1 Introduction

In this paper, we investigate the initial value problem for the following semi-linear wave equation:

$$ u_{tt}-\Delta u-\Delta u_{tt}+u_{t}= \Psi(u) $$
(1.1)

with the initial value

$$ t=0: u=u_{0}(x),\qquad u_{t}=u_{1}(x), $$
(1.2)

where \(u = u(x, t) \) is the unknown function of \(x = (x_{1}, \ldots, x_{n})\in\mathbb{R}^{n}\) and \(t > 0\). The term \(u_{t}\) represents a frictional dissipation, and the term \(\Delta u_{tt}\) corresponds to the rotational inertia effects. \(\Psi(u)\) is a smooth function of u and satisfies \(\Psi(u) = O(|u|^{\theta})\) for \(u \rightarrow0\).

Equation (1.1) is an inertial model characterized by the term \(\Delta u_{tt}\). Without this inertial term \(\Delta u_{tt}\), (1.1) is reduced to the semi-linear wave equation with a dissipative term,

$$ u_{tt} - \Delta u + u_{t} = \Psi(u). $$
(1.3)

The initial value problem for (1.3) has been extensively studied, we refer to [1–6], and [7]. The analysis for (1.3) will be much easier than for (1.1) since the associated fundamental solutions to (1.3) are similar to the heat kernel and exponential decay in the high frequency region. In fact, the decay structure of (1.3) is characterized by

$$ \operatorname{Re} \lambda(\xi) \leq-c|\xi|^{2}/ \bigl(1 + |\xi|^{2} \bigr). $$
(1.4)

The decay structure of (1.1) is characterized by

$$ \operatorname{Re} \lambda(\xi) \leq-c|\xi|^{2}/ \bigl(1 + |\xi|^{2} \bigr)^{2}. $$
(1.5)

The linear term \(u_{t}\) is relatively weak compared with the one given by the damping term \(-\Delta u_{t}\). For the latter case, the equation becomes

$$u_{tt}-\Delta u-\Delta u_{tt}-\Delta u_{t}= \Psi(u). $$

In the case, the decay structure of in this case is characterized by (1.4). The dissipative structure that is characterized by (1.5) is called the ‘regularity-loss’ type. This dissipative structure is very weak in the high frequency region, so that even in a bounded domain case it does not give an exponential decay but a polynomial decay of the energy.

If the nonlinear term \(\Psi(u)\) is replaced by \(\Psi(u_{t})\), then (1.1) becomes

$$ u_{tt}-\Delta u-\Delta u_{tt}+u_{t}= \Psi(u_{t}). $$
(1.6)

The global existence and the asymptotic behavior of the solutions to the problem (1.6), (1.2) with \(L^{1}\) data were established by Wang et al. [8]. Later, Wang and Wang [9] proved the global existence and the asymptotic behavior of the solutions to the problem (1.6), (1.2) with \(L^{2}\) data. The analysis for the problem (1.1), (1.2) is much harder than the problem (1.6), (1.2), since the decay estimate of \(u_{t}\) is better than that of u.

The main purpose of this paper is to establish the global existence and decay estimates of solutions to (1.1), (1.2). We prove the global existence and the following decay estimates of the solutions to the problem (1.1), (1.2) for \(n\geq1\):

$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq CE_{0}(1+t)^{-\frac {k}{2}-\frac{n}{4}} $$
(1.7)

for \(k\geq0\), \(2k+[\frac{n+1}{2}]\leq s\), and \(s\geq [\frac {n+1}{2}]+2\). Here \(E_{0}:=\|u_{0}\|_{H^{s}\cap L^{1}}+\|u_{1}\|_{H^{s}\cap L^{1}}\) is assumed to be small. Our proof is carried out by a fixed point-contraction mapping argument, relying on the decay estimates for the linear problem.

The study of the global existence and asymptotic behavior of solutions to dissipative hyperbolic-type equations has a long history. We refer to [2–4, 6, 7] for the damped wave equation. Also, we refer to [10, 11] and [8, 9, 12–16] for various aspects of dissipation of the plate equation.

We give some notations which are used in this paper. Let \(F[u]\) denote the Fourier transform of u defined by

$$\hat{u}(\xi)=F[u]=\int_{\mathbb{R}^{n}}e^{-i\xi\cdot x}u(x)\,dx, $$

and we denote its inverse transform by \(F^{-1}\). For a nonnegative integer k, \(\partial^{k}_{x}\) denotes the totality of all the kth order derivatives with respect to \(x \in \mathbb{R}^{n}\).

For \(1\leq p\leq\infty\), \(L^{p}=L^{p}(\mathbb{R}^{n})\) denotes the usual Lebesgue space with the norm \(\|\cdot\|_{L^{p}}\). Let s be a nonnegative integer, \(H^{s} = H^{s}(\mathbb{R}^{n})\) denotes the Sobolev space of \(L^{2}\) functions, equipped with the norm \(\|\cdot\|_{H^{s}}\). Also, \(C^{k}(I;H^{s}) \) denotes the space of k-times continuously differentiable functions on the interval I with values in the Sobolev space \(H^{s} = H^{s}(\mathbb{R}^{n})\).

Finally, in this paper, we denote every positive constant by the same symbol C or c without confusion. \([\cdot]\) is the Gauss symbol.

The paper is organized as follows. The decay property of the solution operators to (1.1) is in Section 2. Then, in Section 3, we discuss the linear problem and show the decay estimates. Finally, we prove the global existence and the decay estimates of solutions for the initial value problem (1.1), (1.2) in Section 4.

2 Decay property of solution operator

The aim of this section is to establish decay estimates of the solution operators to (1.1). Firstly, we derive the solution formula for the problem (1.1), (1.2). We investigate the linear equation of (1.1):

$$ u_{tt}-\Delta u-\Delta u_{tt}+u_{t}=0, $$
(2.1)

with the initial data (1.2).

We apply the Fourier transform to (2.1), (1.2); it yields

$$\begin{aligned}& \bigl(1+|\xi|^{2} \bigr)\hat{u}_{tt}+\hat{u}_{t}+| \xi|^{2}\hat{u}=0, \end{aligned}$$
(2.2)
$$\begin{aligned}& t=0: \hat{u}=\hat{u}_{0}(\xi), \qquad\hat{u}_{t}= \hat{u}_{1}(\xi). \end{aligned}$$
(2.3)

The corresponding characteristic equation of (2.2) is

$$ \bigl(1+|\xi|^{2} \bigr)\lambda^{2}+\lambda+| \xi|^{2}=0. $$
(2.4)

Let \(\lambda=\lambda_{\pm}(\xi)\) be the solutions to (2.4). It is not difficult to find

$$ \lambda_{\pm}(\xi)=\frac{-1\pm\sqrt{1-4|\xi|^{2}(1+|\xi|^{2})}}{2(1+|\xi |^{2})}. $$
(2.5)

The solution to (2.2), (2.3) in the Fourier space is then given explicitly in the form

$$ \hat{u}(\xi, t)=\hat{G}(\xi, t) \bigl(\hat{u}_{0}(\xi)+ \hat{u}_{1}(\xi) \bigr)+\hat {H}(\xi, t)\hat{u}_{0}(\xi), $$
(2.6)

where

$$ \hat{G}(\xi, t)=\frac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)} \bigl(e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t} \bigr) $$
(2.7)

and

$$ \hat{H}(\xi, t)=\frac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)} \bigl( \bigl(1+\lambda _{+}(\xi) \bigr)e^{\lambda_{-}(\xi)t}- \bigl(1+\lambda_{-}(\xi) \bigr)e^{\lambda_{+}(\xi)t} \bigr). $$
(2.8)

Let

$$G(x, t) = F^{-1} \bigl[\hat{G}(\xi, t) \bigr](x) $$

and

$$H(x, t) = F^{-1} \bigl[\hat{H}(\xi, t) \bigr](x), $$

where \(F^{-1}\) denotes the inverse Fourier transform. Then we apply \(F^{-1}\) to (2.6), and it yields

$$ u(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}. $$
(2.9)

Now we return to our nonlinear problem (1.1), (1.2). By the Duhamel principle, the problem (1.1), (1.2) is equivalent to

$$ u(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}+\int ^{t}_{0}G(t-\tau)*(1-\Delta)^{-1}\Psi (u) (\tau)\,d\tau. $$
(2.10)

Next, we consider the linearized problem (2.1), (1.2) and derive the pointwise estimates of solutions in the Fourier space, which were already obtained in [8]. For the reader’s convenience, we give a detailed proof.

Lemma 2.1

Let u be the solution to the linearized problem (2.1), (1.2). Then its Fourier image \(\hat{u}\) verifies the pointwise estimate

$$ \bigl|\hat{u}_{t}(\xi, t)\bigr|^{2}+\bigl|\hat{u}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t} \bigl(\bigl|\hat {u}_{1}(\xi)\bigr|^{2}+\bigl| \hat{u}_{0}(\xi)\bigr|^{2} \bigr), $$
(2.11)

for \(\xi\in\mathbb{R}^{n}\) and \(t\geq0\), where \(\omega(\xi)=\frac{|\xi |^{2}}{(1+|\xi|^{2})^{2}}\).

Proof

By multiplying (2.2) by \(\bar{\hat{u}}_{t}\) and taking the real part, we deduce that

$$ \frac{1}{2}\frac{d}{dt} \bigl\{ \bigl(1+|\xi|^{2} \bigr)| \hat{u}_{t}|^{2}+|\xi|^{2}|\hat{u}|^{2} \bigr\} +|\hat{u}_{t}|^{2}=0. $$
(2.12)

We multiply (2.2) by \(\bar{\hat{u}}\) and take the real part. This yields

$$ \frac{1}{2}\frac{d}{dt} \bigl\{ |\hat{u}|^{2}+2 \bigl(1+| \xi|^{2} \bigr)\operatorname{Re}(\hat{u}_{t}\bar{\hat {u}}) \bigr\} + | \xi|^{2}|\hat{u}|^{2}- \bigl(1+|\xi|^{2} \bigr)| \hat{u}_{t}|^{2}=0. $$
(2.13)

Then by (2.12) and (2.13), we have

$$ \frac{d}{dt}E+F=0, $$
(2.14)

where

$$E= \bigl(1+|\xi|^{2} \bigr)^{2}|\hat{u}_{t}|^{2}+ \biggl[\frac{1}{2}+ \bigl(1+|\xi|^{2} \bigr)|\xi|^{2} \biggr]|\hat {u}|^{2}+ \bigl(1+|\xi|^{2} \bigr)\operatorname{Re}( \hat{u}_{t}\bar{\hat{u}}) $$

and

$$F= \bigl(1+|\xi|^{2} \bigr)|\hat{u}_{t}|^{2}+| \xi|^{2}|\hat{u}|^{2}. $$

Let

$$E_{0}=|\hat{u}_{t}|^{2}+|\hat{u}|^{2}. $$

It is easy to see that

$$ C \bigl(1+|\xi|^{2} \bigr)^{2}E_{0}\leq E\leq C \bigl(1+|\xi|^{2} \bigr)^{2}E_{0}. $$
(2.15)

Noting that \(F \geq|\xi|^{2}E_{0}\) and with (2.15), we obtain

$$ F\geq c\omega(\xi)E, $$
(2.16)

where

$$\omega(\xi)=\frac{|\xi|^{2}}{(1+|\xi|^{2})^{2}}. $$

Combining (2.14) and (2.16) yields

$$\frac{d}{dt}E+c\omega(\xi)E\leq0. $$

Thus

$$E(\xi, t)\leq e^{-cw(\xi)t}E(\xi, 0), $$

which together with (2.15) proves the desired estimate (2.11). Then we have completed the proof of the lemma. □

Lemma 2.2

Let \(\hat{G}(\xi, t)\) and \(\hat{H}(\xi, t)\) be the fundamental solution of (2.1) in the Fourier space, which are given in (2.7) and (2.8), respectively. Then we have the estimates

$$ \bigl|\hat{G}(\xi, t)\bigr|^{2}+\bigl|\hat{G}_{t}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t} $$
(2.17)

and

$$ \bigl|\hat{H}(\xi, t)\bigr|^{2}+\bigl|\hat{H}_{t}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t} $$
(2.18)

for \(\xi\in\mathbb{R}^{n}\) and \(t\geq0\), where \(\omega(\xi)=\frac{|\xi |^{2}}{(1+|\xi|^{2})^{2}}\).

Proof

If \(\hat{u}_{0}(\xi)=0\), from (2.5), we obtain

$$\hat{u}(\xi, t)=\hat{G}(\xi, t)\hat{u}_{1}(\xi),\qquad \hat{u}_{t}( \xi, t)=\hat{G}_{t}(\xi, t)\hat{u}_{1}(\xi). $$

Substituting the equalities into (2.11) with \(\hat{u}_{0}(\xi)=0\), we get (2.17).

In the following, we consider \(\hat{u}_{1}(\xi)=0\); it follows from (2.5) that

$$\hat{u}(\xi, t)= \bigl(\hat{G}(\xi, t)+\hat{H}(\xi, t) \bigr) \hat{u}_{0}( \xi),\qquad \hat {u}_{t}(\xi, t)= \bigl( \hat{G}_{t}(\xi, t)+ \hat{H}_{t}(\xi, t) \bigr) \hat{u}_{0}(\xi). $$

Substituting the equalities into (2.11) with \(\hat{u}_{1}(\xi)=0\), we get

$$\bigl|\hat{G}(\xi, t)+\hat{H}(\xi, t)\bigr|^{2}+\bigl|\hat{G}_{t}(\xi, t)+ \hat{H}_{t}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t}, $$

which together with (2.17) proves the desired estimate (2.18). The lemma is proved. □

Using the Taylor formula and (2.7), (2.8), we immediately obtain the following.

Lemma 2.3

Let \(\hat{G}(\xi, t)\) and \(\hat{H}(\xi, t)\) be the fundamental solutions of (2.1) in the Fourier space, which are given in (2.7) and (2.8), respectively. Then there is a small positive number \(R_{0}\) such that if \(|\xi|\leq R_{0}\) and \(t\geq0\), we have the following estimates:

$$\begin{aligned}& \bigl|\hat{H}(\xi, t)\bigr|\leq C|\xi|^{2}e^{-c|\xi|^{2}t}+Ce^{-ct}, \end{aligned}$$
(2.19)
$$\begin{aligned}& \bigl|\hat{G}_{t}(\xi, t)\bigr|\leq C|\xi|^{2}e^{-c|\xi|^{2}t}+Ce^{-ct}, \end{aligned}$$
(2.20)

and

$$ \bigl|\hat{H}_{t}(\xi, t)\bigr|\leq C|\xi|^{4}e^{-c|\xi|^{2}t}+Ce^{-ct}. $$
(2.21)

Lemma 2.4

Let \(1\leq p\leq2\) and \(k\geq0\). Then for \(l\geq0\) we have

$$ \bigl\| \partial^{k}_{x}G(t)*\phi\bigr\| _{L^{2}}\leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}}\|\phi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\phi\bigr\| _{L^{2}} $$
(2.22)

and

$$ \bigl\| \partial^{k}_{x}H(t)*\psi\bigr\| _{L^{2}}\leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}-1}\|\psi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\psi\bigr\| _{L^{2}}. $$
(2.23)

Moreover, we have

$$ \bigl\| \partial^{k}_{x}G_{t}(t)*\phi\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}-1}\|\phi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\phi\bigr\| _{L^{2}} $$
(2.24)

and

$$ \bigl\| \partial^{k}_{x}H_{t}(t)*\psi\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}-2}\|\psi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\psi\bigr\| _{L^{2}}. $$
(2.25)

Proof

The proof method of Lemma 2.4 has been used in many papers, we refer to [8] and [17].

We only prove (2.23). It follows from the Plancherel theorem that

$$\begin{aligned} \bigl\| \partial^{k}_{x}H(t)*\psi\bigr\| ^{2}_{L^{2}} =& \int_{\mathbb{R}^{n}}|\xi|^{2k}\bigl|\hat{H}(\xi, t)\bigr|^{2} \bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ =& \int_{|\xi|\leq R_{0}}|\xi|^{2k}\bigl|\hat{H}(\xi, t)\bigr|^{2}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ &{} + \int_{|\xi|\geq R_{0}}|\xi|^{2k}\bigl|\hat{H}(\xi, t)\bigr|^{2}\bigl|\hat {\psi}(\xi)\bigr|^{2}\,d\xi \\ \triangleq& I_{1}+I_{2}, \end{aligned}$$
(2.26)

where \(R_{0}\) is a positive constant in Lemma 2.3.

In the following, we estimate \(I_{1}\). Using (2.19) and the Hölder inequality, we obtain

$$\begin{aligned} I_{1} \leq& C\int_{|\xi|\leq R_{0}}|\xi |^{2k+4}e^{-c|\xi|^{2}t}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi+ C\int_{|\xi|\leq R_{0}}|\xi |^{2k}e^{-ct}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ \leq& C\bigl\| |\xi|^{2k+4}e^{-c|\xi|^{2}t}\bigr\| _{L^{q}(|\xi|\leq R_{0})}\bigl\| \hat{\psi}(\xi)\bigr\| ^{2}_{L^{p'}} +C\bigl\| |\xi|^{2k}e^{-ct}\bigr\| _{L^{q}(|\xi |\leq R_{0})}\bigl\| \hat{\psi}(\xi)\bigr\| ^{2}_{L^{p'}}, \end{aligned}$$

where \(\frac{1}{q}+\frac{2}{p'}=1\) and \(\frac{1}{p}+\frac{1}{p'}=1\).

By a straightforward computation, we get

$$\bigl\| |\xi|^{2k+4}e^{-c|\xi|^{2}t}\bigr\| _{L^{q}(|\xi|\leq R_{0})} \leq C(1+t)^{-\frac{n}{2q}-k-2} $$

and

$$\bigl\| |\xi|^{2k}e^{-c|\xi|^{2}t}\bigr\| _{L^{q}(|\xi|\leq R_{0})} \leq Ce^{-ct}. $$

The Hausdorff-Young inequality gives

$$\bigl\| \hat{\psi}(\xi)\bigr\| _{L^{p'}}\leq C\|\psi\|_{L^{p}}. $$

Thus, we have

$$\begin{aligned} I_{1} \leq& C(1+t)^{-\frac{n}{2q}-k-2}\| \psi\|^{2}_{L^{p}} \\ =&C(1+t)^{-\frac{n}{2}(\frac{2}{p}-1)-k-2}\|\psi\|^{2}_{L^{p}}. \end{aligned}$$
(2.27)

Note that \(\omega(\xi)\geq c|\xi|^{-2}\) when \(|\xi|\geq R_{0}\). By (2.17), we arrive at

$$\begin{aligned} I_{2} \leq& C\int_{|\xi|\geq R_{0}}|\xi |^{2k}e^{-c|\xi|^{-2}t}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ \leq& C\sup_{|\xi|\geq R_{0}}\bigl(|\xi|^{-2l}e^{-c|\xi |^{-2}t}\bigr)\int_{|\xi|\geq R_{0}}|\xi|^{2(k+l)}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ \leq& C(1+t)^{-l}\bigl\| \partial^{k+l}_{x}\psi\bigr\| ^{2}_{L^{2}}. \end{aligned}$$
(2.28)

Combining (2.27) and (2.28) yields (2.23). Thus we have completed the proof of the lemma. □

3 Decay estimates of solutions to (2.1), (1.2)

In the previous section, we observed that the decay structure of the linear equation (2.1) is of the regularity-loss type. The purpose of this section is to show the decay estimates of the solutions to (2.1), (1.2) when the initial data are in \(H^{s}\cap L^{1}\).

Theorem 3.1

Let \(s\geq0\) and suppose that \(u_{0}, u_{1} \in H^{s}(\mathbb{R}^{n})\cap L^{1}({\mathbb{R}^{n}})\). Put

$$E_{0}=\|u_{0}\|_{H^{s}\cap L^{1}}+\|u_{1} \|_{H^{s}\cap L^{1}}. $$

Then the solution \(u(x, t)\) of the linear problem (2.1), (1.2), which is given by (2.9), satisfies the decay estimate

$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq CE_{0}(1+t)^{-\frac {n}{4}-\frac{k}{2}} $$
(3.1)

for \(k\geq0\) and \(2k+[\frac{n+1}{2}]\leq s\). Moreover, for each j with \(0\leq j\leq1\), we have

$$ \bigl\| \partial^{k}_{x}u_{t}(t)\bigr\| _{H^{s-2k-[\frac{n+1}{2}]-2j}}\leq CE_{0}(1+t)^{-\frac{n}{4}-\frac{k}{2}-j} $$
(3.2)

for \(k\geq0\) and \(2k+[\frac{n+1}{2}]+2j\leq s\).

Remark 3.1

In addition to the above decay estimates, we also have

$$\begin{aligned}& \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-2j}}\leq CE_{0}(1+t)^{-\frac{k}{2}-j}, \quad0\leq j\leq \biggl[\frac{n}{4} \biggr], \end{aligned}$$
(3.3)
$$\begin{aligned}& \bigl\| \partial^{k}_{x}u_{t}(t)\bigr\| _{H^{s-2k-2j}}\leq CE_{0}(1+t)^{-\frac{k}{2}-j}, \quad0\leq j\leq \biggl[\frac{n}{4} \biggr]+1. \end{aligned}$$
(3.4)

Proof

The proof of (3.1)-(3.4) is similar. We only prove (3.1). Owing to (2.9), to prove (3.1), it suffices to prove the following estimates:

$$\begin{aligned}& \bigl\| \partial^{k}_{x}G(t)*u_{1}\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq C(1+t)^{-\frac {n}{4}-\frac{k}{2}}\|u_{1}\|_{H^{s}\cap L^{1}}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \bigl\| \partial^{k}_{x}G(t)*u_{0}\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq C(1+t)^{-\frac {n}{4}-\frac{k}{2}}\|u_{0}\|_{H^{s}\cap L^{1}}, \end{aligned}$$
(3.6)

and

$$ \bigl\| \partial^{k}_{x}H(t)*u_{0}\bigr\| _{H^{s-2k-[\frac{n+1}{2}]-2j}} \leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}-j}\|u_{0}\|_{H^{s}\cap L^{1}},\quad 0\leq j\leq1. $$
(3.7)

Firstly, we prove (3.5). Let \(k\geq0\) and \(h\geq0\), we have from (2.22) with k replaced by \(k+h\) and with \(p=1\)

$$\bigl\| \partial^{k+h}_{x}G(t)*u_{1}\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{4}-\frac {k+h}{2}}\|u_{1}\|_{L^{1}}+C(1+t)^{-\frac{l}{2}} \bigl\| \partial^{k+h+l}_{x}u_{1}\bigr\| _{L^{2}}, $$

where \(l\geq0\) and \(k+h+l\leq s\). We choose l as the smallest integer satisfying \(\frac{l}{2}\geq\frac{n}{4}+\frac{k}{2}\), i.e. \(l\geq\frac{n}{2}+k\). Thus, we take \(l=[\frac{n+1}{2}]+k\). It follows from \(k+h+l\leq s\) that \(h\leq s-2k-[\frac{n+1}{2}]\). Then we have

$$\bigl\| \partial^{k+h}_{x}G(t)*u_{1}\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{4}-\frac {k}{2}}\|u_{1}\|_{H^{s}\cap L^{1}} $$

for \(0\leq h\leq s-2k-[\frac{n+1}{2}]\). Then (3.5) is proved.

Similarly, we can prove (3.6).

Finally, we prove (3.7). Let \(k\geq0\) and \(h\geq0\). It follows from (2.23) that

$$\bigl\| \partial^{k+h}_{x}H(t)*u_{0}\bigr\| _{L^{2}} \leq c(1+t)^{-\frac{n}{4}-\frac {k+h}{2}-1}\|u_{0}\|_{L^{1}}+C(1+t)^{-\frac{l}{2}} \bigl\| \partial^{k+h+l}_{x}u_{0}\bigr\| _{L^{2}}, $$

where \(l\geq0\) and \(k+h+l\leq s\).

Let \(0\leq j\leq1\). We choose l as the smallest integer satisfying \(\frac{l}{2}\geq\frac{n}{4}+\frac{k}{2}+j\). Thus we take \(l=[\frac{n+1}{2}]+k+2j\). From \(k+h+l\leq s\), we obtain

$$0\leq h\leq s-2k- \biggl[\frac{n+1}{2} \biggr]-2j. $$

Then we have

$$\bigl\| \partial^{k+h}_{x}H(t)*u_{0}\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{4}-\frac {k}{2}-j}\|u_{0}\|_{H^{s}} $$

for \(0\leq h\leq s-2k-[\frac{n+1}{2}]- 2j\) and \(0\leq j\leq1\). This proves (3.7). The theorem is proved. □

4 Global existence and decay estimates

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. □