- Research
- Open Access
Local well-posedness of generalized BBM equations with generalized damping on 1D torus
- Junjun Kang1,
- Yantao Guo2 and
- Yanbin Tang1Email author
- Received: 24 September 2015
- Accepted: 20 November 2015
- Published: 3 December 2015
Abstract
We consider the periodic initial value problem associated to the generalized Benjamin-Bona-Mahony equation with generalized damping on the one dimensional torus. In contrast to the classical BBM equation, the main difference is that the generalized equation contains two nonlocal operators, and the main difficulty comes from two nonlocal operators. By the fixed point theorem, we prove that the periodic initial value problem is locally well-posed. We also prove that if the solution exists globally in time, it exhibits some asymptotic behavior.
Keywords
- generalized Benjamin-Bona-Mahony equation
- nonlocal operator
- generalized damping
- local well-posedness
MSC
- 35A07
- 35Q53
1 Introduction
The classical Benjamin-Bona-Mahony (BBM) equation was proposed in [1] as a model for propagation of long waves which incorporates nonlinear dispersive and dissipative effects. It has extensively been studied in the recent literature; see for example [1–10] on the existence and uniqueness of solutions and [11–17] on the global attractors and references therein.
$$\begin{aligned} u_{t}-u_{xxt}+u_{x}+uu_{x}=0 \end{aligned}$$
(1.1)
In this paper, we consider the periodic initial value problem of generalized BBM equations with generalized damping on the 1D torus \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\): where the two nonlocal operators are defined by and \(\widehat{u}(k)\) is the kth Fourier coefficient of \(u(t,x)\) in x.
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} u_{t}+L_{p}u_{t}+u_{x}+uu_{x}+M_{\alpha}u=0,\quad x\in\mathbb {T}, t\in(0,T),\\ u(0,x)=u_{0}(x),\quad x\in\mathbb{T},\\ \int_{\mathbb{T}}u(t,x)\,dx=0, \end{array}\displaystyle \right . \end{aligned}$$
(1.2)
$$\begin{aligned} &\widehat{L_{p}u}(k)=|k|^{p+1}\widehat{u}(k),\quad p>0; \\ &\widehat{M_{\alpha}u}(k)=|k|^{2\alpha}\widehat{u}(k), \quad \alpha\in \biggl(0,\frac{p+1}{2}\biggr], \end{aligned}$$
(1.3)
For \(\alpha=1\), the generalized damping becomes a parabolic damping,
$$\begin{aligned} M_{1}u=-u_{xx} \quad\mbox{as } \widehat{M_{1}u}(k)=|k|^{2} \widehat{u}(k). \end{aligned}$$
For \(\alpha=0\), it is a weak damping, For example, Wang [18] considered the damped BBM equation \(u_{t}-u_{txx}+\gamma(u-u_{xx})+uu_{x}=f(x)\) (Introduction, p.134) and the BBM equation with different damping coefficients \(u_{t}-u_{txx}+\gamma u-\nu u_{xx}+uu_{x}=f(x)\) (Remark 3.2, p.142).
$$\begin{aligned} M_{0}u=u \quad\mbox{as } \widehat{M_{0}u}(k)=\widehat{u}(k). \end{aligned}$$
In fact, one can consider more general damping terms. For example, Chehab et al. [19] studied the long-time behavior of the solution of a damped BBM equation with and \((\gamma_{k})_{k\in\mathbb{Z}}\) are positive real numbers.
$$\begin{aligned} u_{t}-u_{xxt}+u_{x}+uu_{x}+M_{\alpha}u=0,\quad x\in\mathbb{T}[0,L], t\in(0,T), \end{aligned}$$
(1.4)
$$\begin{aligned} \widehat{M_{\alpha}u}(k)=\gamma_{k}\widehat{u}(k), \end{aligned}$$
In the absence of fractional damping \(M_{\alpha}u\), Carvajal and Panthee [20] proved that the Cauchy problem is ill-posed for data with lower order Sobolev regularity and in a certain range of the Sobolev regularity, even if the solution exists globally in time, it fails to be smooth.
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} u_{t}+L_{p}u_{t}+u_{x}+(u^{k+1})_{x}=0, \quad x\in\mathbb{R}, t\in(0,T), k\in\mathbb{Z}^{+},\\ u(0,x)=u_{0}(x),\quad x\in\mathbb{R}, \end{array}\displaystyle \right . \end{aligned}$$
(1.5)
In this paper we study the generalized BBM equations with the fractional damping terms. In contrast to the classical BBM equation, the main difference is that equation (1.2) contains two nonlocal operators, its dissipation is weaker than the classical BBM equation. In the study of the periodic initial value problem (1.2), the main difficulty is that \(L_{p}\) and \(M_{\alpha}\) are nonlocal operators. By the fixed point theorem and the Fourier analysis method, similar to [21–23], we prove the local well-posedness of the solution to the problem (1.2).
2 Local well-posedness
We define a space with the norm
$$\begin{aligned} \dot{H}^{\beta}(\mathbb{T})=\biggl\{ u\in L^{2}(\mathbb{T}): \int_{\mathbb {T}}u\,dx=0, \sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\beta}\bigl| \widehat {u}(k)\bigr|^{2}< +\infty\biggr\} \end{aligned}$$
$$\begin{aligned} |u|_{\beta}^{2}=\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\beta}\bigl| \widehat {u}(k)\bigr|^{2}. \end{aligned}$$
Then we can obtain the local existence and uniqueness of the solution to the periodic initial value problem (1.2).
Theorem 2.1
Assume
\(\alpha\in(\frac{1}{2}, \frac {p+1}{2}]\). If
\(u_{0}(x)\in\dot{H}^{\alpha}(\mathbb{T})\), there exist a constant
\(T=C_{0}(|u_{0}|_{\alpha})>0\)
and a unique solution
\(u(t,x)\in C([0,T], \dot{H}^{\alpha}(\mathbb{T}))\). Moreover, for any constant
\(M>0\), \(|u_{0}|_{\alpha}\leq M\), \(|v_{0}|_{\alpha}\leq M\), there exists a constant
\(C_{1}>0\)
such that the solutions
\(u(t,x)\), \(v(t,x)\)
of the periodic initial value problem (1.2) with the initial data
\(u_{0}(x)\in\dot{H}^{\alpha}(\mathbb{T})\)
and
\(v_{0}(x)\in\dot{H}^{\alpha}(\mathbb{T})\), respectively, satisfy
$$\begin{aligned} \bigl|u(t)-v(t)\bigr|_{\alpha}\leq C_{1}|u_{0}-v_{0}|_{\alpha},\quad \forall t\leq \frac{1}{C_{0}M}. \end{aligned}$$
(2.1)
Proof
We first write equation (1.2) in the following form: where Then we get where
$$\begin{aligned} u_{t} =&-(I+L_{p})^{-1} \partial_{x}u-(I+L_{p})^{-1}\partial_{x} \biggl(\frac {u^{2}}{2}\biggr)-(I+L_{p})^{-1}M_{\alpha}u \\ =&-i\varphi(D_{x})u-i\varphi(D_{x}) \biggl( \frac{u^{2}}{2}\biggr)-\varphi_{\alpha }(D_{x})u \\ =&-\bigl[i\varphi(D_{x})+\varphi_{\alpha}(D_{x}) \bigr]u-i\varphi(D_{x}) \biggl(\frac{u^{2}}{2}\biggr), \end{aligned}$$
(2.2)
$$\begin{aligned} &\widehat{\varphi(D_{x})u}(k)=\frac{k}{1+|k|^{p+1}} \widehat {u}(k), \\ &\widehat{\varphi_{\alpha}(D_{x})u}(k)=\frac{|k|^{2\alpha }}{1+|k|^{p+1}} \widehat{u}(k), \quad k\in\mathbb{Z}. \end{aligned}$$
(2.3)
$$\begin{aligned} u(t)=S_{t}u_{0}-i \int_{0}^{t}S_{t-\tau}\varphi(D_{x}) \biggl(\frac {u^{2}}{2}\biggr)\,d\tau, \end{aligned}$$
(2.4)
$$\begin{aligned} S_{t}u_{0}=\sum _{k\in\mathbb{Z}\setminus\{0\}}e^{ikx}e^{-\frac {ik+|k|^{2\alpha}}{1+|k|^{p+1}}t}\widehat{u_{0}}(k). \end{aligned}$$
(2.5)
We define a map and a closed ball We now prove that Φ has a unique fixed point in \(\overline {B}(\mathbb{T})\).
$$\begin{aligned} \Phi\bigl(u(t)\bigr)=S_{t}u_{0}-i \int_{0}^{t}S_{t-\tau}\varphi(D_{x}) \biggl(\frac {u^{2}}{2}\biggr)\,d\tau \end{aligned}$$
(2.6)
$$\begin{aligned} \overline{B}(\mathbb{T})= \bigl\{ u(t,x)\in C\bigl([0,T], \dot{H}^{\alpha }(\mathbb{T})\bigr): \bigl|u(t)-u_{0}\bigr|_{\alpha} \leq3|u_{0}|_{\alpha} \bigr\} . \end{aligned}$$
(2.7)
Step one: Φ is onto, that is, for \(u(t)\in\overline{B}(\mathbb {T})\) we have \(\Phi(u(t))\in\overline{B}(\mathbb{T})\).
According to the definition of the norm \(|u(t)|_{\alpha}\), we get and the last inequality comes from the fact that \(\dot{H}^{\alpha}(\mathbb {T})\) is an algebra for \(\alpha>\frac{1}{2}\).
$$\begin{aligned} |S_{t}u_{0}|_{\alpha}^{2} =&\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{2\alpha}\bigl|\widehat{S_{t}u_{0}}(k)\bigr|^{2} \\ =&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha}\bigl|e^{-\frac {ik+|k|^{2\alpha}}{1+|k|^{p+1}}t} \widehat{u_{0}}(k)\bigr|^{2} \\ =&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha}e^{-\frac {2|k|^{2\alpha}}{1+|k|^{p+1}}t}\bigl| \widehat{u_{0}}(k)\bigr|^{2} \\ \leq&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha}\bigl|\widehat {u_{0}}(k)\bigr|^{2}=|u_{0}|_{\alpha}^{2} \end{aligned}$$
$$\begin{aligned} \biggl|S_{t-\tau}\varphi(D_{x}) \biggl(\frac{u^{2}}{2} \biggr)\biggr|_{\alpha}^{2} =&\sum_{k\in \mathbb{Z}\setminus\{0\}}|k|^{2\alpha}\bigl| \widehat{S_{t-\tau}\varphi (D_{x}) \bigl(u^{2}/2 \bigr)}(k)\bigr|^{2} \\ =&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha}\biggl|e^{-\frac {ik+|k|^{2\alpha}}{1+|k|^{p+1}}(t-\tau)} \frac{k}{1+|k|^{p+1}}\widehat {\bigl(u^{2}/2\bigr)}(k)\biggr|^{2} \\ \leq&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha}\bigl|\widehat { \bigl(u^{2}/2\bigr)}(k)\bigr|^{2} \\ =&\frac{1}{4}\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha}\bigl| \widehat {u^{2}}(k)\bigr|^{2} \\ =&\frac{1}{4}\bigl|u^{2}\bigr|_{\alpha}^{2}\leq C|u|_{\alpha}^{4}, \end{aligned}$$
Putting the above two inequalities into (2.6) we have Since \(u(t)\in\overline{B}(\mathbb{T})\) and we have \(|u(t)|_{\alpha}\leq4|u_{0}|_{\alpha}\) and Therefore, for \(T\in(0,\frac{1}{16C|u_{0}|_{\alpha}})\), we have \(\Phi (u(t))\in\overline{B}(\mathbb{T})\) for \(u(t)\in\overline{B}(\mathbb{T})\).
$$\begin{aligned} \bigl|\Phi(u)\bigr|_{\alpha} \leq& |S_{t}u_{0}|_{\alpha}+ \int_{0}^{t}\biggl|S_{t-\tau }\varphi(D_{x}) \biggl(\frac{u^{2}}{2}\biggr)\biggr|_{\alpha}\,d\tau \\ \leq& |u_{0}|_{\alpha}+C \int_{0}^{t}\bigl|u(\tau)\bigr|^{2}_{\alpha}\,d\tau \\ \leq& |u_{0}|_{\alpha}+CT\sup_{t\in[0,T]}\bigl|u(t)\bigr|^{2}_{\alpha}. \end{aligned}$$
(2.8)
$$\begin{aligned} \bigl|u(t)\bigr|_{\alpha}-|u_{0}|_{\alpha}\leq\bigl|u(t)-u_{0}\bigr|_{\alpha} \leq 3|u_{0}|_{\alpha}, \end{aligned}$$
$$\begin{aligned} \bigl|\Phi\bigl(u(t)\bigr)-u_{0}\bigr|_{\alpha} \leq& \bigl|\Phi\bigl(u(t) \bigr)\bigr|_{\alpha}+|u_{0}|_{\alpha } \leq 2|u_{0}|_{\alpha}+16CT|u_{0}|_{\alpha}^{2} \\ \leq& 3|u_{0}|_{\alpha}, \quad\mbox{if } 0< T< \frac{1}{16C|u_{0}|_{\alpha}}. \end{aligned}$$
Step two: Φ is a contractive mapping on \(\overline{B}(\mathbb{T})\).
Let \(u(t), v(t)\in\overline{B}(\mathbb{T})\). Since due to \(u(t), v(t)\in\overline{B}(\mathbb{T})\), we have that is,
$$\begin{aligned} \bigl|\Phi\bigl(u(t)\bigr)-\Phi\bigl(v(t)\bigr)\bigr|_{\alpha}^{2} &=\biggl| \int_{0}^{t}S_{t-\tau}\varphi(D_{x}) \biggl(\frac{u^{2}}{2}-\frac {v^{2}}{2}\biggr)\,d\tau\biggr|_{\alpha}^{2} \\ &\leq \int_{0}^{t}\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha }\biggl|e^{-\frac{ik+|k|^{2\alpha}}{1+|k|^{p+1}}(t-\tau)} \frac {k}{1+|k|^{p+1}}\widehat{\biggl(\frac{u^{2}}{2}-\frac{v^{2}}{2} \biggr)}(k)\biggr|^{2}\,d\tau \\ &\leq C \int_{0}^{t}\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha }\bigl| \widehat{\bigl(u^{2}-v^{2}\bigr)}(k)\bigr|^{2}\,d\tau \\ &=C \int_{0}^{t}\bigl|u^{2}-v^{2}\bigr|^{2}_{\alpha}\,d\tau \\ &\leq 2C \int_{0}^{t}\bigl(|u|^{2}_{\alpha}+|v|^{2}_{\alpha } \bigr)|u-v|^{2}_{\alpha}\,d\tau, \end{aligned}$$
$$\begin{aligned} \bigl|\Phi\bigl(u(t)\bigr)-\Phi\bigl(v(t)\bigr)\bigr|_{\alpha}^{2} \leq32C\bigl(|u_{0}|^{2}_{\alpha }+|v_{0}|^{2}_{\alpha} \bigr)T\sup_{t\in[0,T]}\bigl|u(t)-v(t)\bigr|^{2}_{\alpha}, \end{aligned}$$
(2.9)
$$\begin{aligned} \sup_{t\in[0,T]}\bigl|\Phi\bigl(u(t)\bigr)-\Phi\bigl(v(t) \bigr)\bigr|_{\alpha}\leq4\sqrt {2CT\bigl(|u_{0}|^{2}_{\alpha}+|v_{0}|^{2}_{\alpha} \bigr)}\sup_{t\in [0,T]}\bigl|u(t)-v(t)\bigr|_{\alpha}. \end{aligned}$$
(2.10)
Therefore, Φ is a contractive mapping on \(\overline{B}(\mathbb {T})\) if \(4\sqrt{2CT(|u_{0}|^{2}_{\alpha}+|v_{0}|^{2}_{\alpha})}<1\), that is, Thanks to the Banach fixed point theorem, Φ has a unique fixed point \(u(t)\) such that \(u(t)=\Phi(u(t))\), that is, there exists a unique solution of the periodic initial value problem (1.2).
$$0< T< \frac{1}{32C(|u_{0}|^{2}_{\alpha}+|v_{0}|^{2}_{\alpha})}. $$
Step three: the continuity of solution with the initial data.
Let \(u(t)\) and \(v(t)\) be solutions of the periodic initial value problem (1.2) with the initial data \(u_{0}\) and \(v_{0}\), respectively, such that \(|u_{0}|_{\alpha}\leq M\), \(|v_{0}|_{\alpha}\leq M\). For \(t\in[0,T]\), the Duhamel principle gives us the following formula: hence if \(T<\frac{1}{8C_{0}M}\), there exists a constant \(C_{1}=\frac {1}{1-8C_{0}MT}>0\) such that
$$\begin{aligned} u(t)-v(t)=S_{t}(u_{0}-v_{0})-i \int_{0}^{t}S_{t-\tau}\varphi(D_{x}) \biggl(\frac {u^{2}}{2}-\frac{v^{2}}{2}\biggr)\,d\tau, \end{aligned}$$
(2.11)
$$\begin{aligned} \bigl|u(t)-v(t)\bigr|_{\alpha} \leq& \bigl|S_{t}(u_{0}-v_{0})\bigr|_{\alpha}+ \int _{0}^{t}\biggl|S_{t-\tau}\varphi(D_{x}) \biggl(\frac{u^{2}}{2}-\frac {v^{2}}{2}\biggr)\biggr|_{\alpha}\,d\tau \\ \leq& |u_{0}-v_{0}|_{\alpha}+C_{0} \bigl(\bigl|u(t)\bigr|_{\alpha}+\bigl|v(t)\bigr|_{\alpha }\bigr)T\sup_{t\in[0,T]}\bigl|u(t)-v(t)\bigr|_{\alpha} \\ \leq& |u_{0}-v_{0}|_{\alpha}+4C_{0}\bigl(|u_{0}|_{\alpha}+|v_{0}|_{\alpha }\bigr)T \sup_{t\in[0,T]}\bigl|u(t)-v(t)\bigr|_{\alpha} \\ \leq& |u_{0}-v_{0}|_{\alpha}+8C_{0}MT \sup_{t\in [0,T]}\bigl|u(t)-v(t)\bigr|_{\alpha}, \end{aligned}$$
$$\begin{aligned} \bigl|u(t)-v(t)\bigr|_{\alpha}\leq\sup_{t\in[0,T]} \bigl|u(t)-v(t)\bigr|_{\alpha} \leq C_{1}|u_{0}-v_{0}|_{\alpha}. \end{aligned}$$
(2.12)
The proof is complete. □
3 Asymptotic behavior of the solution
We first consider the corresponding problem with the linear equation If \(u(t)\in L^{2}(\mathbb{T})\), \(\forall t>0\), then the kth Fourier coefficient \(\widehat{u}_{k}(t)\) of \(u(t,x)\) in x satisfies that is, Therefore, we have
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} u_{t}+L_{p}u_{t}+u_{x}+M_{\alpha}u=0,\quad x\in\mathbb {T}, t\in(0,T),\\ u(0,x)=u_{0}(x),\quad x\in\mathbb{T},\\ \int_{\mathbb{T}}u(t,x)\,dx=0. \end{array}\displaystyle \right . \end{aligned}$$
(3.1)
$$\begin{aligned} \bigl(1+|k|^{p+1}\bigr)\widehat{u}_{k}'(t)+ \bigl(ik+|k|^{2\alpha}\bigr)\widehat {u}_{k}(t)=0, \quad k\in \mathbb{Z}, \end{aligned}$$
$$\widehat{u}_{k}(t)=e^{-\frac{ik+|k|^{2\alpha}}{1+|k|^{p+1}}t}\widehat{u}_{k}(0). $$
$$\begin{aligned} |u|^{2}_{\frac{p+1}{2}}=\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}\bigl|\widehat{u}_{k}(t)\bigr|^{2}= \sum_{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}e^{-\frac{2|k|^{2\alpha}}{1+|k|^{p+1}}t}\bigl| \widehat{u}_{k}(0)\bigr|^{2}. \end{aligned}$$
(3.2)
Theorem 3.1
If
\(u_{0}(x)\in\dot{H}^{\frac {p+1}{2}}(\mathbb{T})\), then the unique solution
\(u(t,x)\)
of the periodic initial value problem (3.1) satisfies
$$\begin{aligned} \bigl|u(t)\bigr|_{\frac{p+1}{2}}^{2}\leq|u_{0}|_{\frac{p+1}{2}}^{2},\quad 0< \alpha \leq\frac{p+1}{2}. \end{aligned}$$
(3.3)
Furthermore, we have
$$\begin{aligned}& \bigl|u(t)\bigr|_{\alpha}^{2}\leq \frac{1}{et}|u_{0}|_{\frac{p+1}{2}}^{2},\quad 0< \alpha< \frac{p+1}{2}, \forall t>0, \end{aligned}$$
(3.4)
$$\begin{aligned}& \bigl|u(t)\bigr|_{\alpha}^{2}\leq e^{-t}|u_{0}|_{\frac{p+1}{2}}^{2},\quad \alpha =\frac{p+1}{2}, \forall t>0. \end{aligned}$$
(3.5)
Proof
Equation (3.2) implies that
$$\begin{aligned} |u|^{2}_{\frac{p+1}{2}} =&\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}e^{-\frac{2|k|^{2\alpha}}{1+|k|^{p+1}}t}\bigl|\widehat {u}_{k}(0)\bigr|^{2} \\ \leq&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{p+1}\bigl|\widehat {u}_{k}(0)\bigr|^{2} \\ =&|u_{0}|_{\frac{p+1}{2}}^{2},\quad 0< \alpha\leq \frac{p+1}{2}. \end{aligned}$$
(3.6)
On the other hand, for \(0<\alpha<\frac{p+1}{2}\) we have where Since the function \(xe^{-xt}\) is uniformly bounded by \(\frac{1}{et}\), we have
$$\begin{aligned} |u|^{2}_{\alpha} =&\sum _{k\in\mathbb{Z}\setminus\{0\}}|k|^{2\alpha }e^{-\frac{2|k|^{2\alpha}}{1+|k|^{p+1}}t}\bigl|\widehat {u}_{k}(0)\bigr|^{2} \\ =&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{p+1}\frac {1+|k|^{p+1}}{|k|^{p+1}} \frac{|k|^{2\alpha}}{1+|k|^{p+1}}e^{-\frac {2|k|^{2\alpha}}{1+|k|^{p+1}}t}\bigl|\widehat{u}_{k}(0)\bigr|^{2} \\ \leq&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{p+1} \varphi_{\alpha }(k)e^{-\varphi_{\alpha}(k)t}\bigl|\widehat{u}_{k}(0)\bigr|^{2}, \end{aligned}$$
(3.7)
$$\begin{aligned} \varphi_{\alpha}(k)=\frac{2|k|^{2\alpha}}{1+|k|^{p+1}},\quad 0< \alpha< \frac {p+1}{2}, k\in\mathbb{Z}\setminus\{0\}. \end{aligned}$$
(3.8)
$$\begin{aligned} |u|^{2}_{\alpha} \leq&\frac{1}{et}\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}\bigl|\widehat{u}_{k}(0)\bigr|^{2}= \frac{1}{et}|u_{0}|_{\frac{p+1}{2}}^{2}. \end{aligned}$$
(3.9)
For \(\alpha=\frac{p+1}{2}\), we have
$$\begin{aligned} |u|^{2}_{\frac{p+1}{2}} =&\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}e^{-\frac{2|k|^{p+1}}{1+|k|^{p+1}}t}\bigl|\widehat {u}_{k}(0)\bigr|^{2} \\ \leq&\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{p+1}e^{-t}\bigl| \widehat {u}_{k}(0)\bigr|^{2} \\ =&e^{-t}\sum_{k\in\mathbb{Z}\setminus\{0\}}|k|^{p+1}\bigl| \widehat {u}_{k}(0)\bigr|^{2} \\ =&e^{-t}|u_{0}|_{\frac{p+1}{2}}^{2}. \end{aligned}$$
(3.10)
The proof is complete. □
We now deal with the nonlinear equation (1.2), that is, \(u_{t}+L_{p}u_{t}+u_{x}+uu_{x}+M_{\alpha}u=0\). We can find similar kind of decreasing properties but less explicit than in the linear case.
Theorem 3.2
If
\(u_{0}(x)\in\dot{H}^{\frac {p+1}{2}}(\mathbb{T})\), then the unique solution
\(u(t,x)\)
of the periodic initial value problem (1.2) satisfies
$$\begin{aligned} \lim_{t\rightarrow+\infty}\bigl|u(t)\bigr|_{\frac{p+1}{2}}^{2}=0. \end{aligned}$$
(3.11)
Proof
Since \(\int_{\mathbb{T}}uu_{x}\,dx=0\), \(\int_{\mathbb {T}}u^{2}u_{x}\,dx=0\), and The equation and zero mean condition in (1.2) imply that hence It implies that \(\sum_{k\in\mathbb{Z}\setminus\{0\} }(1+|k|^{p+1})|\widehat{u}_{k}(t)|^{2}\) is decreasing in t, so we have Therefore, Equation (3.15) and \(u_{0}(x)\in\dot{H}^{\frac{p+1}{2}}\) lead to \(u(t)\in\dot{H}^{\frac{p+1}{2}}\) and then \(u(t)\in\dot{H}^{\alpha }\) for \(\frac{1}{2}<\alpha\leq\frac{p+1}{2}\).
$$\begin{aligned}& \begin{aligned}[b] \int_{\mathbb{T}}u(u_{t}+L_{p}u_{t})\,dx&= \pi\frac{d}{dt} \biggl(\sum_{k\in \mathbb{Z}\setminus\{0\}}\bigl| \widehat{u}_{k}(t)\bigr|^{2}+\sum_{k\in\mathbb {Z}\setminus\{0\}}|k|^{p+1}\bigl| \widehat{u}_{k}(t)\bigr|^{2} \biggr) \\ &=\pi\frac{d}{dt}\sum_{k\in\mathbb{Z}\setminus\{0\} } \bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(t)\bigr|^{2}, \end{aligned} \\& \int_{\mathbb{T}}uM_{\alpha}u\,dx=2\pi\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{2\alpha}\bigl|\widehat{u}_{k}(t)\bigr|^{2}=2 \pi\bigl|u(t)\bigr|_{\alpha}^{2}. \end{aligned}$$
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\sum _{k\in\mathbb{Z}\setminus\{0\} }\bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(t)\bigr|^{2}+\bigl|u(t)\bigr|_{\alpha}^{2}=0, \end{aligned}$$
(3.12)
$$\begin{aligned} \frac{d}{dt}\sum_{k\in\mathbb{Z}\setminus\{0\}} \bigl(1+|k|^{p+1}\bigr)\bigl|\widehat {u}_{k}(t)\bigr|^{2}=-2\bigl|u(t)\bigr|_{\alpha}^{2} \leq0. \end{aligned}$$
(3.13)
$$\begin{aligned} \sum_{k\in\mathbb{Z}\setminus\{0\}}\bigl(1+|k|^{p+1} \bigr)\bigl|\widehat {u}_{k}(t)\bigr|^{2}\leq\sum _{k\in\mathbb{Z}\setminus\{0\} }\bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(0)\bigr|^{2},\quad \forall t\geq0. \end{aligned}$$
(3.14)
$$\begin{aligned} \bigl|u(t)\bigr|^{2}_{\frac{p+1}{2}} =&\sum _{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}\bigl|\widehat{u}_{k}(t)\bigr|^{2} \\ \leq&\sum_{k\in\mathbb{Z}\setminus\{0\}}\bigl(1+|k|^{p+1}\bigr)\bigl| \widehat {u}_{k}(0)\bigr|^{2} \\ \leq& 2|u_{0}|^{2}_{\frac{p+1}{2}},\quad \forall t\geq0. \end{aligned}$$
(3.15)
On the other hand, (3.13) implies that \(\sum_{k\in\mathbb {Z}\setminus\{0\}}(1+|k|^{p+1})|\widehat{u}_{k}(t)|^{2}\) is decreasing in t and bounded below by zero, then the limit exists, we denote it by A.
$$\lim_{t\rightarrow+\infty}\sum_{k\in\mathbb{Z}\setminus\{0\} } \bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(t)\bigr|^{2} $$
Denote If \(B>0\), for large enough t, we have \(|u(t)|_{\alpha}^{2}>B/2\), then there is a constant \(T>0\) such that hence and then This contradiction leads to \(\lim_{t\rightarrow+\infty}|u(t)|_{\alpha }^{2}=B=0\), that is, then \(\forall k\in\mathbb{Z}\), \(\lim_{t\rightarrow+\infty}|\widehat {u}_{k}(t)|^{2}=0\), therefore we have and The proof is complete. □
$$\lim_{t\rightarrow+\infty}\bigl|u(t)\bigr|_{\alpha}^{2}=B. $$
$$\begin{aligned} \frac{d}{dt}\sum_{k\in\mathbb{Z}\setminus\{0\}}\bigl(1+|k|^{p+1} \bigr)\bigl|\widehat {u}_{k}(t)\bigr|^{2}< -B, \quad t>T, \end{aligned}$$
$$\begin{aligned} \sum_{k\in\mathbb{Z}\setminus\{0\}}\bigl(1+|k|^{p+1}\bigr)\bigl|\widehat {u}_{k}(t)\bigr|^{2}< \sum_{k\in\mathbb{Z}\setminus\{0\} } \bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(T)\bigr|^{2}-B(t-T), \end{aligned}$$
$$\begin{aligned} \lim_{t\rightarrow+\infty}\sum_{k\in\mathbb{Z}\setminus\{0\} } \bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(t)\bigr|^{2}\leq- \infty. \end{aligned}$$
$$\begin{aligned} \lim_{t\rightarrow+\infty}\sum_{k\in\mathbb{Z}\setminus\{0\} }|k|^{2\alpha}\bigl| \widehat{u}_{k}(t)\bigr|^{2}=0, \end{aligned}$$
$$\begin{aligned} \lim_{t\rightarrow+\infty}\sum_{k\in\mathbb{Z}\setminus\{0\} } \bigl(1+|k|^{p+1}\bigr)\bigl|\widehat{u}_{k}(t)\bigr|^{2}=A=0 \end{aligned}$$
$$\begin{aligned} \lim_{t\rightarrow+\infty}\sum_{k\in\mathbb{Z}\setminus\{0\} }|k|^{p+1}\bigl| \widehat{u}_{k}(t)\bigr|^{2}=0, \quad\textit{i.e.} \lim _{t\rightarrow+\infty}\bigl|u(t)\bigr|_{\frac{p+1}{2}}^{2}=0. \end{aligned}$$
Remark 3.1
This paper gives the local well-posedness for the subcritical index \(\alpha>\frac{1}{2}\). The interesting case would be to consider the supercritical case \(0<\alpha<\frac{1}{2}\) and the critical case \(\alpha=\frac{1}{2}\). In the supercritical case \(0<\alpha<\frac{1}{2}\), there will be less dissipation, so the dispersive part comes to play a principal role. In the cases \(0<\alpha <\frac{1}{2}\) and \(\alpha=\frac{1}{2}\), \(\dot{H}^{\alpha}(\mathbb{T})\) is not an algebra, we must find another way to establish the estimates on the nonlinear term. We will consider the supercritical case in future work.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for helpful comments and suggestions that greatly improved the presentation of this paper. This work was supported by National Natural Science Foundation of China (Grant Nos. 11471129 and 11272277).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China
(2)
School of Mathematics and Statistics, Xuchang University, Xuchang, Henan, 461000, P.R. China
References
- Benjamin, TB, Bona, JL, Mahony, JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. 272, 47-78 (1972) View ArticleMathSciNetMATHGoogle Scholar
- Avrin, J: The generalized Benjamin-Bona-Mahony equation in \(R^{n}\) with singular initial data. Nonlinear Anal. 11, 139-147 (1987) View ArticleMathSciNetMATHGoogle Scholar
- Avrin, J, Goldstein, JA: Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions. Nonlinear Anal. 9, 861-865 (1985) View ArticleMathSciNetMATHGoogle Scholar
- Bona, JL, Bryant, PJ: A mathematical model for long waves generated by wavemakers in nonlinear dispersive systems. Proc. Camb. Philos. Soc. 73, 391-405 (1973) View ArticleMathSciNetMATHGoogle Scholar
- Bona, JL, Dougalis, VA: An initial and boundary value problem for a model equation for propagation of long waves. J. Math. Anal. Appl. 75, 503-522 (1980) View ArticleMathSciNetMATHGoogle Scholar
- Chen, YM: Remark on the global existence for the generalized Benjamin-Bona-Mahony equations in arbitrary dimension. Appl. Anal. 30, 1-15 (1988) View ArticleMathSciNetMATHGoogle Scholar
- Goldstein, JA, Wichnoski, BJ: On the Benjamin-Bona-Mahony equation in higher dimensions. Nonlinear Anal. 4, 665-675 (1980) View ArticleMathSciNetMATHGoogle Scholar
- Guo, YT, Wang, M, Tang, YB: Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R. Appl. Anal. 94(9), 1766-1783 (2015) View ArticleMathSciNetGoogle Scholar
- Medeiros, LA, Miranda, MM: Weak solutions for a nonlinear dispersive equation. J. Math. Anal. Appl. 59, 432-441 (1977) View ArticleMathSciNetMATHGoogle Scholar
- Medeiros, LA, Menzala, GP: Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation. SIAM J. Math. Anal. 8, 792-799 (1977) View ArticleMathSciNetMATHGoogle Scholar
- Astaburuaga, MA, Bisognin, E, Bisognin, V, Fernandez, C: Global attractors and finite dimensionality for a class of dissipative equations of BBM’s type. Electron. J. Differ. Equ. 1998, 25 (1998) MathSciNetGoogle Scholar
- Celebi, AO, Kalantarov, VK, Polat, M: Attractors for the generalized Benjamin-Bona-Mahony equation. J. Differ. Equ. 157, 439-451 (1999) View ArticleMathSciNetMATHGoogle Scholar
- Chueshov, I, Polat, M, Siegmund, S: Gevrey regularity of global attractor for generalized Benjamin-Bona-Mahony equation. Mat. Fiz. Anal. Geom. 11, 226-242 (2004) MathSciNetMATHGoogle Scholar
- Stanislavova, M, Stefanov, A, Wang, B: Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on \(R^{3}\). J. Differ. Equ. 219, 451-483 (2005) View ArticleMathSciNetMATHGoogle Scholar
- Wang, B: Regularity of attractors for the Benjamin-Bona-Mahony equation. J. Phys. A 31, 7635-7645 (1998) View ArticleMathSciNetMATHGoogle Scholar
- Wang, B, Fussner, DW, Bi, C: Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains. J. Phys. A 40, 10491-10504 (2007) View ArticleMathSciNetMATHGoogle Scholar
- Wang, B, Yang, W: Finite dimensional behaviour for the Benjamin-Bona-Mahony equation. J. Phys. A 30, 4877-4885 (1997) View ArticleMathSciNetMATHGoogle Scholar
- Wang, M: Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity. Nonlinear Anal. TMA 105, 134-144 (2014) View ArticleMATHGoogle Scholar
- Chehab, JP, Garnier, P, Mammeri, Y: Long-time behavior of solutions of a BBM equation with generalized damping (2014) arXiv:1402.5009
- Carvajal, X, Panthee, M: On ill-posedness for the generalized Benjamin-Bona-Mahony equation. Discrete Contin. Dyn. Syst. 34(11), 4565-4576 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Wang, M, Tang, YB: Attractors in \(H^{2}\) and \(L^{2p-2}\) for reaction diffusion equations on unbounded domains. Commun. Pure Appl. Anal. 12(2), 1111-1121 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Wang, M, Tang, YB: On dimension of the global attractor for 2D quasi-geostrophic equations. Nonlinear Anal., Real World Appl. 14, 1887-1895 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Wang, M, Tang, YB: Long time dynamics of 2D quasi-geostrophic equations with damping in \(L^{p}\). J. Math. Anal. Appl. 412, 866-877 (2014) View ArticleMathSciNetMATHGoogle Scholar
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