The purpose of this section is to prove the global existence and the decay estimates of solutions to the initial value problem (1.1), (1.2). Based on the existence and the decay estimates of the solutions to the linear problem (2.1), (1.2), we define
$$\mathcal{E}= \bigl\{ u\in C \bigl([0, \infty ); H^{s} \bigl( \mathbb{R}^{n} \bigr)\bigr): \|u\|_{\mathcal{E}}<\infty \bigr\} , $$
where
$$\begin{aligned} \|u\|_{\mathcal{E}}= \sum_{\varrho (k,n)\leq s}\sup_{t\geq0}(1+t)^{\frac{n}{4}+\frac{k}{2}}\bigl\| \partial ^{k}_{x}u(t)\bigr\| _{H^{s-\varrho(k, n)}} + \sum^{[\frac{n}{4}]}_{j=0}\sum_{2k+2j\leq s} \sup_{t\geq0}(1+t)^{\frac{k}{2}+j}\bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-2j}}, \end{aligned}$$
where
$$\varrho(k, n)=2k+ \biggl[\frac{n+1}{2} \biggr]. $$
For \(\mathfrak{R}>0\), we define
$$\mathcal{E}_{\mathfrak{R}}=\bigl\{ u\in X: \|u\|_{\mathcal{E}}\leq\mathfrak {R}\bigr\} , $$
where ℜ depends on the initial value, which is chosen in the proof of the main result.
Theorem 4.1
Let
\(n\geq1\), \(\theta>1+\frac{2}{n}\), and
\(s\geq [\frac{n+1}{2}]+2\). Assume that
\(u_{0}, u_{1} \in H^{s}(\mathbb {R}^{n})\cap L^{1}(\mathbb{R}^{n})\). Put
$$E_{0}=\|u_{0}\|_{H^{s}\cap L^{1}}+\|u_{1} \|_{H^{s}\cap L^{1}}. $$
If
\(E_{0}\)
is suitably small, the initial value problem (1.1), (1.2) has a unique global solution
$$u\in C^{0} \bigl([0, \infty ); H^{s} \bigl( \mathbb{R}^{n} \bigr)\bigr). $$
Moreover, the solution satisfies the decay estimate
$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-2j}}\leq CE_{0}(1+t)^{-\frac{k}{2}-j} $$
(4.1)
for
\(k\geq0\), \(0\leq j\leq[\frac{n}{4}]\), and
\(2k+2j\leq s\). Also, we have
$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-\varrho(k, n)}}\leq CE_{0}(1+t)^{-\frac {n}{4}-\frac{k}{2}} $$
(4.2)
for
\(k\geq0\)
and
\(\varrho(k, n)\leq s\).
Proof
We shall prove Theorem 4.1 by the Banach fixed point theorem. Thus we define the mapping
$$\mathcal{M}[u](t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}+ \int^{t}_{0}G(t-\tau)*(1-\Delta )^{-1} \Psi(u) (\tau)\,d\tau $$
and
$$\mathcal{M}_{0}(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}. $$
For \(\forall v, w\in\mathcal{E}\), we arrive at
$$\mathcal{M}[v](t)-\mathcal{M}[w](t)=\int^{t}_{0}G(t- \tau)*(1-\Delta )^{-1} \bigl(\Psi(v)-\Psi(w) \bigr) (\tau)\,d\tau. $$
For \(\Psi(u)=O(|u|^{\theta})\), it is not difficult to get the following nonlinear estimates (see [18]):
$$\begin{aligned} &\bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{1}} \\ &\quad\leq C\bigl\| (v, w)\bigr\| ^{\theta -2}_{L^{\infty}} \bigl(\bigl\| (v, w) \bigr\| _{L^{2}}\bigl\| \partial^{k}_{x}(v-w)\bigr\| _{L^{2}}+\bigl\| \partial _{x}^{k}(v, w)\bigr\| _{L^{2}}\|v-w \|_{L^{2}} \bigr) \end{aligned}$$
(4.3)
and
$$\begin{aligned} &\bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{2}} \\ &\quad\leq C\bigl\| (v, w)\bigr\| ^{\theta -2}_{L^{\infty}} \bigl(\bigl\| (v, w) \bigr\| _{L^{\infty}}\bigl\| \partial^{k}_{x}(v-w)\bigr\| _{L^{2}}+\bigl\| \partial_{x}^{k}(v, w)\bigr\| _{L^{2}}\|v-w \|_{L^{\infty}} \bigr). \end{aligned}$$
(4.4)
If \(u\in X\), the Gagliardo-Nirenberg inequality gives
$$ \bigl\| u(\tau)\bigr\| _{L^{\infty}}\leq C\|u\|_{\mathcal{E}}(1+ \tau)^{-\frac {n}{2}}. $$
(4.5)
Firstly, we shall prove
$$ \bigl\| \partial^{k}_{x} \bigl(\mathcal{M}[v]-\mathcal{M}[w] \bigr) \bigr\| _{H^{s-2k-2j}}\leq C(1+t)^{-\frac{k}{2}-j} \bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \|v-w\| _{\mathcal{E}}, $$
(4.6)
where \(0\leq j\leq[\frac{n}{4}]\), \(k\geq0\), and \(2k+2j\leq s\).
Assume that k, j, m are non-positive integers. Let \(j\leq[\frac {n}{4}]\) and \(2k+2j\leq s\). We apply \(\partial^{k+m}_{x}\) to \(\mathcal {M}[v]-\mathcal{M}[w]\). This yields
$$\begin{aligned} & \bigl\| \partial^{k+m}_{x}\bigl(\mathcal{M}[v]-\mathcal {M}[w]\bigr)(t)\bigr\| _{L^{2}} \\ &\quad\leq \int^{\frac{t}{2}}_{0}\bigl\| \partial^{k+m}_{x}G(t-\tau )*(1-\Delta)^{-1}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{2}}\,d\tau \\ &\qquad{}+ \int^{t}_{\frac{t}{2}}\bigl\| \partial^{k+m}_{x}G(t-\tau )*(1-\Delta)^{-1}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{2}}\,d\tau \\ &\quad\triangleq J_{1}+J_{2}. \end{aligned}$$
(4.7)
By (2.22), we obtain
$$\begin{aligned} J_{1} \leq& C\int^{\frac {t}{2}}_{0}(1+t-\tau)^{-\frac{n}{4}-\frac{k+m}{2}}\bigl\| \Psi(v)-\Psi(w)\bigr\| _{L^{1}}\,d\tau \\ &{}+ C\int^{\frac{t}{2}}_{0}(1+t-\tau)^{-\frac{l}{2}}\bigl\| \partial_{x}^{k+m+l-2}\bigl(\Psi(v)-\Psi(w)\bigr)\bigr\| _{L^{2}}\,d\tau \\ \triangleq& J_{11}+J_{12}. \end{aligned}$$
(4.8)
Using (4.3) and (4.5), we arrive at
$$ \bigl\| \Psi(v)-\Psi(w)\bigr\| _{L^{1}}\leq C\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \|v-w\| _{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta-1)}. $$
(4.9)
Noting that \(\theta> 1+\frac{2}{n}\) for \(n=1, 2\) and \(\theta\geq2\) for \(n\geq3\), using (4.9), we deduce that
$$ J_{11}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.10)
If \(k+m+l-2\leq s\), (4.4) gives
$$ \bigl\| \partial^{k+m+l-2}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) (\tau) \bigr\| _{L^{2}}\leq C\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac {n}{2}(\theta-1)}. $$
(4.11)
Take \(l=k+2j+2\). By (4.11) and noticing \(\theta\geq2\), we obtain
$$J_{12}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}} $$
with \(0\leq m\leq s-2k-2j\).
Combining (4.10) and (4.11) yields
$$J_{1}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}} $$
with \(0\leq m\leq s-2k-j\). We apply \(l=2\) and \(p=1\) to the term \(J_{2}\). This yields
$$\begin{aligned} J_{2} \leq& C\int^{t}_{\frac{t}{2}}(1+t-\tau )^{-\frac{n}{4}-\frac{m}{2}}\bigl\| \partial^{k}_{x}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{1}}\,d\tau \\ &{}+ C\int^{t}_{\frac{t}{2}}(1+t-\tau)^{-1}\bigl\| \partial ^{k+m}_{x}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{2}}\,d\tau=:J_{21}+J_{22}. \end{aligned}$$
(4.12)
It follows from (4.3) and (4.5) that
$$ \bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{1}}\leq C\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta -2)-\frac{n}{4}-\frac{k}{2}-j}, $$
(4.13)
where \(0\leq j\leq[\frac{n}{4}]\) and \(2k+2j\leq s\).
Since \(\theta>1+\frac{2}{n}\) for \(n=1, 2\) and \(\theta\geq2\) for \(n\geq 3\), using (4.13), we obtain
$$J_{21}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}} $$
for \(0\leq j\leq[\frac{n}{4}]\) and \(2k+2j\leq s\).
If \(m\leq s-2k-2j\), we apply (4.4) to get
$$ \bigl\| \partial^{k+m}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) (\tau) \bigr\| _{L^{2}}\leq C\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-d(\theta-1)-\frac {k}{2}-j}, $$
(4.14)
which yields
$$J_{22}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v ,w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}}, $$
where \(0\leq m\leq s-2k-2j\).
Thus
$$ J_{2} \leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v ,w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}}. $$
(4.15)
We immediately get (4.6).
In the following we prove
$$ \bigl\| \partial^{k}_{x} \bigl(\mathcal{M}[v]-\mathcal{M}[w] \bigr) (t)\bigr\| _{H^{s-\varrho(k, n)}}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v, w) \bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}, $$
(4.16)
where \(\varrho(k, n)\leq s\).
Assume that k and m are nonnegative integers and \(\varrho(k, n)\leq s\). Applying \(\partial^{k+m}_{x}\) to \(\mathcal{M}[v]-\mathcal{M}[w]\), then (4.7), (4.8), and (4.10) still hold. Now we estimate \(J_{12}\).
If \(k+m+l-2\leq s\), we still have (4.11). Taking \(l=\varrho(k, n)-k+2\), then we obtain
$$ J_{12}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.17)
Combining (4.10) and (4.17) yields
$$ J_{1}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.18)
Similarly, we have (4.12). Since \(\varrho(k, n)\leq s\), in view of (4.3) and (4.5), we get
$$\bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{1}}\leq C\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta -1)-\frac{k}{2}}, $$
which yields
$$J_{21}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|. $$
Since \(\|\partial^{k+m}_{x}(\Psi(v)-\Psi(w))\|_{L^{2}}\leq\|\partial ^{k}_{x}(\Psi(v)-\Psi(w))\|_{H^{m}}\), by using (4.4), (4.5) and noticing that \(\varrho(k, n)\leq s\), we have
$$\bigl\| \partial^{k+m}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{2}}\leq C\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta -1)-\frac{n}{4}-\frac{k}{2}}, $$
which yields
$$J_{22}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
Thus
$$ J_{2}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.19)
Equations (4.18) and (4.19) give (4.16).
It follows from (4.6) and (4.16) that
$$ \bigl\| \mathcal{M}[v]-\mathcal{M}[w]\bigr\| _{\mathcal{E}} \leq C\bigl\| (v ,w)\bigr\| ^{\theta-1}_{\mathcal{E}}\|v-w\|_{\mathcal{E}}. $$
(4.20)
Thus for \(v, w \in X_{R}\), we have
$$\bigl\| \mathcal{M}[v]-\mathcal{M}[w]\bigr\| _{\mathcal{E}} \leq C_{1} \mathfrak {R}^{\theta-1}\|v-w\|_{\mathcal{E}}. $$
On the other hand, \(\mathcal{M}[0](t)=\mathcal{M}_{0}(t)\) is a solution of the problem (2.1), (1.2). From Theorem 3.1, we know that
$$\bigl\| \mathcal{M}_{0}(t)\bigr\| _{\mathcal{E}}\leq C_{2}E_{0}, $$
if \(E_{0}\) is suitably small.
We take \(\mathfrak{R}=2C_{2}E_{0}\). If \(E_{0}\) is suitably small such that \(\mathfrak{R}< 1\) and \(C_{1}\mathfrak{R}\leq\frac{1}{2}\), then we arrive at
$$\bigl\| \mathcal{M}[v]-\mathcal{M}[w]\bigr\| _{X} \leq\frac{1}{2}\|v-w \|_{\mathcal{E}}. $$
Therefore, for \(v \in\mathcal{E}_{\mathfrak{R}}\), we have
$$\bigl\| \mathcal{M}[v]\bigr\| _{\mathcal{E}} \leq\bigl\| \mathcal{M}_{0}(t) \bigr\| _{\mathcal{E}}+\frac{1}{2}\mathfrak{R}\leq C_{2}E_{0}+ \frac{1}{2}\mathfrak{R}=\mathfrak {R}. $$
Thus \(u\rightarrow\mathcal{M}[u]\) is a contraction mapping on \(\mathcal{E}_{\mathfrak{R}}\), so there exists a unique \(u\in\mathcal {E}_{\mathfrak{R}}\) satisfying \(\mathcal{M}[u]=u\). Therefore, the initial value problem (1.1), (1.2) has a unique solution u satisfying the decay estimates (4.1) and (4.2). We have completed the proof of the theorem. □