By standard PDE theory [21], we can establish the existence of local solutions for (3.1).
Lemma 4.1
(Local existence)
For
\(s\geq1\) (\(d=1\)) and
\(s\geq2\) (\(d=2,3\)), there exists a
\(T_{0}>0\)
such that (3.1) with
\(u(\cdot,0), v(\cdot,0)\in{H^{s}}\)
has a unique solution
\(\mathbf{w}(\cdot,t)\)
on
\((0,T_{0})\)
which satisfies
$$\bigl\Vert \mathbf{w}(t) \bigr\Vert _{H^{s}}\leq{C} { \bigl\Vert \mathbf{w}(0) \bigr\Vert _{H^{s}}},\quad 0< t< T_{0}, $$
where
C
is a positive constant depending on
\(d_{1}\), \(d_{2}\), k, α, β, r, χ.
Lemma 4.2
Let
\([u(\mathbf{x},t),v(\mathbf{x},t)]\)
be a solution of (3.1). Then
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\sum_{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \bigl\{ \bigl\vert D^{\sigma}u \bigr\vert ^{2}+ K \bigl\vert D^{\sigma}v \bigr\vert ^{2} \bigr\} \, d \mathbf{x} \\& \qquad {} +\sum_{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \biggl\{ \frac {d_{1}k^{n}}{4} \bigl\vert \nabla{D^{\sigma}}u \bigr\vert ^{2}+ \frac{d_{2}K}{2} \bigl\vert \nabla{D^{\sigma}}v \bigr\vert ^{2} \biggr\} \, d \mathbf{x} \\& \qquad {} +r\sum_{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \bigl\vert D^{\sigma}u \bigr\vert ^{2}\, d \mathbf{x} +\frac{K}{2}\sum _{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \bigl\vert D^{\sigma}v \bigr\vert ^{2}\, d \mathbf{x} \\& \quad \leq \hat{C}_{2} \Biggl(\chi+d_{1}\sum ^{n} _{i=1}C^{i}_{n}k^{n-i}+ \frac {2r}{k} \Biggr) \Biggl(\sum^{n} _{i=1} \Vert \mathbf{w} \Vert ^{i}_{H^{2}} \Biggr) \bigl\Vert \nabla^{3}\mathbf{w} \bigr\Vert ^{2}+ \hat{C}_{3} \Vert u \Vert ^{2}, \end{aligned}$$
where
\(\hat{C}_{3}=\frac{C^{3}_{0}\chi^{6}\alpha ^{6}}{2d^{5}_{1}d^{3}_{2}k^{5n-6}}\).
Proof
Notice that if \(\mathbf{w}(\mathbf{x},t)\) is a solution of (3.1) on \(\mathbb{T}^{d}\), then the even extension of \(\mathbf{w}(\mathbf{x},t)\) on \(2\mathbb{T}^{d} =(-\pi,\pi)^{d}\) (\(d=1,2,3\)) is also the solution of (3.1) which satisfies homogeneous Neumann boundary conditions and periodical boundary conditions on \(2\mathbb{T}^{d}=(-\pi,\pi)^{d}\) (\(d=1,2,3\)). From this, we have
$$ \textstyle\begin{cases} \tilde{u}_{t}=d_{1}k^{n}\nabla^{2}{\tilde{u}}-\chi k\nabla^{2}{\tilde {v}}+d_{1}\nabla [ (\sum^{n} _{i=1}C^{i}_{n} k^{n-i}\tilde {u}^{i} )\nabla\tilde{u} ] \\ \hphantom{\tilde{u}_{t}={}}{}-\chi\nabla(\tilde{u}\nabla{\tilde{v}})-r\tilde{u}-\frac{r}{k}\tilde {u}^{2}, \\ \tilde{v}_{t}=d_{2}\nabla^{2}{\tilde{v}}+\alpha\tilde{u}-\beta\tilde {v}, \\ \frac{\partial{\tilde{u}}}{\partial{x_{i}}}=\frac{\partial{\tilde {v}}}{\partial{x_{i}}}=0,\quad \mbox{at }x_{i}=-\pi,0,\pi, \mbox{for }1\leq{i}\leq{d}, \end{cases} $$
(4.1)
where \([\tilde{u}(\mathbf{x},t),\tilde{v}(\mathbf{x},t)]\) is the even extension of \([u(\mathbf{x},t),v(\mathbf{x},t)]\) on \(2\mathbb{T}^{d}\). By (4.1), we can easily deduce that
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{2\mathbb{T}^{d}} \vert \partial_{x_{i}x_{j}}\tilde{u} \vert ^{2}\,d \mathbf{x}+d_{1}k^{n} \int_{2\mathbb {T}^{d}} \bigl\vert \nabla(\partial_{x_{i}x_{j}} \tilde{u}) \bigr\vert ^{2}\,d \mathbf{x} \\& \qquad {}-\chi k \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \nabla(\partial_{x_{i}x_{j}}\tilde{v})\,d \mathbf{x} +r \int_{2\mathbb{T}^{d}} \vert \partial_{x_{i}x_{j}}\tilde{u} \vert ^{2}\,d \mathbf{x} \\& \quad =\chi \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}(\tilde{u}\nabla\tilde{v})\,d \mathbf{x} -d_{1}n k^{n-1} \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \partial_{x_{i}x_{j}}(\tilde{u}\nabla\tilde{u})\,d \mathbf{x} \\& \qquad {}-\frac{1}{2}d_{1}n(n-1)k^{n-2} \int_{2\mathbb{T}^{d}}\nabla(\partial _{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{2}\nabla \tilde{u}\bigr)\,d \mathbf{x} \\& \qquad {}-\frac{1}{6}d_{1}n(n-1) (n-2)k^{n-3} \int_{2\mathbb{T}^{d}}\nabla (\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde {u}^{3}\nabla\tilde{u}\bigr)\,d \mathbf{x}-\cdot\cdot\cdot \\& \qquad {}-d_{1}nk \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{n-1}\nabla\tilde{u}\bigr)\,d \mathbf {x} \\& \qquad {}-d_{1} \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{n}\nabla\tilde{u}\bigr)\,d \mathbf{x} \\& \qquad {}-\frac{2r}{k} \int_{2\mathbb{T}^{d}}\tilde{u} \vert \partial _{x_{i}x_{j}}\tilde{u} \vert ^{2}\,d \mathbf{x} -\frac{2r}{k} \int_{2\mathbb{T}^{d}}\partial_{x_{i}}\tilde{u}\cdot \partial_{x_{j}}\tilde{u}\cdot\partial_{x_{i}x_{j}}\tilde{u}\,d \mathbf{x} \end{aligned}$$
(4.2)
and
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{2\mathbb{T}^{d}} K|\partial_{x_{i}x_{j}}\tilde{v}|^{2}\, d \mathbf{x}+Kd_{2} \int_{2\mathbb {T}^{d}}\bigl|\nabla(\partial_{x_{i}x_{j}}\tilde{v})\bigr|^{2} \, d \mathbf{x} +K\beta \int_{2\mathbb{T}^{d}}|\partial_{x_{i}x_{j}}\tilde{v}|^{2}\, d \mathbf{x} \\& \quad =K\alpha \int_{2\mathbb{T}^{d}}\partial_{x_{i}x_{j}}\tilde{u}\cdot \partial_{x_{i}x_{j}}\tilde{v}\, d \mathbf{x}. \end{aligned}$$
(4.3)
It follows from (4.2) and (4.3) that
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{2\mathbb{T}^{d}} \bigl( \vert \partial_{x_{i}x_{j}}\tilde{u} \vert ^{2}+K \vert \partial_{x_{i}x_{j}}\tilde {v} \vert ^{2} \bigr)\,d\mathbf{x} \\& \qquad {}+ \int_{2\mathbb{T}^{d}} \bigl(d_{1}k^{n} \bigl\vert \nabla(\partial _{x_{i}x_{j}}\tilde{u}) \bigr\vert ^{2}+ Kd_{2} \bigl\vert \nabla(\partial_{x_{i}x_{j}}\tilde{v}) \bigr\vert ^{2} -\chi k\nabla(\partial_{x_{i}x_{j}}\tilde{u})\cdot \nabla(\partial _{x_{i}x_{j}}\tilde{v}) \bigr)\,d\mathbf{x} \\& \qquad {}+r \int_{2\mathbb{T}^{d}} \vert \partial_{x_{i}x_{j}}\tilde{u} \vert ^{2}\,d\mathbf{x} +\beta K \int_{2\mathbb{T}^{d}} \vert \partial_{x_{i}x_{j}}\tilde{v} \vert ^{2}\,d\mathbf{x} \\& \quad =\chi \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}(\tilde{u}\nabla\tilde{v})\,d\mathbf{x} -d_{1}nk^{n-1} \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \partial_{x_{i}x_{j}}(\tilde{u}\nabla\tilde{u})\,d\mathbf{x} \\& \qquad {}-\frac{1}{2}d_{1}n(n-1)k^{n-2} \int_{2\mathbb{T}^{d}}\nabla(\partial _{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{2}\nabla \tilde{u}\bigr)\,d \mathbf{x} \\& \qquad {}-\frac{1}{6}d_{1}n(n-1) (n-2)k^{n-3} \int_{2\mathbb{T}^{d}}\nabla (\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde {u}^{3}\nabla\tilde{u}\bigr)\,d \mathbf{x} \\& \qquad {}-\cdots-d_{1}nk \int_{2\mathbb{T}^{d}}\nabla(\partial _{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{n-1}\nabla \tilde{u}\bigr)\,d \mathbf{x} \\& \qquad {}-d_{1} \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{n}\nabla\tilde{u}\bigr)\,d \mathbf{x} +\alpha K \int_{2\mathbb{T}^{d}}\partial_{x_{i}x_{j}}\tilde{u}\cdot \partial_{x_{i}x_{j}}\tilde{v}\,d\mathbf{x} \\& \qquad {}-\frac{2r}{k} \int_{2\mathbb{T}^{d}}\bigl(\tilde{u} \vert \partial _{x_{i}x_{j}} \tilde{u} \vert ^{2} +\partial_{x_{i}}\tilde{u}\cdot\partial _{x_{j}}\tilde{u}\cdot\partial_{x_{i}x_{j}}\tilde{u}\bigr)\,d \mathbf{x} \\& \quad :=J_{\chi}+J_{1}+J_{2}+J_{3}+ \cdots+J_{n-1}+J_{n}+J_{\alpha}+J_{r}. \end{aligned}$$
(4.4)
Thanks to Young’s inequality and (3.10),
$$ -\chi k\nabla(\partial_{x_{i}x_{j}}\tilde{u})\cdot\nabla(\partial _{x_{i}x_{j}}\tilde{v}) \geq-\frac{d_{1}k^{n}}{2} \bigl\vert \nabla( \partial_{x_{i}x_{j}}\tilde {u}) \bigr\vert ^{2}-\frac{d_{2}K}{2} \bigl\vert \nabla(\partial_{x_{i}x_{j}}\tilde{v}) \bigr\vert ^{2}. $$
(4.5)
The nonlinear term \(J_{\chi}\) is bounded by
$$\begin{aligned} J_{\chi} :=&\chi \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \partial_{x_{i}x_{j}}(\tilde{u}\nabla\tilde{v})\,d\mathbf{x} \\ \leq&\chi\biggl\{ \int_{2\mathbb{T}^{d}} \bigl\vert \nabla(\partial _{x_{i}x_{j}} \tilde{u})\cdot\partial_{x_{i}x_{j}}\tilde{u}\cdot\nabla \tilde{v} \bigr\vert \,d\mathbf{x} + \int_{2\mathbb{T}^{d}} \bigl\vert \nabla(\partial_{x_{i}x_{j}} \tilde{u})\cdot \partial_{x_{j}}\tilde{u}\cdot\nabla(\partial_{x_{i}} \tilde{v}) \bigr\vert \,d\mathbf{x} \\ &{}+ \int_{2\mathbb{T}^{d}} \bigl\vert \nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot\partial_{x_{i}}\tilde{u}\cdot\nabla(\partial_{x_{j}} \tilde {v}) \bigr\vert \,d\mathbf{x} + \int_{2\mathbb{T}^{d}} \bigl\vert \tilde{u}\cdot\nabla( \partial_{x_{i}x_{j}}\tilde {u})\cdot\nabla(\partial_{x_{i}x_{j}}\tilde{v}) \bigr\vert \,d\mathbf{x}\biggr\} \\ \leq&\chi \Vert \nabla\tilde{v} \Vert _{L^{\infty}} \bigl\Vert \nabla( \partial _{x_{i}x_{j}}\tilde{u}) \bigr\Vert \Vert \partial_{x_{i}x_{j}} \tilde{u} \Vert +2\chi\sum^{d} _{i=1} \Vert \nabla\tilde{u} \Vert _{L^{\infty}} \Vert \partial _{x_{i}x_{j}} \tilde{v} \Vert \bigl\Vert \nabla(\partial_{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\ &{}+\chi \Vert \tilde{u} \Vert _{L^{\infty}} \bigl\Vert \nabla( \partial_{x_{i}x_{j}}\tilde {u}) \bigr\Vert \bigl\Vert \nabla( \partial_{x_{i}x_{j}}\tilde{v}) \bigr\Vert . \end{aligned}$$
(4.6)
By applying the Sobolev embedding to control the \(L^{\infty}\) norm for \(d\leq3\), there exists a constant \(C_{5}>0\) such that
$$ \|g\|_{L^{\infty}(2\mathbb{T}^{d})}\leq{C_{5}\|g\|_{H^{2}(2\mathbb {T}^{d})}}. $$
(4.7)
Moreover, notice that
$$ \int_{2\mathbb{T}^{d}}\nabla{\tilde{u}}\,d\mathbf{x}= \int_{2\mathbb {T}^{d}}\nabla{\tilde{v}}\,d\mathbf{x}=0,\qquad \int_{2\mathbb{T}^{d}}\partial_{x_{i}x_{j}}\tilde{u}\,d\mathbf{x}= \int _{2\mathbb{T}^{d}}\partial_{x_{i}x_{j}}\tilde{v}\,d\mathbf{x}=0. $$
(4.8)
Using the Poincaré inequality, there exists a constant \(C_{6}>0\) such that if \(g\in H^{1}(2\mathbb{T}^{d})\) and \(\int_{2\mathbb {T}^{d}}g\, d \mathbf{x}=0\), then
$$ \|g\|_{L^{4}(2\mathbb{T}^{d})}\leq{C_{6}\|\nabla{g}\|}. $$
(4.9)
From (4.8) and (4.9), it follows that
$$\|\partial_{x_{i}}g\|\leq{C_{7}\|\nabla\partial_{x_{i}}g} \|,\qquad \|\partial _{x_{i}x_{j}}g\|\leq{C_{8}\|\nabla \partial_{x_{i}x_{j}}g\|} $$
and
$$ \Vert \nabla{g} \Vert \leq{C_{9} \biggl(\sum _{|\sigma|=2} \bigl\Vert \nabla{D^{\sigma}}g \bigr\Vert ^{2} \biggr)^{\frac{1}{2}}} \leq{{C^{2}_{9}} \biggl(\sum_{|\alpha|=2} \bigl\Vert \nabla \bigl(D^{\sigma}\bigr)g \bigr\Vert ^{2} \biggr)^{\frac{1}{2}}}. $$
(4.10)
Using (4.7), we get
$$ \Vert \nabla{g} \Vert _{L^{\infty}}\leq C_{10} \Vert \nabla{g} \Vert _{H^{2}}\leq C_{11} \bigl\Vert \nabla^{3}{g} \bigr\Vert . $$
(4.11)
Thus from (4.7) and (4.11), one can derive that
$$ J_{\chi}\leq\chi C_{12} \Vert {\tilde{\mathbf{w}}} \Vert _{H^{2}} \bigl\Vert \nabla^{3}{\tilde {\mathbf{w}}} \bigr\Vert ^{2}. $$
(4.12)
Similarly, one knows that
$$ J_{1}\leq d_{1}n k^{n-1}C_{12} \Vert {\tilde{\mathbf{w}}} \Vert _{H^{2}} \bigl\Vert \nabla ^{3}{ \tilde{\mathbf{w}}} \bigr\Vert ^{2}. $$
(4.13)
Applying interpolation inequalities, one has
$$ \Vert \partial_{x_{i}x_{j}}g \Vert ^{2}\leq C_{0} \biggl(a \bigl\Vert \nabla(\partial _{x_{i}x_{j}}g) \bigr\Vert ^{2}+\frac{ \Vert g \Vert ^{2}}{4a^{2}} \biggr),\quad \forall a>0. $$
(4.14)
By the choice of \(a=\frac{d^{2}_{1}d_{2}k^{2n-2}}{2\alpha^{2}\chi ^{2}C_{0}}\) in (4.14),
$$\begin{aligned} J_{\alpha} :=&\alpha K \int_{2\mathbb{T}^{d}}\partial_{x_{i}x_{j}}\tilde {u}\cdot \partial_{x_{i}x_{j}}\tilde{v}\,d\mathbf{x} \\ \leq&\frac{K}{2}\sum_{|\sigma|=2} \int_{2\mathbb{T}^{d}} \bigl\vert D^{\sigma }\tilde{v} \bigr\vert ^{2}\,d\mathbf{x} +\frac{d_{1}k^{n}}{4}\sum _{|\sigma|=2} \int_{2\mathbb{T}^{d}} \bigl\vert \nabla \bigl(D^{\sigma}\tilde{u} \bigr) \bigr\vert ^{2}\,d\mathbf{x}+\hat{C}_{3}\|\tilde{u} \|^{2}, \end{aligned}$$
(4.15)
where \(\hat{C}_{3}=\frac{C^{3}_{0}\chi^{6}\alpha^{6}}{2d^{5}_{1}d^{3}_{2}k^{5n-6}}\).
Now, according to Hölder’s inequality, (4.7), (4.9) and (4.11), one can verify that
$$\begin{aligned}& J_{2}:=-\frac{1}{2}d_{1}n(n-1)k^{n-2} \int_{2\mathbb{T}^{d}}\nabla(\partial _{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{2}\nabla \tilde{u}\bigr)\,d \mathbf{x} \\& \hphantom{J_{2}}\leq d_{1}n(n-1)k^{n-2} \Vert \nabla \tilde{u} \Vert _{L^{\infty}} \Vert \nabla{\tilde {u}} \Vert ^{2}_{L^{4}} \bigl\Vert \nabla(\partial_{x_{i}x_{j}} \tilde{u}) \bigr\Vert \\& \hphantom{J_{2}:={}}{}+d_{1}n(n-1)k^{n-2} \Vert \nabla\tilde{u} \Vert _{L^{\infty}} \Vert \tilde{u} \Vert _{L^{\infty}} \Vert \partial_{x_{i}x_{j}}\tilde{u} \Vert \bigl\Vert \nabla(\partial _{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\& \hphantom{J_{2}:={}}{}+2d_{1}n(n-1)k^{n-2}\sum ^{d} _{i=1} \Vert \tilde{u} \Vert _{L^{\infty}} \Vert \nabla \tilde{u} \Vert _{L^{\infty}} \Vert \partial_{x_{i}x_{j}} \tilde{u} \Vert \bigl\Vert \nabla (\partial_{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\& \hphantom{J_{2}:={}}{}+\frac{1}{2}d_{1}n(n-1)k^{n-2} \Vert \tilde{u} \Vert ^{2}_{L^{\infty}} \bigl\Vert \nabla ( \partial_{x_{i}x_{j}}\tilde{u}) \bigr\Vert ^{2} \\& \hphantom{J_{2}}\leq\frac{1}{2}d_{1}n(n-1)k^{n-2} C_{13} \Vert {\tilde{\mathbf{w}}} \Vert ^{2}_{H^{2}} \bigl\Vert \nabla^{3}{\tilde{\mathbf{w}}} \bigr\Vert ^{2}, \end{aligned}$$
(4.16)
$$\begin{aligned}& J_{3}:=-\frac{1}{6}d_{1}n(n-1) (n-2)k^{n-3} \int_{2\mathbb{T}^{d}}\nabla (\partial_{x_{i}x_{j}}\tilde{u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde {u}^{3}\nabla\tilde{u}\bigr)\,d \mathbf{x} \\& \hphantom{J_{3}}\leq d_{1}n(n-1) (n-2)k^{n-3} \Vert \tilde{u} \Vert _{L^{\infty}} \Vert \nabla\tilde{u} \Vert _{L^{\infty}} \Vert \nabla{\tilde{u}} \Vert ^{2}_{L^{4}} \bigl\Vert \nabla(\partial _{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\& \hphantom{J_{3}:={}}{}+\frac{3}{2}d_{1}n(n-1) (n-2)k^{n-3} \Vert \tilde{u} \Vert ^{2}_{L^{\infty}} \Vert \nabla \tilde{u} \Vert _{L^{\infty}} \Vert \partial_{x_{i}x_{j}}\tilde{u} \Vert \bigl\Vert \nabla (\partial_{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\& \hphantom{J_{3}:={}}{}+\frac{1}{6}d_{1}n(n-1) (n-2)k^{n-3} \Vert \tilde{u} \Vert ^{3}_{L^{\infty}} \bigl\Vert \nabla ( \partial_{x_{i}x_{j}}\tilde{u}) \bigr\Vert ^{2} \\& \hphantom{J_{3}}\leq\frac{1}{6}d_{1}n(n-1) (n-2)k^{n-3} C_{14} \Vert {\tilde{\mathbf{w}}} \Vert ^{3}_{H^{2}} \bigl\Vert \nabla^{3}{\tilde{\mathbf{w}}} \bigr\Vert ^{2}, \end{aligned}$$
(4.17)
$$\begin{aligned}& J_{n-1}:=-d_{1}nk \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{n-1}\nabla\tilde{u}\bigr)\,d \mathbf {x} \\& \hphantom{J_{n-1}}\leq d_{1}nk\bigl\{ (n-1) (n-2) \Vert \tilde{u} \Vert ^{n-3}_{L^{\infty}} \Vert \nabla \tilde{u} \Vert _{L^{\infty}} \Vert \nabla{\tilde{u}} \Vert ^{2}_{L^{4}} \bigl\Vert \nabla(\partial _{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\& \hphantom{J_{n-1}:={}}{}+3(n-1) \Vert \tilde{u} \Vert ^{n-2}_{L^{\infty}} \Vert \nabla\tilde{u} \Vert _{L^{\infty}} \Vert \partial_{x_{i}x_{j}} \tilde{u} \Vert \bigl\Vert \nabla(\partial _{x_{i}x_{j}}\tilde{u}) \bigr\Vert + \Vert \tilde{u} \Vert ^{n-1}_{L^{\infty}} \bigl\Vert \nabla(\partial_{x_{i}x_{j}}\tilde {u}) \bigr\Vert ^{2}\bigr\} \\& \hphantom{J_{n-1}}\leq d_{1}nkC_{1n} \Vert {\tilde{ \mathbf{w}}} \Vert ^{n-1}_{H^{2}} \bigl\Vert \nabla^{3}{\tilde {\mathbf{w}}} \bigr\Vert ^{2}, \end{aligned}$$
(4.18)
and
$$\begin{aligned} J_{n} :=&-d_{1} \int_{2\mathbb{T}^{d}}\nabla(\partial_{x_{i}x_{j}}\tilde {u})\cdot \partial_{x_{i}x_{j}}\bigl(\tilde{u}^{n}\nabla\tilde{u}\bigr)\,d \mathbf {x} \\ \leq& d_{1}\bigl\{ n(n-1) \Vert \tilde{u} \Vert ^{n-2}_{L^{\infty}} \Vert \nabla\tilde {u} \Vert _{L^{\infty}} \Vert \nabla{\tilde{u}} \Vert ^{2}_{L^{4}} \bigl\Vert \nabla(\partial _{x_{i}x_{j}}\tilde{u}) \bigr\Vert \\ &{}+3n \Vert \tilde{u} \Vert ^{n-1}_{L^{\infty}} \Vert \nabla \tilde{u} \Vert _{L^{\infty}} \Vert \partial_{x_{i}x_{j}}\tilde{u} \Vert \bigl\Vert \nabla(\partial_{x_{i}x_{j}}\tilde {u}) \bigr\Vert + \Vert \tilde{u} \Vert ^{n}_{L^{\infty}} \bigl\Vert \nabla( \partial_{x_{i}x_{j}}\tilde{u}) \bigr\Vert ^{2}\bigr\} \\ \leq& d_{1}C_{1n+1} \Vert {\tilde{\mathbf{w}}} \Vert ^{n}_{H^{2}} \bigl\Vert \nabla^{3}{\tilde { \mathbf{w}}} \bigr\Vert ^{2}, \end{aligned}$$
(4.19)
where \(C_{13}=2C^{2}_{6}C_{11}+6C_{5}C_{11}+C^{2}_{5}\), \(C_{14}=6C_{5}C^{2}_{6}C_{11}+9C^{2}_{5}C_{11}+C^{3}_{5}\), \(C_{1n}=(n-1)(n-2)C^{n-3}_{5}C^{2}_{6}C_{11}+3(n-1)C^{n-2}_{5}C_{11}+C^{n-1}_{5}\) and \(C_{1,n+1}=n(n-1)C^{n-2}_{5}C^{2}_{6}C_{11}+3nC^{n-2}_{5}C_{11}+C^{n}_{5}\).
Again by (4.7), (4.10) and (4.11), we can estimate
$$J_{r}\leq\frac{2r}{k} \bigl\{ \|\tilde{u}\|_{L^{\infty}}\| \partial_{x_{i}x_{j}}\tilde{u}\|^{2} +\|\nabla\tilde{u} \|_{L^{\infty}}\|\nabla\tilde{u}\|\|\partial _{x_{i}x_{j}}\tilde{u}\| \bigr\} , $$
further,
$$ \sum_{|\alpha|=2}J_{r}\leq\frac{2r}{k}C_{1r} \Vert {\tilde{\mathbf{w}}} \Vert _{H^{2}} \bigl\Vert \nabla^{3}{\tilde{\mathbf{w}}} \bigr\Vert ^{2}, $$
(4.20)
where \(C_{1r}=C_{8}(C_{5}C_{8}+C_{11})\).
Recall that \([\tilde{u},\tilde{v}]\) is the even extension of \([u,v]\). Plugging (4.12), (4.13), (4.15), (4.16), (4.17), (4.18), (4.19), (4.20) into (4.4),
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\sum_{ \vert \sigma \vert =2} \int_{2\mathbb{T}^{d}} \bigl( \bigl\vert D^{\sigma}\tilde{u} \bigr\vert ^{2}+ K \bigl\vert D^{\sigma}\tilde{v} \bigr\vert ^{2} \bigr)\,d\mathbf{x} \\& \qquad {}+\sum_{ \vert \sigma \vert =2} \int_{2\mathbb{T}^{d}} \biggl(\frac{d_{1}k^{n}}{4} \bigl\vert \nabla{D^{\sigma}\tilde{u}} \bigr\vert ^{2}+\frac {d_{2}K}{2} \bigl\vert \nabla\bigl(D^{\sigma}\tilde{v}\bigr) \bigr\vert ^{2} \biggr)\,d\mathbf{x} \\& \qquad {}+r\sum_{ \vert \sigma \vert =2} \int_{2\mathbb{T}^{d}} \bigl\vert D^{\sigma}\tilde{u} \bigr\vert ^{2}\,d\mathbf{x} +\frac{K}{2}\sum_{ \vert \sigma \vert =2} \int_{2\mathbb{T}^{d}} \bigl\vert D^{\sigma}\tilde {v} \bigr\vert ^{2}\,d\mathbf{x} \\& \quad \leq\hat{C}_{2} \Biggl(\chi+d_{1}\sum ^{n} _{i=1}C^{i}_{n}k^{n-i}+ \frac {2r}{k} \Biggr) \Biggl(\sum^{n} _{i=1} \Vert {\tilde{\mathbf{w}}} \Vert ^{i}_{H^{2}} \Biggr) \bigl\Vert \nabla^{3}{\tilde{\mathbf{w}}} \bigr\Vert ^{2}+\hat{C}_{3} \Vert \tilde {u} \Vert ^{2}, \end{aligned}$$
where \(\hat{C}_{2}=C_{12}+\sum^{n+1} _{i=2}C_{1i}+C_{1r}\). This completes the proof. □
Lemma 4.3
Let
\(\mathbf{w}(\mathbf{x},t)\)
be a solution of (3.1) such that, for
\(0\leq{t}\leq{T}< T_{0}\),
$$ \sum^{n} _{i=1} \bigl\Vert \mathbf{w}( \mathbf {\cdot},t) \bigr\Vert ^{i}_{H^{2}} \leq\frac{1}{\hat{C}_{2}} \min{ \biggl\{ \frac{d_{1}k^{n}}{4}, \frac {d_{2}K}{2} \biggr\} }, $$
(4.21)
and
$$ \bigl\Vert \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert \leq{\hat{C}_{1}}e^{\lambda_{\max}t} \bigl\Vert \mathbf {w}(\mathbf {\cdot},0) \bigr\Vert . $$
(4.22)
Then
$$\bigl\Vert \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{n}_{H^{2}} \leq{\hat{C}_{4}} \bigl\{ \bigl\Vert \mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2}_{H^{2}}+e^{2\lambda _{\max}t} \bigl\Vert \mathbf{w}( \mathbf {\cdot},0) \bigr\Vert ^{2} \bigr\} ^{\frac{n}{2}}, \quad 0\leq {t} \leq{T}, $$
where
\(\hat{C}_{4}=\max{ \{ ((1+C^{2}_{9})K )^{\frac{n}{2}}, [\hat{C}^{2}_{1} (1+\frac{(1+C^{2}_{9})\hat{C}^{2}_{3}}{\lambda_{\max }} ) ]^{\frac{n}{2}} \}}\), if
\(K\geq1\)
and
\(\hat{C}_{4}=\max \{ (\frac{1+C^{2}_{9}}{K} )^{\frac{n}{2}}, [\hat{C}^{2}_{1} (1+\frac{(1+C^{2}_{9})\hat{C}^{2}_{3}}{K\lambda_{\max }} ) ]^{\frac{n}{2}} \}\), if
\(K<1\).
Proof
From (4.10), one knows that
$$ \bigl\Vert \nabla\mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{2}\leq C^{2}_{9}\sum_{|\sigma|=2} \bigl\Vert D^{\sigma}\mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{2}, $$
(4.23)
which leads to
$$ \bigl\Vert \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{n}_{H^{2}} \leq \biggl( \bigl\Vert \mathbf{w}(\mathbf {\cdot },t) \bigr\Vert ^{2}+ \bigl( C^{2}_{9}+1\bigr)\sum_{|\sigma|=2} \bigl\Vert D^{\sigma}\mathbf{w}(\mathbf {\cdot },t) \bigr\Vert ^{2} \biggr)^{\frac{n}{2}}. $$
(4.24)
By Lemma 4.2 and (4.21), we can see that
$$ \frac{1}{2}\frac{d}{dt}\sum_{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \bigl( \bigl\vert D^{\sigma}u \bigr\vert ^{2}+K \bigl\vert D^{\sigma}v \bigr\vert ^{2} \bigr)\,d\mathbf{x} \leq{\hat{C}_{3}} \Vert u \Vert ^{2} \leq{\hat{C}_{3}} \bigl\Vert \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{2}. $$
(4.25)
From this and (4.22), it follows that
$$\begin{aligned}& \sum_{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \bigl( \bigl\vert D^{\sigma}u(\mathbf {\cdot},t) \bigr\vert ^{2}+ K \bigl\vert D^{\sigma}v(\mathbf {\cdot},t) \bigr\vert ^{2} \bigr)\,d\mathbf{x} \\& \quad \leq\sum_{ \vert \sigma \vert =2} \int_{\mathbb{T}^{d}} \bigl( \bigl\vert D^{\sigma}u(\mathbf {\cdot},0) \bigr\vert ^{2} +K \bigl\vert D^{\sigma}v(\mathbf {\cdot},0) \bigr\vert ^{2} \bigr)\,d\mathbf{x} +\frac{\hat{C}^{2}_{1}\hat{C}_{3}}{\lambda_{\max}}e^{2\lambda_{\max}t} \bigl\Vert \mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2}. \end{aligned}$$
(4.26)
We will proceed in the following two cases: \(K\geq1\), \(K<1\).
(1) If \(K\geq1\), using (4.26) yields
$$\sum_{|\sigma|=2} \bigl\Vert D^{\sigma}\mathbf{w}( \mathbf {\cdot},t) \bigr\Vert ^{2} \leq K\sum _{|\sigma|=2} \bigl\Vert D^{\sigma}\mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2} +\frac{\hat{C}^{2}_{1}\hat{C}_{3}}{\lambda_{\max}}e^{2\lambda_{\max}t} \bigl\Vert \mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2}. $$
Then it is not hard to verify from (4.22), (4.24) and (4.26) that
$$\bigl\Vert \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{n}_{H^{2}} \leq{\hat{C}_{4}} \bigl( \bigl\Vert \mathbf {w}(\mathbf {\cdot},0) \bigr\Vert ^{2}_{H^{2}}+ \bigl\Vert \mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2}e^{2\lambda _{\max}t} \bigr)^{\frac{n}{2}}, $$
where \(\hat{C}_{4}=\max{ \{ ((1+C^{2}_{9})K )^{\frac{n}{2}}, [\hat{C}^{2}_{1} (1+\frac{(1+C^{2}_{9})\hat{C}^{2}_{3}}{\lambda_{\max }} ) ]^{\frac{n}{2}} \}}\).
(2) If \(K<1\), notice that
$$K\sum_{|\sigma|=2} \bigl\Vert D^{\sigma} \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{2}\leq \sum _{|\sigma|=2} \bigl\Vert D^{\sigma}\mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2} +\frac{\hat{C}^{2}_{1}\hat{C}_{3}}{\lambda_{\max}}e^{2\lambda_{\max}t} \bigl\Vert \mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2}. $$
Moreover, applying (4.22) and (4.24), we can see that
$$\bigl\Vert \mathbf{w}(\mathbf {\cdot},t) \bigr\Vert ^{n}_{H^{2}} \leq{\hat{C}_{4}} \bigl( \bigl\Vert \mathbf {w}(\mathbf {\cdot},0) \bigr\Vert ^{2}_{H^{2}}+ \bigl\Vert \mathbf{w}(\mathbf {\cdot},0) \bigr\Vert ^{2}e^{2\lambda _{\max}t} \bigr)^{\frac{n}{2}}, $$
where \(\hat{C}_{4}=\max{ \{ (\frac{1+C^{2}_{9}}{K} )^{\frac{n}{2}}, [\hat{C}^{2}_{1} (1+\frac{(1+C^{2}_{9})\hat{C}^{2}_{3}}{\lambda_{\max }K} ) ]^{\frac{n}{2}} \}}\) and thereby we complete the proof. □