Research
Open Access
Decay estimate of solutions for a semi-linear wave equation
- Hengjun Zhao1Email author and
- Jiya Nuen2
Boundary Value Problems20152015:10
https://doi.org/10.1186/s13661-014-0272-6
© Zhao and Nuen; licensee Springer 2015
Received: 20 September 2014
Accepted: 20 December 2014
Published: 16 January 2015
Abstract
In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Under a smallness condition on the initial value, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates for the spatial derivatives of the solution are provided. The proof is carried out by means of the decay property of the solution operator and a fixed point-contraction mapping argument.
Keywords
semi-linear wave equationglobal existenceasymptotic decay
MSC
35L3035L75
1 Introduction
In this paper, we investigate the initial value problem for the following semi-linear wave equation:
with the initial value
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 \(u \rightarrow0\).
$$ u_{tt}-\Delta u-\Delta u_{tt}+u_{t}= \Psi(u) $$
(1.1)
$$ t=0: u=u_{0}(x),\qquad u_{t}=u_{1}(x), $$
(1.2)
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,
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
The decay structure of (1.1) is characterized by
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
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.
$$ u_{tt} - \Delta u + u_{t} = \Psi(u). $$
(1.3)
$$ \operatorname{Re} \lambda(\xi) \leq-c|\xi|^{2}/ \bigl(1 + |\xi|^{2} \bigr). $$
(1.4)
$$ \operatorname{Re} \lambda(\xi) \leq-c|\xi|^{2}/ \bigl(1 + |\xi|^{2} \bigr)^{2}. $$
(1.5)
$$u_{tt}-\Delta u-\Delta u_{tt}-\Delta u_{t}= \Psi(u). $$
If the nonlinear term \(\Psi(u)\) is replaced by \(\Psi(u_{t})\), then (1.1) becomes
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.
$$ u_{tt}-\Delta u-\Delta u_{tt}+u_{t}= \Psi(u_{t}). $$
(1.6)
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 \(n\geq1\):
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.
$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq CE_{0}(1+t)^{-\frac {k}{2}-\frac{n}{4}} $$
(1.7)
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
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 \(x \in \mathbb{R}^{n}\).
$$\hat{u}(\xi)=F[u]=\int_{\mathbb{R}^{n}}e^{-i\xi\cdot x}u(x)\,dx, $$
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):
with the initial data (1.2).
$$ u_{tt}-\Delta u-\Delta u_{tt}+u_{t}=0, $$
(2.1)
We apply the Fourier transform to (2.1), (1.2); it yields
The corresponding characteristic equation of (2.2) is
Let \(\lambda=\lambda_{\pm}(\xi)\) be the solutions to (2.4). It is not difficult to find
The solution to (2.2), (2.3) in the Fourier space is then given explicitly in the form
where
and
Let
and
where \(F^{-1}\) denotes the inverse Fourier transform. Then we apply \(F^{-1}\) to (2.6), and it yields
Now we return to our nonlinear problem (1.1), (1.2). By the Duhamel principle, the problem (1.1), (1.2) is equivalent to
$$\begin{aligned}& \bigl(1+|\xi|^{2} \bigr)\hat{u}_{tt}+\hat{u}_{t}+| \xi|^{2}\hat{u}=0, \end{aligned}$$
(2.2)
$$\begin{aligned}& t=0: \hat{u}=\hat{u}_{0}(\xi), \qquad\hat{u}_{t}= \hat{u}_{1}(\xi). \end{aligned}$$
(2.3)
$$ \bigl(1+|\xi|^{2} \bigr)\lambda^{2}+\lambda+| \xi|^{2}=0. $$
(2.4)
$$ \lambda_{\pm}(\xi)=\frac{-1\pm\sqrt{1-4|\xi|^{2}(1+|\xi|^{2})}}{2(1+|\xi |^{2})}. $$
(2.5)
$$ \hat{u}(\xi, t)=\hat{G}(\xi, t) \bigl(\hat{u}_{0}(\xi)+ \hat{u}_{1}(\xi) \bigr)+\hat {H}(\xi, t)\hat{u}_{0}(\xi), $$
(2.6)
$$ \hat{G}(\xi, t)=\frac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)} \bigl(e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t} \bigr) $$
(2.7)
$$ \hat{H}(\xi, t)=\frac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)} \bigl( \bigl(1+\lambda _{+}(\xi) \bigr)e^{\lambda_{-}(\xi)t}- \bigl(1+\lambda_{-}(\xi) \bigr)e^{\lambda_{+}(\xi)t} \bigr). $$
(2.8)
$$G(x, t) = F^{-1} \bigl[\hat{G}(\xi, t) \bigr](x) $$
$$H(x, t) = F^{-1} \bigl[\hat{H}(\xi, t) \bigr](x), $$
$$ u(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}. $$
(2.9)
$$ u(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}+\int ^{t}_{0}G(t-\tau)*(1-\Delta)^{-1}\Psi (u) (\tau)\,d\tau. $$
(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
for
\(\xi\in\mathbb{R}^{n}\)
and
\(t\geq0\), where
\(\omega(\xi)=\frac{|\xi |^{2}}{(1+|\xi|^{2})^{2}}\).
$$ \bigl|\hat{u}_{t}(\xi, t)\bigr|^{2}+\bigl|\hat{u}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t} \bigl(\bigl|\hat {u}_{1}(\xi)\bigr|^{2}+\bigl| \hat{u}_{0}(\xi)\bigr|^{2} \bigr), $$
(2.11)
Proof
By multiplying (2.2) by \(\bar{\hat{u}}_{t}\) and taking the real part, we deduce that
We multiply (2.2) by \(\bar{\hat{u}}\) and take the real part. This yields
Then by (2.12) and (2.13), we have
where
and
Let
It is easy to see that
Noting that \(F \geq|\xi|^{2}E_{0}\) and with (2.15), we obtain
where
Combining (2.14) and (2.16) yields
Thus
which together with (2.15) proves the desired estimate (2.11). Then we have completed the proof of the lemma. □
$$ \frac{1}{2}\frac{d}{dt} \bigl\{ \bigl(1+|\xi|^{2} \bigr)| \hat{u}_{t}|^{2}+|\xi|^{2}|\hat{u}|^{2} \bigr\} +|\hat{u}_{t}|^{2}=0. $$
(2.12)
$$ \frac{1}{2}\frac{d}{dt} \bigl\{ |\hat{u}|^{2}+2 \bigl(1+| \xi|^{2} \bigr)\operatorname{Re}(\hat{u}_{t}\bar{\hat {u}}) \bigr\} + | \xi|^{2}|\hat{u}|^{2}- \bigl(1+|\xi|^{2} \bigr)| \hat{u}_{t}|^{2}=0. $$
(2.13)
$$ \frac{d}{dt}E+F=0, $$
(2.14)
$$E= \bigl(1+|\xi|^{2} \bigr)^{2}|\hat{u}_{t}|^{2}+ \biggl[\frac{1}{2}+ \bigl(1+|\xi|^{2} \bigr)|\xi|^{2} \biggr]|\hat {u}|^{2}+ \bigl(1+|\xi|^{2} \bigr)\operatorname{Re}( \hat{u}_{t}\bar{\hat{u}}) $$
$$F= \bigl(1+|\xi|^{2} \bigr)|\hat{u}_{t}|^{2}+| \xi|^{2}|\hat{u}|^{2}. $$
$$E_{0}=|\hat{u}_{t}|^{2}+|\hat{u}|^{2}. $$
$$ C \bigl(1+|\xi|^{2} \bigr)^{2}E_{0}\leq E\leq C \bigl(1+|\xi|^{2} \bigr)^{2}E_{0}. $$
(2.15)
$$ F\geq c\omega(\xi)E, $$
(2.16)
$$\omega(\xi)=\frac{|\xi|^{2}}{(1+|\xi|^{2})^{2}}. $$
$$\frac{d}{dt}E+c\omega(\xi)E\leq0. $$
$$E(\xi, t)\leq e^{-cw(\xi)t}E(\xi, 0), $$
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
and
for
\(\xi\in\mathbb{R}^{n}\)
and
\(t\geq0\), where
\(\omega(\xi)=\frac{|\xi |^{2}}{(1+|\xi|^{2})^{2}}\).
$$ \bigl|\hat{G}(\xi, t)\bigr|^{2}+\bigl|\hat{G}_{t}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t} $$
(2.17)
$$ \bigl|\hat{H}(\xi, t)\bigr|^{2}+\bigl|\hat{H}_{t}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t} $$
(2.18)
Proof
If \(\hat{u}_{0}(\xi)=0\), from (2.5), we obtain
Substituting the equalities into (2.11) with \(\hat{u}_{0}(\xi)=0\), we get (2.17).
$$\hat{u}(\xi, t)=\hat{G}(\xi, t)\hat{u}_{1}(\xi),\qquad \hat{u}_{t}( \xi, t)=\hat{G}_{t}(\xi, t)\hat{u}_{1}(\xi). $$
In the following, we consider \(\hat{u}_{1}(\xi)=0\); it follows from (2.5) that
Substituting the equalities into (2.11) with \(\hat{u}_{1}(\xi)=0\), we get
which together with (2.17) proves the desired estimate (2.18). The lemma is proved. □
$$\hat{u}(\xi, t)= \bigl(\hat{G}(\xi, t)+\hat{H}(\xi, t) \bigr) \hat{u}_{0}( \xi),\qquad \hat {u}_{t}(\xi, t)= \bigl( \hat{G}_{t}(\xi, t)+ \hat{H}_{t}(\xi, t) \bigr) \hat{u}_{0}(\xi). $$
$$\bigl|\hat{G}(\xi, t)+\hat{H}(\xi, t)\bigr|^{2}+\bigl|\hat{G}_{t}(\xi, t)+ \hat{H}_{t}(\xi, t)\bigr|^{2} \leq Ce^{-c\omega(\xi)t}, $$
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:
and
$$\begin{aligned}& \bigl|\hat{H}(\xi, t)\bigr|\leq C|\xi|^{2}e^{-c|\xi|^{2}t}+Ce^{-ct}, \end{aligned}$$
(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)
$$ \bigl|\hat{H}_{t}(\xi, t)\bigr|\leq C|\xi|^{4}e^{-c|\xi|^{2}t}+Ce^{-ct}. $$
(2.21)
Lemma 2.4
Let
\(1\leq p\leq2\)
and
\(k\geq0\). Then for
\(l\geq0\)
we have
and
Moreover, we have
and
$$ \bigl\| \partial^{k}_{x}G(t)*\phi\bigr\| _{L^{2}}\leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}}\|\phi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\phi\bigr\| _{L^{2}} $$
(2.22)
$$ \bigl\| \partial^{k}_{x}H(t)*\psi\bigr\| _{L^{2}}\leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}-1}\|\psi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\psi\bigr\| _{L^{2}}. $$
(2.23)
$$ \bigl\| \partial^{k}_{x}G_{t}(t)*\phi\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}-1}\|\phi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\phi\bigr\| _{L^{2}} $$
(2.24)
$$ \bigl\| \partial^{k}_{x}H_{t}(t)*\psi\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{2}(\frac {1}{p}-\frac{1}{2})-\frac{k}{2}-2}\|\psi\|_{L^{p}} +C(1+t)^{-\frac{l}{2}}\bigl\| \partial^{k+l}_{x}\psi\bigr\| _{L^{2}}. $$
(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
where \(R_{0}\) is a positive constant in Lemma 2.3.
$$\begin{aligned} \bigl\| \partial^{k}_{x}H(t)*\psi\bigr\| ^{2}_{L^{2}} =& \int_{\mathbb{R}^{n}}|\xi|^{2k}\bigl|\hat{H}(\xi, t)\bigr|^{2} \bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ =& \int_{|\xi|\leq R_{0}}|\xi|^{2k}\bigl|\hat{H}(\xi, t)\bigr|^{2}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ &{} + \int_{|\xi|\geq R_{0}}|\xi|^{2k}\bigl|\hat{H}(\xi, t)\bigr|^{2}\bigl|\hat {\psi}(\xi)\bigr|^{2}\,d\xi \\ \triangleq& I_{1}+I_{2}, \end{aligned}$$
(2.26)
In the following, we estimate \(I_{1}\). Using (2.19) and the Hölder inequality, we obtain
where \(\frac{1}{q}+\frac{2}{p'}=1\) and \(\frac{1}{p}+\frac{1}{p'}=1\).
$$\begin{aligned} I_{1} \leq& C\int_{|\xi|\leq R_{0}}|\xi |^{2k+4}e^{-c|\xi|^{2}t}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi+ C\int_{|\xi|\leq R_{0}}|\xi |^{2k}e^{-ct}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ \leq& C\bigl\| |\xi|^{2k+4}e^{-c|\xi|^{2}t}\bigr\| _{L^{q}(|\xi|\leq R_{0})}\bigl\| \hat{\psi}(\xi)\bigr\| ^{2}_{L^{p'}} +C\bigl\| |\xi|^{2k}e^{-ct}\bigr\| _{L^{q}(|\xi |\leq R_{0})}\bigl\| \hat{\psi}(\xi)\bigr\| ^{2}_{L^{p'}}, \end{aligned}$$
By a straightforward computation, we get
and
The Hausdorff-Young inequality gives
Thus, we have
$$\bigl\| |\xi|^{2k+4}e^{-c|\xi|^{2}t}\bigr\| _{L^{q}(|\xi|\leq R_{0})} \leq C(1+t)^{-\frac{n}{2q}-k-2} $$
$$\bigl\| |\xi|^{2k}e^{-c|\xi|^{2}t}\bigr\| _{L^{q}(|\xi|\leq R_{0})} \leq Ce^{-ct}. $$
$$\bigl\| \hat{\psi}(\xi)\bigr\| _{L^{p'}}\leq C\|\psi\|_{L^{p}}. $$
$$\begin{aligned} I_{1} \leq& C(1+t)^{-\frac{n}{2q}-k-2}\| \psi\|^{2}_{L^{p}} \\ =&C(1+t)^{-\frac{n}{2}(\frac{2}{p}-1)-k-2}\|\psi\|^{2}_{L^{p}}. \end{aligned}$$
(2.27)
Note that \(\omega(\xi)\geq c|\xi|^{-2}\) when \(|\xi|\geq R_{0}\). By (2.17), we arrive at
Combining (2.27) and (2.28) yields (2.23). Thus we have completed the proof of the lemma. □
$$\begin{aligned} I_{2} \leq& C\int_{|\xi|\geq R_{0}}|\xi |^{2k}e^{-c|\xi|^{-2}t}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ \leq& C\sup_{|\xi|\geq R_{0}}\bigl(|\xi|^{-2l}e^{-c|\xi |^{-2}t}\bigr)\int_{|\xi|\geq R_{0}}|\xi|^{2(k+l)}\bigl|\hat{\psi}(\xi)\bigr|^{2}\,d\xi \\ \leq& C(1+t)^{-l}\bigl\| \partial^{k+l}_{x}\psi\bigr\| ^{2}_{L^{2}}. \end{aligned}$$
(2.28)
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
Then the solution
\(u(x, t)\)
of the linear problem (2.1), (1.2), which is given by (2.9), satisfies the decay estimate
for
\(k\geq0\)
and
\(2k+[\frac{n+1}{2}]\leq s\). Moreover, for each
j
with
\(0\leq j\leq1\), we have
for
\(k\geq0\)
and
\(2k+[\frac{n+1}{2}]+2j\leq s\).
$$E_{0}=\|u_{0}\|_{H^{s}\cap L^{1}}+\|u_{1} \|_{H^{s}\cap L^{1}}. $$
$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq CE_{0}(1+t)^{-\frac {n}{4}-\frac{k}{2}} $$
(3.1)
$$ \bigl\| \partial^{k}_{x}u_{t}(t)\bigr\| _{H^{s-2k-[\frac{n+1}{2}]-2j}}\leq CE_{0}(1+t)^{-\frac{n}{4}-\frac{k}{2}-j} $$
(3.2)
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:
and
Firstly, we prove (3.5). Let \(k\geq0\) and \(h\geq0\), we have from (2.22) with k replaced by \(k+h\) and with \(p=1\)
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
for \(0\leq h\leq s-2k-[\frac{n+1}{2}]\). Then (3.5) is proved.
$$\begin{aligned}& \bigl\| \partial^{k}_{x}G(t)*u_{1}\bigr\| _{H^{s-2k-[\frac{n+1}{2}]}}\leq C(1+t)^{-\frac {n}{4}-\frac{k}{2}}\|u_{1}\|_{H^{s}\cap L^{1}}, \end{aligned}$$
(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)
$$ \bigl\| \partial^{k}_{x}H(t)*u_{0}\bigr\| _{H^{s-2k-[\frac{n+1}{2}]-2j}} \leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}-j}\|u_{0}\|_{H^{s}\cap L^{1}},\quad 0\leq j\leq1. $$
(3.7)
$$\bigl\| \partial^{k+h}_{x}G(t)*u_{1}\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{4}-\frac {k+h}{2}}\|u_{1}\|_{L^{1}}+C(1+t)^{-\frac{l}{2}} \bigl\| \partial^{k+h+l}_{x}u_{1}\bigr\| _{L^{2}}, $$
$$\bigl\| \partial^{k+h}_{x}G(t)*u_{1}\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{4}-\frac {k}{2}}\|u_{1}\|_{H^{s}\cap L^{1}} $$
Similarly, we can prove (3.6).
Finally, we prove (3.7). Let \(k\geq0\) and \(h\geq0\). It follows from (2.23) that
where \(l\geq0\) and \(k+h+l\leq s\).
$$\bigl\| \partial^{k+h}_{x}H(t)*u_{0}\bigr\| _{L^{2}} \leq c(1+t)^{-\frac{n}{4}-\frac {k+h}{2}-1}\|u_{0}\|_{L^{1}}+C(1+t)^{-\frac{l}{2}} \bigl\| \partial^{k+h+l}_{x}u_{0}\bigr\| _{L^{2}}, $$
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
Then we have
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. □
$$0\leq h\leq s-2k- \biggl[\frac{n+1}{2} \biggr]-2j. $$
$$\bigl\| \partial^{k+h}_{x}H(t)*u_{0}\bigr\| _{L^{2}} \leq C(1+t)^{-\frac{n}{4}-\frac {k}{2}-j}\|u_{0}\|_{H^{s}} $$
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
where
where
$$\mathcal{E}= \bigl\{ u\in C \bigl([0, \infty ); H^{s} \bigl( \mathbb{R}^{n} \bigr)\bigr): \|u\|_{\mathcal{E}}<\infty \bigr\} , $$
$$\begin{aligned} \|u\|_{\mathcal{E}}= \sum_{\varrho (k,n)\leq s}\sup_{t\geq0}(1+t)^{\frac{n}{4}+\frac{k}{2}}\bigl\| \partial ^{k}_{x}u(t)\bigr\| _{H^{s-\varrho(k, n)}} + \sum^{[\frac{n}{4}]}_{j=0}\sum_{2k+2j\leq s} \sup_{t\geq0}(1+t)^{\frac{k}{2}+j}\bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-2j}}, \end{aligned}$$
$$\varrho(k, n)=2k+ \biggl[\frac{n+1}{2} \biggr]. $$
For \(\mathfrak{R}>0\), we define
where ℜ depends on the initial value, which is chosen in the proof of the main result.
$$\mathcal{E}_{\mathfrak{R}}=\bigl\{ u\in X: \|u\|_{\mathcal{E}}\leq\mathfrak {R}\bigr\} , $$
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
If
\(E_{0}\)
is suitably small, the initial value problem (1.1), (1.2) has a unique global solution
Moreover, the solution satisfies the decay estimate
for
\(k\geq0\), \(0\leq j\leq[\frac{n}{4}]\), and
\(2k+2j\leq s\). Also, we have
for
\(k\geq0\)
and
\(\varrho(k, n)\leq s\).
$$E_{0}=\|u_{0}\|_{H^{s}\cap L^{1}}+\|u_{1} \|_{H^{s}\cap L^{1}}. $$
$$u\in C^{0} \bigl([0, \infty ); H^{s} \bigl( \mathbb{R}^{n} \bigr)\bigr). $$
$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-2k-2j}}\leq CE_{0}(1+t)^{-\frac{k}{2}-j} $$
(4.1)
$$ \bigl\| \partial^{k}_{x}u(t)\bigr\| _{H^{s-\varrho(k, n)}}\leq CE_{0}(1+t)^{-\frac {n}{4}-\frac{k}{2}} $$
(4.2)
Proof
We shall prove Theorem 4.1 by the Banach fixed point theorem. Thus we define the mapping
and
For \(\forall v, w\in\mathcal{E}\), we arrive at
For \(\Psi(u)=O(|u|^{\theta})\), it is not difficult to get the following nonlinear estimates (see [18]):
and
$$\mathcal{M}[u](t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}+ \int^{t}_{0}G(t-\tau)*(1-\Delta )^{-1} \Psi(u) (\tau)\,d\tau $$
$$\mathcal{M}_{0}(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}. $$
$$\mathcal{M}[v](t)-\mathcal{M}[w](t)=\int^{t}_{0}G(t- \tau)*(1-\Delta )^{-1} \bigl(\Psi(v)-\Psi(w) \bigr) (\tau)\,d\tau. $$
$$\begin{aligned} &\bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{1}} \\ &\quad\leq C\bigl\| (v, w)\bigr\| ^{\theta -2}_{L^{\infty}} \bigl(\bigl\| (v, w) \bigr\| _{L^{2}}\bigl\| \partial^{k}_{x}(v-w)\bigr\| _{L^{2}}+\bigl\| \partial _{x}^{k}(v, w)\bigr\| _{L^{2}}\|v-w \|_{L^{2}} \bigr) \end{aligned}$$
(4.3)
$$\begin{aligned} &\bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{2}} \\ &\quad\leq C\bigl\| (v, w)\bigr\| ^{\theta -2}_{L^{\infty}} \bigl(\bigl\| (v, w) \bigr\| _{L^{\infty}}\bigl\| \partial^{k}_{x}(v-w)\bigr\| _{L^{2}}+\bigl\| \partial_{x}^{k}(v, w)\bigr\| _{L^{2}}\|v-w \|_{L^{\infty}} \bigr). \end{aligned}$$
(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
where \(0\leq j\leq[\frac{n}{4}]\), \(k\geq0\), and \(2k+2j\leq s\).
$$ \bigl\| \partial^{k}_{x} \bigl(\mathcal{M}[v]-\mathcal{M}[w] \bigr) \bigr\| _{H^{s-2k-2j}}\leq C(1+t)^{-\frac{k}{2}-j} \bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \|v-w\| _{\mathcal{E}}, $$
(4.6)
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
By (2.22), we obtain
Using (4.3) and (4.5), we arrive at
Noting that \(\theta> 1+\frac{2}{n}\) for \(n=1, 2\) and \(\theta\geq2\) for \(n\geq3\), using (4.9), we deduce that
If \(k+m+l-2\leq s\), (4.4) gives
Take \(l=k+2j+2\). By (4.11) and noticing \(\theta\geq2\), we obtain
with \(0\leq m\leq s-2k-2j\).
$$\begin{aligned} & \bigl\| \partial^{k+m}_{x}\bigl(\mathcal{M}[v]-\mathcal {M}[w]\bigr)(t)\bigr\| _{L^{2}} \\ &\quad\leq \int^{\frac{t}{2}}_{0}\bigl\| \partial^{k+m}_{x}G(t-\tau )*(1-\Delta)^{-1}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{2}}\,d\tau \\ &\qquad{}+ \int^{t}_{\frac{t}{2}}\bigl\| \partial^{k+m}_{x}G(t-\tau )*(1-\Delta)^{-1}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{2}}\,d\tau \\ &\quad\triangleq J_{1}+J_{2}. \end{aligned}$$
(4.7)
$$\begin{aligned} J_{1} \leq& C\int^{\frac {t}{2}}_{0}(1+t-\tau)^{-\frac{n}{4}-\frac{k+m}{2}}\bigl\| \Psi(v)-\Psi(w)\bigr\| _{L^{1}}\,d\tau \\ &{}+ C\int^{\frac{t}{2}}_{0}(1+t-\tau)^{-\frac{l}{2}}\bigl\| \partial_{x}^{k+m+l-2}\bigl(\Psi(v)-\Psi(w)\bigr)\bigr\| _{L^{2}}\,d\tau \\ \triangleq& J_{11}+J_{12}. \end{aligned}$$
(4.8)
$$ \bigl\| \Psi(v)-\Psi(w)\bigr\| _{L^{1}}\leq C\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \|v-w\| _{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta-1)}. $$
(4.9)
$$ J_{11}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.10)
$$ \bigl\| \partial^{k+m+l-2}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) (\tau) \bigr\| _{L^{2}}\leq C\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac {n}{2}(\theta-1)}. $$
(4.11)
$$J_{12}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}} $$
Combining (4.10) and (4.11) yields
with \(0\leq m\leq s-2k-j\). We apply \(l=2\) and \(p=1\) to the term \(J_{2}\). This yields
It follows from (4.3) and (4.5) that
where \(0\leq j\leq[\frac{n}{4}]\) and \(2k+2j\leq s\).
$$J_{1}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}} $$
$$\begin{aligned} J_{2} \leq& C\int^{t}_{\frac{t}{2}}(1+t-\tau )^{-\frac{n}{4}-\frac{m}{2}}\bigl\| \partial^{k}_{x}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{1}}\,d\tau \\ &{}+ C\int^{t}_{\frac{t}{2}}(1+t-\tau)^{-1}\bigl\| \partial ^{k+m}_{x}\bigl(\Psi(v)-\Psi(w)\bigr)(\tau)\bigr\| _{L^{2}}\,d\tau=:J_{21}+J_{22}. \end{aligned}$$
(4.12)
$$ \bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{1}}\leq C\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta -2)-\frac{n}{4}-\frac{k}{2}-j}, $$
(4.13)
Since \(\theta>1+\frac{2}{n}\) for \(n=1, 2\) and \(\theta\geq2\) for \(n\geq 3\), using (4.13), we obtain
for \(0\leq j\leq[\frac{n}{4}]\) and \(2k+2j\leq s\).
$$J_{21}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}} $$
If \(m\leq s-2k-2j\), we apply (4.4) to get
which yields
where \(0\leq m\leq s-2k-2j\).
$$ \bigl\| \partial^{k+m}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) (\tau) \bigr\| _{L^{2}}\leq C\bigl\| (v, w)\bigr\| ^{\theta-1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-d(\theta-1)-\frac {k}{2}-j}, $$
(4.14)
$$J_{22}\leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v ,w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}}, $$
Thus
We immediately get (4.6).
$$ J_{2} \leq C(1+t)^{-\frac{k}{2}-j}\bigl\| (v ,w)\bigr\| ^{\theta-1}_{\mathcal{E}} \| v-w\|_{\mathcal{E}}. $$
(4.15)
In the following we prove
where \(\varrho(k, n)\leq s\).
$$ \bigl\| \partial^{k}_{x} \bigl(\mathcal{M}[v]-\mathcal{M}[w] \bigr) (t)\bigr\| _{H^{s-\varrho(k, n)}}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v, w) \bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}, $$
(4.16)
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
Combining (4.10) and (4.17) yields
Similarly, we have (4.12). Since \(\varrho(k, n)\leq s\), in view of (4.3) and (4.5), we get
which yields
$$ J_{12}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.17)
$$ J_{1}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.18)
$$\bigl\| \partial^{k}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{1}}\leq C\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta -1)-\frac{k}{2}}, $$
$$J_{21}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|. $$
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
which yields
Thus
Equations (4.18) and (4.19) give (4.16).
$$\bigl\| \partial^{k+m}_{x} \bigl(\Psi(v)-\Psi(w) \bigr) \bigr\| _{L^{2}}\leq C\bigl\| (v, w)\bigr\| ^{\theta -1}_{\mathcal{E}}\|v-w \|_{\mathcal{E}}(1+\tau)^{-\frac{n}{2}(\theta -1)-\frac{n}{4}-\frac{k}{2}}, $$
$$J_{22}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
$$ J_{2}\leq C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\bigl\| (v ,w)\bigr\| ^{\theta -1}_{\mathcal{E}} \|v-w\|_{\mathcal{E}}. $$
(4.19)
It follows from (4.6) and (4.16) that
Thus for \(v, w \in X_{R}\), we have
$$ \bigl\| \mathcal{M}[v]-\mathcal{M}[w]\bigr\| _{\mathcal{E}} \leq C\bigl\| (v ,w)\bigr\| ^{\theta-1}_{\mathcal{E}}\|v-w\|_{\mathcal{E}}. $$
(4.20)
$$\bigl\| \mathcal{M}[v]-\mathcal{M}[w]\bigr\| _{\mathcal{E}} \leq C_{1} \mathfrak {R}^{\theta-1}\|v-w\|_{\mathcal{E}}. $$
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
if \(E_{0}\) is suitably small.
$$\bigl\| \mathcal{M}_{0}(t)\bigr\| _{\mathcal{E}}\leq C_{2}E_{0}, $$
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
Therefore, for \(v \in\mathcal{E}_{\mathfrak{R}}\), we have
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. □
$$\bigl\| \mathcal{M}[v]-\mathcal{M}[w]\bigr\| _{X} \leq\frac{1}{2}\|v-w \|_{\mathcal{E}}. $$
$$\bigl\| \mathcal{M}[v]\bigr\| _{\mathcal{E}} \leq\bigl\| \mathcal{M}_{0}(t) \bigr\| _{\mathcal{E}}+\frac{1}{2}\mathfrak{R}\leq C_{2}E_{0}+ \frac{1}{2}\mathfrak{R}=\mathfrak {R}. $$
Declarations
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
(1)
School of Science, Henan Institute of Engineering, Zhengzhou, China
(2)
Department of Physiology, Hetao College, Bayanzhuoer, China
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