x,t)=0$ for

$$\begin{aligned} 0<t\le C(p,N)B(x)^{-[m(p-1)-1]}. \end{aligned}$$

(2.5)

Lemma 2

(Propagation Speed Estimate) Let u(x, t) be the weak solution of Problem (1.1)–(1.2) with the initial value $0\le u_{0}\in W_{\sigma }(\mathbb {R}^{N})$ and $0<t_{1}<t_{2}<\infty$, we have

$$\begin{aligned} \varOmega (t_{2})\subset \varOmega _{\rho (t_{2}-t_{1})}(t_{1}), \end{aligned}$$

where

$$\begin{aligned} \varOmega (t)\equiv \{x\in \mathbb {R}^{N};u(x,t)>0\}, \end{aligned}$$

$$\begin{aligned} \varOmega _{\rho (t_{2}-t_{1})}(t_{1})\equiv \{x\in \mathbb {R}^{N};d(x,\varOmega (t_1))<\rho (t_{2}-t_{1})\} \end{aligned}$$

and

$$\begin{aligned} \rho (t_{2}-t_{1})=C(t_{2}-t_{1})^{\frac{1}{\sigma [m(p-1)-1]+p}}\Vert u_{0}\Vert ^{\frac{m(p-1)-1}{\sigma [m(p-1)-1]+p}}_{W_{\sigma }(\mathbb {R}^{N})}. \end{aligned}$$

Proof

We can consider the case of $t=0$. Let $x_{0}\in \mathbb {R}^{N}$ and $B_R(x_0)>0$, for $R<d(x_0)\equiv d(x_{0},\varOmega (0)),$ we have

$$\begin{aligned} \int _{B_{R}(x_{0})}u_{0}(y)dy=0. \end{aligned}$$

(2.6)

For $R\ge d(x_{0}),$ if $|x_0|<2R,$ we have

$$\begin{aligned} B_{R}(x_{0})\subset B_{3R}(0), \end{aligned}$$

so

$$\begin{aligned}&R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{B_{R}(x_{0})}u_{0}(y)dy\le \nonumber C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{B_{3R}(0)}|y|^{-\sigma }dy\\&\quad =C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}R^{-\frac{p}{m(p-1)-1}-\sigma } \le C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}d(x_{0})^{-\frac{p}{m(p-1)-1}-\sigma }. \end{aligned}$$

(2.7)

If $|x_{0}|\ge 2R$ and $y\in B_{R}(x_{0})$, then

$$\begin{aligned} |y|\ge |x_{0}|-R\ge R. \end{aligned}$$

Since $0<\sigma <N,$ we have

$$\begin{aligned} |y|^{-\sigma }\le R^{-\sigma }. \end{aligned}$$

Therefore,

$$\begin{aligned}&R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{B_{R}(x_{0})}u_{0}(y)dy\le \nonumber C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}-\sigma }\int _{B_{R}(0)}dy\\&\quad =C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}R^{-\frac{p}{m(p-1)-1}-\sigma } \le C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}d(x_{0})^{-\frac{p}{m(p-1)-1}-\sigma }. \end{aligned}$$

Combining this with (2.6) and (2.7), we have

$$\begin{aligned} B(x_{0})=\sup _{R>0}R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{B_{R}(x_{0})}u_{0}(y)dy\le C\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}d(x_{0})^{-\frac{\sigma [m(p-1)-1]+p}{m(p-1)-1}}. \end{aligned}$$

Therefore, Lemma 1 implies

$$\begin{aligned} u(x_{0},t)=0 \ \ for \ all \ 0\le t\le C\Vert u_{0}\Vert ^{-m(p-1)-1}_{W_{\sigma }(\mathbb {R}^{N})}d(x_{0})^{\sigma [m(p-1)-1]+p}. \end{aligned}$$

Hence,

$$\begin{aligned} \varOmega (t)\subset \varOmega _{\rho (t)}(0), \end{aligned}$$

where

$$\begin{aligned} \rho (t)=C\Vert u_{0}\Vert ^{\frac{m(p-1)-1}{\sigma [m(p-1)-1]+p}}_{W_{\sigma }(\mathbb {R}^{N})}t^{\frac{1}{\sigma [m(p-1)-1]+p}}. \end{aligned}$$

So we complete the proof of this lemma. $\square$

Lemma 3

Let $w\in C(\mathbb {R}^{N}\backslash \{0\})$ be a homogeneous function of degree 0. If $0<\sigma <N$ and $u_{0}(x)=|x|^{-\sigma }w(x)$, then there exists a function $g(x)\in C^{\alpha }(\mathbb {R}^{N})$ such that

$$\begin{aligned} S(t)u_{0}(x)=t^{-\frac{\sigma }{\sigma [m(p-1)-1]+p}}g(t^{-\frac{1}{\sigma [m(p-1)-1]+p}}x), \end{aligned}$$

(2.8)

and $|x|^{\sigma }g(x)-w(x)\rightarrow 0$ as $|x|\rightarrow \infty$.

Proof

From the definition of the initial value $u_{0}$ and (2.2), we have

$$\begin{aligned}&\Gamma ^{\sigma }_{\lambda }[S(s)u_{0}(x)]=\lambda ^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}[S(\lambda ^{2}s)u_{0}(x)](\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}x) \nonumber \\&\quad =S(s)[\lambda ^{\frac{2\sigma }{\sigma [m(p-1)-1]+p}}u_{0}(\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}\cdot )](x)=S(s)u_{0}(x). \end{aligned}$$

(2.9)

Since $0<\sigma <N$, then

$$\begin{aligned} \Vert u_{0}\Vert _{r}=\sup _{R\ge r}R^{-\frac{N[m(p-1)-1]+p}{m(p-1)-1}}\int _{B_{R}}|u_{0}(x)|dx\le Cr^{-\sigma -\frac{p}{m(p-1)-1}}\rightarrow 0 \ as\ r\rightarrow \infty . \end{aligned}$$

So $u_{0}\in X_{0}$. From this we can get

$$\begin{aligned} S(s)u_{0}\in C^{\frac{\alpha }{2},\alpha }((0,\infty )\times \mathbb {R}^{N}) \end{aligned}$$

for some $0<\alpha <1$ [7, 15]. In particular,

$$\begin{aligned} S(1)u_{0}\in C^{\alpha }(\mathbb {R}^{N}). \end{aligned}$$

Taking $g(x)=S(1)u_{0}\in C^{\alpha }(\mathbb {R}^{N})$ and $s=1,\lambda =t^{\frac{1}{2}}$ in the expression (2.9), we obtain

$$\begin{aligned} S(t)u_{0}(x)=t^{-\frac{\sigma }{\sigma [m(p-1)-1]+p}}g(t^{-\frac{1}{\sigma [m(p-1)-1]+p}}x). \end{aligned}$$

The fact $S(t)u_{0}(x)\in C([0,\infty )\times \mathbb {R}^N\setminus \{0,0\})$ clearly implies that for $|x|=1$,

$$\begin{aligned} t^{-\frac{\sigma }{\sigma [m(p-1)-1]+p}}g(t^{-\frac{1}{\sigma [m(p-1)-1]+p}}x)=S(t)u_{0}(x)\rightarrow \varphi (x)=|x|^{-\sigma }w(x)=w(x) \end{aligned}$$

as $t\rightarrow 0$. Let

$$\begin{aligned} y=t^{-\frac{1}{\sigma [m(p-1)-1]+p}}x. \end{aligned}$$

So $|y|\rightarrow \infty$ as $t\rightarrow 0$. Therefore,

$$\begin{aligned} |y|^{\sigma }g(y)-w(y)\rightarrow 0\quad \mathrm{as} \quad |y|\rightarrow \infty . \end{aligned}$$

Here we have used the fact that $w\in C(\mathbb {R}^{N}\backslash \{0\})$ is a homogeneous function of degree 0. So we complete the proof of this theorem. $\square$

Lemma 4

(Space-Time Decay Estimates) Let $0<\sigma <N$ and $M>0$ be a constant, if $0\le u_{0}\in W_{\sigma }(\mathbb {R}^{N})$ such that

$$\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}\le M,$$

then there exists a constant C such that

$$\begin{aligned} S(t)u_{0}(x)\le C(t^{\frac{2}{\sigma [m(p-1)-1]+p}}+|x|^{2})^{-\frac{\sigma }{2}} \end{aligned}$$

(2.10)

for all $t>0$ and all $x\in \mathbb {R}^{N}$.

Proof

Let $g(x)=S(1)\varphi (x)$ and $\varphi (x)=M|x|^{-\sigma }$. By Lemma 3, we have that there exists a constant C such that

$$\begin{aligned} |g(x)|\le C(1+|x|^{2})^{-\frac{\sigma }{2}}. \end{aligned}$$

By the comparison principle [6, 15], we have

$$\begin{aligned} S(t)u_{0}\le S(t)\varphi (x)\le C(t^{\frac{2}{\sigma [m(p-1)-1]+p}}+|x|^{2})^{-\frac{\sigma }{2}}. \end{aligned}$$

The proof is completed. $\square$

3 Relation Between Solutions and Initial Values

In this section, we consider the relation between the solutions and the initial value of the problem (1.1)–(1.2).

Theorem 1

If $0\le u_{0}\in W_{\sigma }$, then

$$\begin{aligned} \omega ^{\sigma }(u_{0})=S(1)\varOmega ^{\sigma }(u_{0}), \end{aligned}$$

(3.1)

where

$$\begin{aligned} S(1)\varOmega ^{\sigma }(u_{0})\equiv \{f:f=S(1)\varphi ,\varphi \in \varOmega ^{\sigma }(u_{0})\}. \end{aligned}$$

Proof

If $f\in S(1)\varOmega ^{\sigma }(u_{0})$, we can find a $\varphi \in \varOmega ^{\sigma }(u_{0})$ and a sequence $\{\lambda _{n}\}^{\infty }_{n=1}$ such that

$$\begin{aligned} f=S(1)\varphi \end{aligned}$$

(3.2)

and

$$\begin{aligned} D^{\sigma }_{\lambda _{n}}u_{0}{\mathop {\rightharpoonup }\limits ^{*}}\varphi \ \mathrm{in} \ W_{\sigma }(\mathbb {R}^{N}) \end{aligned}$$

(3.3)

as $\lambda _{n}\rightarrow \infty .$ The fact $u_{0}\in {W_{\sigma }(\mathbb {R}^{N})}$ implies that there exists a constant $C>0$ such that

$$\begin{aligned} \Vert D^{\sigma }_{\lambda _{n}}u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}=\Vert u_{0}\Vert _{W_{\sigma }(\mathbb {R}^{N})}\le C, \end{aligned}$$

(3.4)

so

$$\begin{aligned} \Vert \varphi \Vert _{W_{\sigma }(\mathbb {R}^{N})}\le C. \end{aligned}$$

(3.5)

By the inequality (2.10), we know that for $\epsilon >0$ and $n\ge 1$, there exists a $R>0$ such that if $|x|\ge R$, then

$$\begin{aligned} |S(t)(D^{\sigma }_{\lambda _{n}}u_{0})(x)|<\frac{\epsilon }{3} \ \mathrm{and} \ |S(t)(\varphi )(x)|<\frac{\epsilon }{3}. \end{aligned}$$

(3.6)

For $R>0$, by Lemma 2 we can get that there exists

$$\begin{aligned} R_1(t)=R+1+C\Vert u_{0}\Vert ^{\frac{m(p-1)-1}{\sigma [m(p-1)-1]+p}}_{W_{\sigma }(\mathbb {R}^{N})}t^{\frac{1}{\sigma [m(p-1)-1]+p}} \end{aligned}$$

such that

$$\begin{aligned} \mathrm{supp}[S(t)(1-\chi _{R_1(t)})D^{\sigma }_{\lambda _{n}}u_{0}]\subset \{x\in \mathbb {R}^{N}; \ |x|\ge R+1\}, \end{aligned}$$

where $\chi _{R_1(t)}(x)$ be the cutoff function defined on $B_{R_1(t)+1}$ relative to $B_{R_1(t)}$. Hence,

$$\begin{aligned} \mathrm{supp}[S(t)(1-\chi _{R(t)})D^{\sigma }_{\lambda _{n}}u_{0}]\bigcap B_{R}=\emptyset . \end{aligned}$$

In other words, for $x\in B_{R}$,

$$\begin{aligned} S(t)(D^{\sigma }_{\lambda _{n}}u_{0})(x)=S(t)[\chi _{R(t)}(D^{\sigma }_{\lambda _{n}}u_{0})](x). \end{aligned}$$

(3.7)

Similarly, for $S(t)\varphi (x)$, we can get that there exists $R_2(t)$ such that if $x\in B_{R}$, then

$$\begin{aligned} S(t)\varphi (x)=S(t)[\chi _{R_2(t)}\varphi ](x). \end{aligned}$$

(3.8)

Let

$$\begin{aligned} R(t)=\min (R_1(t),\ R_2(t)). \end{aligned}$$

Since (3.3) and (3.4), we know that

$$\begin{aligned} D^{\sigma }_{\lambda _{n}}u_{0}\rightarrow \varphi \ \mathrm{in} \ \mathscr {D}'(\mathbb {R}^{N}), \end{aligned}$$

let $1<q<\frac{N}{\sigma }$, we can get that

$$\begin{aligned} \chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}\rightharpoonup \chi _{R(t)}\varphi \ \mathrm{in} \ L^{q}(\mathbb {R}^{N}) \end{aligned}$$

as $\lambda _{n}\rightarrow \infty .$ By Lemma 2, we have

$$\begin{aligned} S(\tau )[\chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}]\rightharpoonup S(\tau )[\chi _{R(t)}\varphi ] \ \mathrm{in} \ L^{q}(\mathbb {R}^{N}) \end{aligned}$$

as $\lambda _{n}\rightarrow \infty ,$ where $0<\tau <1.$ This means that

$$\begin{aligned} S(\tau )[\chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}]\rightharpoonup S(\tau )[\chi _{R(t)}\varphi ] \ \mathrm{in} \ \mathscr {D}'(\mathbb {R}^{N}) \end{aligned}$$

as $\lambda _{n}\rightarrow \infty .$ By Lemma 4, we can find a constant $C(\tau )$ such that

$$\begin{aligned} \Vert S(\tau )[\chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}]\Vert _{L^{\infty }(\mathbb {R}^{N})}\le C(\tau ) \end{aligned}$$

for $n\ge 1$, hence

$$\begin{aligned} S(\tau )[\chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}]{\mathop {\rightharpoonup }\limits ^{w*}}S(\tau )[\chi _{R(t)}\varphi ] \ \mathrm{in} \ L^{\infty }(\mathbb {R}^{N}) \end{aligned}$$

as $\lambda _{n}\rightarrow \infty .$ We can use $0<\tau <1$ and the regularity of the semigroup operators S(t) to get

$$\begin{aligned} \Vert S(1)[\chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}]-S(1)[\chi _{R(t)}\varphi ]\Vert _{L^{\infty }_\mathrm{loc}(\mathbb {R}^{N})}\rightarrow 0 \end{aligned}$$

as $\lambda _{n}\rightarrow \infty .$ By (3.7) and (3.8), we have

$$\begin{aligned} \Vert S(1)[D^{\sigma }_{\lambda _{n}}u_{0}]-S(1)(\varphi )\Vert _{L^{\infty }(\mathbb {B}_{R})}=\Vert S(1)[\chi _{R(t)}D^{\sigma }_{\lambda _{n}}u_{0}]-S(1)[\chi _{R(t)}\varphi ]\Vert _{L^{\infty }(\mathbb {B}_{R})}\rightarrow 0 \end{aligned}$$

as $\lambda _{n}\rightarrow \infty .$ Then, it follows from (3.2) and (3.6) that

$$\begin{aligned} \Vert S(1)[D^{\sigma }_{\lambda _{n}}u_{0}]-f\Vert _{L^{\infty }(\mathbb {R}^{N})}=\Vert S(1)[D^{\sigma }_{\lambda _{n}}u_{0}]-S(1)(\varphi )\Vert _{L^{\infty }(\mathbb {R}^{N})}\rightarrow 0 \end{aligned}$$

(3.9)

as $\lambda _{n}\rightarrow \infty .$ By the commutative relation (2.2), we have

$$\begin{aligned} \Gamma ^{\sigma }_{\sqrt{t}}[S(1)u_{0}]=D^{\sigma }_{\sqrt{t}}[S(t)U_{0}]=S(1)[D^{\sigma }_{\sqrt{t}}u_{0}]. \end{aligned}$$

(3.10)

Taking $t_{n}=\lambda ^{2}_{n}$ in (3.9), it follows from (3.10) that $f\in \omega ^{\sigma }(u_{0}).$ This means that

$$\begin{aligned} S(1)\varOmega ^{\sigma }(u_{0})\subset \omega ^{\sigma }(u_{0}). \end{aligned}$$

(3.11)

On the other hand, if $f\in \omega ^{\sigma }(u_{0})$, then we can find a sequence $t_{n}\rightarrow \infty$ such that

$$\begin{aligned} \Gamma ^{\sigma }_{\sqrt{t_{n}}}[S(1)u_{0}]=D^{\sigma }_{\sqrt{t_{n}}}[S(t_{n})u_{0}]=S(1)[D^{\sigma }_{\sqrt{t_{n}}}u_{0}]\rightarrow f \ \mathrm{in} \ L^{\infty }(\mathbb {R}^{N}). \end{aligned}$$

(3.12)

If $n\ge 1$, we have

$$\begin{aligned} \Vert D^{\sigma }_{\sqrt{t_{n}}}u_{0}\Vert _{W_\sigma (\mathbb {R}^{N})}=\Vert u_{0}\Vert _{W_\sigma (\mathbb {R}^{N})}\le M. \end{aligned}$$

We can find a function $\varphi \in W_\sigma (\mathbb {R}^{N})$ and a subsequence $t_{n_{k}}\rightarrow \infty$ such that

$$\begin{aligned} D^{\sigma }_{\sqrt{t_{n_{k}}}}u_{0}{\mathop {\rightharpoonup }\limits ^{w*}}\varphi \ \mathrm{in} \ W_\sigma (\mathbb {R}^{N}), \end{aligned}$$

hence

$$\begin{aligned} \varphi \in \varOmega ^{\sigma }(u_{0}). \end{aligned}$$

(3.13)

We can use the similar proof method of (3.9) to get that there exists a subsequence $t_{n_{k}}\rightarrow \infty$ such that

$$\begin{aligned} S(1)D^{\sigma }_{\sqrt{t_{n_{k}}}}u_{0}\rightarrow S(1)\varphi \ \mathrm{in} \ L^{\infty }(\mathbb {R}^{N}). \end{aligned}$$

(3.14)

It follows from (3.12), (3.13) and (3.14) that

$$\begin{aligned} f=S(1)\varphi \in S(1)\varOmega ^{\sigma }(u_{0}). \end{aligned}$$

This means

$$\begin{aligned}\omega ^{\sigma }(u_{0})\subset S(1)\varOmega ^{\sigma }(u_{0}). \end{aligned}$$

Therefore, by (3.11), we get

$$\begin{aligned} \omega ^{\sigma }(u_{0})=S(1)\varOmega ^{\sigma }(u_{0}). \end{aligned}$$

So we completed this proof. $\square$

4 Different Asymptotic Behaviors

In this section, we first use the relation (3.1) to prove that $\omega ^{\sigma }(u_{0})$ may contain infinite functions. In other word, these solutions have complicated asymptotic behavior.

Theorem 2

Let $\{\varphi _i\}_{i=1}^\infty$ be a nonnegative sequence of functions such that

$$\begin{aligned} \Vert \varphi _i\Vert _{W_{\sigma }(\mathbb {R}^{N})}\le M\quad {\text{ f }or }\quad i\ge 1, \end{aligned}$$

where $M>0$ is a constant. Then, there exists a nonnegative initial value $U_{0}\in W_{\sigma }(\mathbb {R}^{N})$ such that

$$\begin{aligned} S(1)\overline{(\cup \varOmega ^{\sigma }(\varphi _{i}))_{i=1}^\infty }\subset \omega ^{\sigma }(U_{0}), \end{aligned}$$

(4.1)

where the closure is taken in the weak-star topology of ${W_{\sigma }(\mathbb {R}^{N})}$.

Proof

By the definition of $\{\varphi _i\}_{i=1}^\infty$, there exists a sequence $\{\phi _{n}\}$ such that for every $\varphi _{i}$, we can find a subsequence $\{\phi _{i_{n}}\}$ of $\{\phi _{n}\}$ satisfying that

$$\begin{aligned} \phi _{i_{n}}=\varphi _{i} \end{aligned}$$

for all $i_{n}\ge 1$. Let $h>1$ be a constant and

$$\begin{aligned} H_{n}\equiv \{x\in \mathbb {R}^{N};\ h^{-n}<|x|<h^{n}\}, \end{aligned}$$

$$\begin{aligned} H_{n-1}\equiv \{x\in \mathbb {R}^{N};\ h^{-n+1}<|x|<h^{n-1}\}. \end{aligned}$$

Then, we define the initial value

$$\begin{aligned} U_{0}(x)=\sum ^{\infty }_{n=0}D^{-\sigma }_{\lambda _{n}}(\chi _{n}(x)\phi _{n}(x)), \end{aligned}$$

where

$$\begin{aligned} \lambda _{n}=\left\{ \begin{aligned}&h,&\ if \ n=1,\\&h^{2n(\sigma (m(p-1)-1)+p)}\lambda _{n-1},&\ if \ n>1 \end{aligned}\right. \end{aligned}$$

and $\chi _{n}(x)$ is the cutoff function defined on the set $H_{n}$ relative to the set $H_{n-1}$. Since $h>1$ and

$$\begin{aligned} \text {supp}(D^{-\sigma }_{\lambda _{n}}(\chi _{n}(x)\phi _{n}(x))) \subset \{x\in \mathbb {R}^{N};\lambda _n^{\frac{1}{\sigma [m(p-1)-1]+p}}h^{-n}<|x|<\lambda _n^{\frac{1}{\sigma [m(p-1)-1]+p}}h^{n}\} \end{aligned}$$

for $n\ge 1,$ we have

$$\begin{aligned} \lambda _n^{\frac{1}{\sigma [m(p-1)-1]+p}}h^{-n}=\lambda _{n-1}^{\frac{1}{\sigma [m(p-1)-1]+p}} h^{n} > \lambda _{n-1}^{\frac{1}{\sigma [m(p-1)-1]+p}}h^{n-1}, \end{aligned}$$

hence

$$\begin{aligned} \text {supp} D^{-\sigma }_{\lambda _{i}}[\chi _{i}(x)\phi _{i}(x)]\bigcap \text {supp}D^{-\sigma }_{\lambda _{j}}[\chi _{j}(x)\phi _{j}(x)]=\emptyset \end{aligned}$$

for $i\ne j,$ so

$$\begin{aligned} U_{0}\in {W_{\sigma }(\mathbb {R}^{N})}. \end{aligned}$$

For $\varphi _{n}\in \{\varphi _i\}_{i=1}^\infty ,$ we can find a subsequence $\{\lambda _{n_{i}}\}$ of $\{\lambda _i\}$ such that

$$\begin{aligned} D^{\sigma }_{\lambda _{n_{i}}}U_{0}=\varphi _{n} \end{aligned}$$

in $H_{n_{i}-1}$ for all $i\ge 1$ by the definition of $U_{0}$. Since the sets $H_{i_{i}}$ expands to $\mathbb {R}^{N}\backslash \{0\}$ as $i\rightarrow \infty$, so we can get

$$\begin{aligned} D^{\sigma }_{\lambda _{n_{i}}}U_{0}{\mathop {\rightharpoonup }\limits ^{*}}\varphi _{n} \ \text {in} \ W_{\sigma }(\mathbb {R}^{N})\quad {\text{ a }s}\quad n_{i}\rightarrow \infty . \end{aligned}$$

So $\varphi _{n}\in \varOmega ^{\sigma }(U_{0}),$ hence

$$\begin{aligned} \overline{(\cup \varOmega ^{\sigma }(\varphi _{i}))_{i=1}^\infty }\subset \varOmega ^{\sigma }(U_{0}). \end{aligned}$$

Then through relation (3.1), we can get (4.1), so the proof is completed. $\square$

As another application of the relation (3.1), we can prove asymptotic of solutions to Problem (1.1)–(1.2).

Theorem 3

If $0\le u_{0}\in W_{\sigma }(\mathbb {R}^{N})$ such that

$$\begin{aligned} \lim _{|x|\rightarrow \infty }|x|^{\sigma }u_{0}(x)=A, \end{aligned}$$

(4.2)

where A is a positive constant, let W(x, t) be the solution of equation

$$\begin{aligned} {\left\{ \begin{array}{ll} W_{t}=\mathrm {div}(|\nabla W^{m}|^{p-2}\nabla W^{m}) &{}\mathrm{in} \ R^{N}\times (0,\infty ),\\ W(x,0)=A|x|^{-\sigma } &{}\mathrm{in} \ R^{N}, \end{array}\right. } \end{aligned}$$

then

$$\begin{aligned} \lim _{t\rightarrow \infty }t^{\frac{\sigma }{\sigma [m(p-1)-1]+p}}\Vert u(t^{\frac{1}{\sigma [m(p-1)-1]+p}}x,t)-W(t^{\frac{1}{\sigma [m(p-1)-1]+p}}x,t)\Vert _{L^\infty }=0. \end{aligned}$$

(4.3)

Proof

By (4.2), we have

$$\begin{aligned} \lim _{|y|\rightarrow \infty }||y|^{\sigma }u_{0}(y)-A|=0, \end{aligned}$$

let $|y|=\lambda ^{\frac{2}{\sigma [m(p-1)-1]+p}}|x|$, for $\epsilon >0$ and $|x|\ge \epsilon$,

$$\begin{aligned} |x|^{\sigma }\lim _{\lambda \rightarrow \infty }|D^{\sigma }_{\lambda }u_{0}(x)-A|x|^{-\sigma }|=0. \end{aligned}$$

In other words,

$$\begin{aligned} D^{\sigma }_{\lambda }u_{0}{\mathop {\rightharpoonup }\limits ^{w*}}A|x|^{-\sigma } \ in \ W_{\sigma }(\mathbb {R}^{N}) \end{aligned}$$

as $\lambda \rightarrow \infty$. Then,

$$\begin{aligned} \varOmega (u_{0})=\{\varphi (x)=A|x|^{-\sigma }\}. \end{aligned}$$

By Theorem 1, we get

$$\begin{aligned} \omega (u_{0})=\{S(1)\varphi (x)\}. \end{aligned}$$

(4.4)

We know that $\varphi$ is homogeneous of degree $-\sigma$, it follows from (2.2) that for any $t>0$,

$$\begin{aligned} t^{\frac{\sigma }{\sigma [m(p-1)-1]+p}}W(t^{\frac{1}{\sigma [m(p-1)-1]+p}}x,t) =S(1)[D^{\sigma }_{\sqrt{t}}\varphi ](x)=[S(1)\varphi ](x). \end{aligned}$$

(4.5)

Then, by (4.4) and (4.5), we obtain (4.3), so we completed this proof. $\square$