In 1872, the Boussinesq equation was derived by Boussinesq [1] to describe the propagation of small amplitude long waves on the surface of shallow water. This was also the first to give a scientific explanation of the existence to solitary waves. One of the classical Boussinesq equations takes the form
$$ u_{tt}=-au_{xxxx}+u_{xx}+\beta\bigl(u^{2} \bigr)_{xx}, $$
(1)
where \(u(t,x)\) is an elevation of the free surface of fluid, and the constant coefficients a and β depend on the depth of fluid and the characteristic speed of long waves. Extensive research has been carried out to study the classical Boussinesq equation in various respects. The Cauchy problem of (1) has been discussed in [2–10]. In [11–13], the initial boundary value problem and the Cauchy problem for the Boussinesq equation
$$ u_{tt}=u_{xx}+u_{xxtt}+u_{xxxx}-ku_{xxt}+f(u)_{xx} $$
(2)
were studied.
In order to discuss the water wave problem with surface tension, Schneider and Eugene [14] investigated the following Boussinesq model:
$$ u_{tt}=u_{xx}+u_{xxtt}+\mu u_{xxxx}-u_{xxxxtt}+f(u)_{xx}, $$
(3)
where \(t,x,\mu\in R\) and \(u(t,x)\in R\). Equation (3) can also be derived from the 2D water wave problem. For a degenerate case, Schneider and Eugene [14] have proved that the long wave limit can be described approximately by two decoupled Kawahara equations. In [15, 16], Wang and Mu studied the well-posedness of the local and globally solution, the blow-up of solutions and nonlinear scattering for small amplitude solutions to the Cauchy problem of (3). In [17, 18], the authors investigated the Cauchy problem of the following Rosenau equation:
$$ u_{tt}-u_{xx}+ u_{xxxx}+ u_{xxxxtt}=f(u)_{xx}. $$
(4)
The existence and uniqueness of the global solution and blow-up of the solution for (4) are proved by Wang and Xu [17]. Wang and Wang [18] also proved the global existence and asymptotic behavior of the solution in n-dimensional Sobolev spaces. Recently, Xu et al. [19, 20] proved the global existence and finite-time blow-up of the solutions for (4) by means of the family of potential wells. The results in [20] improve the results obtained by Wang and Xu [17].
This work considers the Cauchy problem for the following equation:
$$ \left \{ \begin{array}{l} u_{tt}-\Delta u_{tt}+\Delta^{2} u_{tt}=-\Delta^{2}u+\Delta u+\Delta f(u),\quad x\in R^{n},t>0 , \\ u(x,0)=\phi(x),\qquad u_{t}(x,0)=\psi(x),\quad x\in R^{n}, \end{array} \right . $$
(5)
where \(f(u)\) satisfies one of the following three assumptions:
$$\begin{aligned} (\mathrm{A}_{1})&\quad f(u)=\pm a|u|^{p} \quad \text{or} \quad -a|u|^{p-1}u,\quad a>0, p>1, \\ (\mathrm{A}_{2})&\quad \left \{ \begin{array}{l} f(u)=\pm a|u|^{p},\quad a>0, p>1, p\neq2k, k=1,2,\ldots\quad \text{or} \\ f(u)=-a|u|^{p-1}u,\quad a>0, p>1, p\neq2k+1, k=1,2,\ldots, \end{array} \right . \\ (\mathrm{A}_{3})&\quad \left \{ \begin{array}{l} f(u)=\pm a|u|^{2k},\quad a>0, p>1, k=1,2,\ldots\quad \text{or} \\ f(u)=-a|u|^{2k+1}u,\quad k=1,2,\ldots. \end{array} \right . \end{aligned}$$
In this paper, we discuss problem (5) in high dimensional space. To our knowledge, there have been few results on the global existence of a solution to problem (5). In [21], Wang and Xue only proved the global existence and finite-time blow-up of the solution to (3) in one space dimension. Though the arguments and methods used in this paper are similar to those in [20], the first equation of problem (5) is different from (3) and (4).
By the Fourier transform and Duhamel’s principle, the solution u of problem (5) can be written as
$$ u(t,x)=\bigl(\partial_{t} S(t)\phi\bigr) (x)+\bigl(S(t)\psi\bigr) (x)+ \int^{t}_{0}\Gamma(t-\tau )f\bigl(u(\tau)\bigr)\, d \tau. $$
(6)
Here \(\Gamma(t)=S(t)(1-\Delta +\Delta ^{2})^{-1}\Delta \) and
$$\begin{aligned}& \bigl(\partial_{t} S(t)\phi\bigr) (x)=\frac{1}{(2\pi)^{n}}\int _{R^{n}}e^{ix\xi}\cos \biggl(\frac{|\xi|\sqrt{1+|\xi|^{2}}t}{\sqrt{1+|\xi|^{2}+|\xi|^{4}}}\biggr)\hat{ \phi}(\xi )\, d\xi, \\& \bigl(\partial_{t} S(t)\psi\bigr) (x)=\frac{1}{(2\pi)^{n}}\int _{R^{n}}e^{ix\xi}\sin \biggl(\frac{|\xi|\sqrt{1+|\xi|^{2}}t}{\sqrt{1+|\xi|^{2}+|\xi|^{4}}}\biggr) \frac{\sqrt{1+|\xi|^{2}+|\xi|^{4}}}{|\xi|\sqrt{1+|\xi|^{2}}}\hat{\psi}(\xi )\, d\xi, \end{aligned}$$
where \(\hat{\phi}(\xi)=F(\phi)(\xi)=\int_{R^{n}}e^{-i(x,\xi)}\phi(x)\, dx\) is the Fourier transform of \(\phi(x)\).
Throughout this paper: \(L^{p}\) denotes the usual Lebesgue space on \(R^{n}\) with norm \(\|\cdot\|_{L^{p}}\), \(H^{s}\) denotes the usual Sobolev space on \(R^{n}\) with norm \(\|u\|_{H^{s}}=\|(I-\Delta )^{\frac{s}{2}}u\|=\|(1+|\xi|^{2})^{\frac {s}{2}}\hat{u}\|\) and \(|\xi|=\sqrt{\xi_{1}^{2}+\xi_{2}^{2}+\cdots+\xi_{1}^{2}}\).
First, by using the contraction mapping theorem, we obtain the following existence and uniqueness of the local solution to problem (5).
Theorem 1.1
Let
\(s>\frac{n}{2}\)
and
\(f\in C^{m}\)
with
\(m\geq s\)
being an integer. Then, for any
\(\phi\in H^{s}\)
and
\(\psi\in H^{s}\), the Cauchy problem (5) has a unique local solution
\(u\in C^{1}([0,T], H^{s})\). Moreover, if
\(T_{m}\)
is the maximal existence time of
u, and
$$ \max_{0\leq t< T_{m}}\bigl[\bigl\Vert u(t)\bigr\Vert _{H^{s}}+ \bigl\Vert u_{t}(t)\bigr\Vert _{H^{s}}\bigr]<\infty, $$
then
\(T_{m}=\infty\).
Theorem 1.2
Let
\(s>\frac{n}{2}\)
and
\(f\in C^{m}\)
with
\(m\geq s\)
being an integer. Assume that
\(\phi\in H^{s}(R^{n})\), \(\psi\in H^{s}(R^{n})\), and
\((-\Delta )^{-\frac{1}{2}}\phi\in L^{2}\), \(F(\phi)\in L^{1}\), \(F(u)=\int^{t}_{0}f(\tau)\, d\tau\). Then, for the local solution
u, we have
\(u\in C^{2}([0,T);H^{s})\), \((-\Delta )^{-\frac{1}{2}}u_{t}\in C^{1}([0, T_{m}), L^{2})\), satisfying
$$\begin{aligned} E(t) =&\frac{1}{2}\bigl[\bigl\Vert (-\Delta )^{-\frac{1}{2}}u_{t} \bigr\Vert ^{2}+\|\nabla u_{t}\| ^{2}+\|\nabla u \|^{2}+\|u\|^{2}+\|u_{t}\|^{2}\bigr]+\int _{R^{n}} F(u)\, dx \\ =&E(0), \quad \forall t\in(0,T_{m}). \end{aligned}$$
(7)
In order to use the potential well method, for \(s>\frac{n}{2}\) (\(s\geq1\)) and \(u\in X_{s}(T)\), we define
$$\begin{aligned}& J(u)=\frac{1}{2}\|u\|^{2}_{H^{1}}+\int _{R^{n}}F(u)\, dx, \\& I(u)=\|u\|^{2}_{H^{1}}+\int_{R^{n}}u f(u)\, dx, \\& d=\inf_{u\in N}J(u),\qquad N=\bigl\{ u\in H^{1}|I(u)=0, \|u\|_{H^{1}}\neq0\bigr\} , \\& W_{1}=\bigl\{ u\in H^{1}|I(u)>0, J(u)< d\bigr\} \cup\{0\}, \\& V_{1}=\bigl\{ u\in H^{1}|I(u)<0,J(u)<d\bigr\} \end{aligned}$$
and
$$\begin{aligned}& W_{2}=\bigl\{ u\in H^{1}|I(u)>0\bigr\} \cup\{0\}, \\& V_{2}=\bigl\{ u\in H^{1}|I(u)<0\bigr\} . \end{aligned}$$
From \(u\in C^{1}([0,T]; H^{s})\), we get \(u\in C^{1}([0,T];L^{\infty})\) and \(u\in C^{1}([0,T];L^{q})\) for all \(2\leq q<\infty\). Hence, \(J(u)\), \(I(u)\), d, \(W_{1}\), and \(V_{1}\) are all well defined. Now, we give the following results for problem (5).
Theorem 1.3
Let
\(s>\frac{n}{2}\)
with
\(s\geq1\), and
\(f(u)\)
satisfy (A2) with
\([p]\geq s\)
or (A3). Assume that
\(\phi\in H^{s}\), \(\psi\in H^{s}\), and
\((-\Delta )^{-\frac{1}{2}}\psi\in L^{2}\), \(E(0)< d\). Then both
\(W_{2}\)
and
\(V_{2}\)
are invariant under the flow of problem (5).
Theorem 1.4
Let
\(n\leq3\)
and
\(f(u)\)
satisfy (A1), where
\(2\leq p<\infty\)
for
\(n=1,2\); \(\frac{7}{3}\leq p\leq5\)
for
\(n=3\), \(\phi\in H^{2}\), \(\psi\in H^{2}\), and
\((-\Delta)^{-\frac{1}{2}}\phi\in L^{2}\). Assume
\(E(0)< d\)
and
\(\phi\in W_{2}\). Then problem (5) admits a unique global solution
\(u\in C^{2}([0, \infty), H^{2})\), with
\((-\Delta)^{-\frac {1}{2}}u_{t}\in C^{1}([0,\infty), L^{2})\)
and
\(u\in W_{1}\)
for
\(0\leq t<\infty\).
Theorem 1.5
Let
\(n\leq3\)
and let
\(f(u)\)
satisfy (A1), where
\(3\leq p<\infty\)
for
\(n=1,2\); \(3\leq p\leq5\)
for
\(n=3\), \(\phi\in H^{3}\), \(\psi\in H^{3}\), and
\((-\Delta)^{-\frac{1}{2}}\phi\in L^{2}\). Assume that
\(E(0)< d\)
and
\(\phi\in W_{2}\). Then problem (5) admits a unique global solution
\(u\in C^{2}([0, \infty), H^{3})\)
with
\((-\Delta)^{-\frac {1}{2}}u_{t}\in C^{1}([0,\infty), L^{2})\)
and
\(u\in W\)
for
\(0\leq t<\infty\).
Theorem 1.6
Let
\(n\leq3\)
and
\(f(u)\)
satisfy (A1), where
\(4\leq p<\infty\)
for
\(n=1,2\); \(4\leq p\leq5\)
for
\(n=3\), \(\phi\in H^{4}\), \(\psi\in H^{4}\), and
\((-\Delta)^{-\frac{1}{2}}\phi\in L^{2}\). Assume that
\(E(0)< d\)
and
\(\phi\in W_{2}\). Then problem (5) admits a unique global solution
\(u\in C^{2}([0, \infty), H^{4})\)
with
\((-\Delta)^{-\frac {1}{2}}u_{t}\in C^{1}([0,\infty), L^{2})\)
and
\(u\in W\)
for
\(0\leq t<\infty\).
Theorem 1.7
Let
\(s>\frac{n}{2}\)
with
\(s\geq1\), and
\(f(u)\)
satisfy (A2) with
\([p]\geq3\)
or
\(\phi,\psi\in H^{s}\), \((-\Delta )^{-\frac{1}{2}}\phi, (-\Delta)^{-\frac{1}{2}}\psi\in L^{2}\). Assume that
\(E(0)< d\)
and
\(I(\phi)<0\). Then the solution of problem (5) blows up in finite time, i.e., the maximal existence time
\(T_{m}\)
of
u
is finite, and
$$ \lim_{t\rightarrow T_{m}}\sup\bigl(\bigl\Vert u(t)\bigr\Vert _{H^{s}}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{s}} \bigr)=\infty. $$
The remainder of this paper is organized as follows. In Section 2, Theorems 1.1 and 1.2 are proved. In Section 3, we give some preliminary lemmas and the proof of Theorem 1.3. The proofs of Theorems 1.4, 1.5 and 1.6 are given in Section 4. Finally, Section 5 is devoted to the proof of Theorem 1.7.