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. □