Abstract
The Cauchy problem for a generalized Boussinesq equation is investigated. The existence and uniqueness for the local solution and global solution of the problem are established under certain conditions. Moreover, the potential well method is used to discuss the finite-time blow-up for the problem.
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1 Introduction
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
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
were studied.
In order to discuss the water wave problem with surface tension, Schneider and Eugene [14] investigated the following Boussinesq model:
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:
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:
where \(f(u)\) satisfies one of the following three assumptions:
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
Here \(\Gamma(t)=S(t)(1-\Delta +\Delta ^{2})^{-1}\Delta \) and
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
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
In order to use the potential well method, for \(s>\frac{n}{2}\) (\(s\geq1\)) and \(u\in X_{s}(T)\), we define
and
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
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.
2 Existence of local solutions
In this section, we consider the local existence and uniqueness of solutions to problem (5).
Lemma 2.1
For the operators \(\partial_{t}S(t)\), \(S(t)\) and \(\Gamma(t)\) defined in Section 1, we have
Proof
We only need to prove (9) and (11), since the proofs of the other inequalities are similar. Using the Plancherel theorem, we have
Therefore (9) and (11) hold. This completes the proof of the lemma. □
Lemma 2.2
([17])
Let \(g\in C^{m}(R)\), where \(m\geq0\) is an integer.
-
(i)
If \(0\leq s\leq m\) and \(u\in H^{s}(R^{n})\cap L^{\infty}(R^{n})\), then \(g(u)\in H^{s}(R^{n})\) and
$$ \bigl\Vert g(u)\bigr\Vert _{H^{s}}\leq C\bigl(\Vert u\Vert _{\infty}\bigr)\|u\|_{H^{s}}. $$(13) -
(ii)
If \(s\leq m\) and \(u,v\in H^{s}(R^{n})\cap L^{\infty}(R^{n})\), then
$$ \bigl\Vert g(u)-g(v)\bigr\Vert _{H^{s}}\leq K\bigl(\Vert u\Vert _{\infty},\|v\|_{\infty}\bigr)\|u-v\|_{H^{s}}. $$(14)
In particular, if \(u,v\in H^{s}\) for some \(s>\frac{n}{2}\), then u and \(v\in L^{\infty}\), (13) and (14) hold.
Proof of Theorem 1.1
Let \(s>\frac{n}{2}\),
and
Similarly to the proofs in [15, 16], we see that for sufficiently small T,
is a contract mapping. Hence by the contracting-mapping principle we obtain the result of Theorem 1.1. □
Corollary 2.3
Under the assumption of Theorem 1.1, if \(T_{m}<\infty\), we have
Corollary 2.4
Let \(s>\frac{n}{2}\) and \(f(u)\) satisfy (A2) or (A3). Then, for any \(\phi\in H^{s}\) and \(\psi\in H^{s}\), problem (5) admits a unique local solution \(u\in C^{1}([0, T_{m}), H^{s_{1}})\), where \(T_{m}\) is the maximal existence time of u. Moreover, either \(T_{m}=+\infty\) or \(T_{m}<\infty\) and
Lemma 2.5
Assume \(s>\frac{n}{2}\), \(f\in C^{m}(R)\), \(\phi\in H^{s}\), and \(\psi\in H^{s}\). Then for the local solution \(u\in C^{1}([0, T_{m}), H^{s})\) given in Theorem 1.1, we have \(u_{tt}\in C([0, T_{m}), H^{s})\).
Proof
Using the Fourier transformation, we have
Since
which together with (15) yields
Furthermore, using
we obtain \(u_{tt}\in C([0, T_{m}); H^{s})\). □
Lemma 2.6
Assume \(s>\frac{n}{2}\), \(f\in C^{m}(R)\), \(\phi\in H^{s}\), \(\psi\in H^{s}\), and \((-\Delta)^{-\frac{1}{2}}\phi\in L^{2}\). Then for the local solution \(u\in C^{1}([0, T_{m}), H^{s})\) given in Theorem 1.1, we have \((-\Delta)^{-\frac{1}{2}}u_{t}\in C^{1}([0, T_{m}), L^{2})\).
Proof
First for the local solution u given in Theorem 1.1, we obtain
From (15), we get
Furthermore, we get \(\|(-\Delta)^{-\frac{1}{2}}u_{tt}(t+\Delta t)-(-\Delta)^{-\frac {1}{2}}u_{tt}(t)\|_{H^{s+1}}\rightarrow0\) as \(\Delta t\rightarrow0\).
Hence, we have
Using
and
we get
□
Proof of Theorem 1.2
Using (5), it follows by straightforward calculation that
where \((\cdot,\cdot)\) denotes the inner product of \(L^{2}\) space, \(\langle \cdot,\cdot\rangle_{{X\ast X}}\) means the usual duality of \(X^{\ast}\) and X with \(X=H^{1}\). Integrating the above equality with respect to t, we have identity (7). The theorem is proved. □
Corollary 2.7
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 \((-\Delta )^{-\frac{1}{2}}\psi\in L^{2}\), problem (5) admits a unique local solution \(u\in C^{2}([0, T_{m}), H^{s})\), with \((-\Delta )^{-\frac{1}{2}}u_{t}\in C^{1}([0, T_{m}), L^{2})\) satisfying (7), where \(T_{m}\) is the maximal existence time of u. Moreover, either \(T_{m}=+\infty\) or \(T_{m}<\infty\) and
Proof
Since \(\phi\in H^{s}\) and \(s>\frac{n}{2}\), we have \(\phi \in L^{\infty}\). Hence, \(\phi\in L^{q}\) for all \(2\leq q\leq\infty\). From \(|F(\phi)|=C|\phi|^{p+1}\), \(2< p+1<\infty\). We obtain \(F(\phi)\in L^{1}\). □
3 Preliminary lemmas and invariant sets
In this section, we will prove several lemmas which are related with the potential well for problem (5). By arguments similar to those in [20], we obtain the following lemmas.
Lemma 3.1
Let \(s>\frac{n}{2}\) with \(s\geq1\) and let \(f(u)\) satisfy (A1), \(u\in H^{s}\) and \(g(\lambda)=-\frac{1}{\lambda } \int_{R^{n}} uf(\lambda u)\, dx\). Assume \(\int_{R^{n}}u f(u)\, dx<0\). Then:
-
(i)
\(g(\lambda)\) is increasing on \(0<\lambda<\infty\).
-
(ii)
\(\lim_{\lambda\rightarrow0}g(\lambda)=0\), \(\lim_{\lambda \rightarrow+\infty}g(\lambda)=+\infty\).
Lemma 3.2
Let \(s>\frac{n}{2}\) with \(s\geq1\), \(u\in H^{s}\), and let \(f(u)\) satisfy (A1), \(u\neq0\). We have:
-
(i)
\(\lim_{\lambda\rightarrow0}J(\lambda u)=0\).
-
(ii)
\(I(\lambda u)=\lambda\frac{d}{d\lambda}J(\lambda u)\), \(\forall\lambda >0\).
Furthermore, if \(\int_{R^{n}}u f(u)\, dx<0\), then:
-
(iii)
\(\lim_{\lambda\rightarrow+\infty}J(\lambda u)=-\infty\).
-
(iv)
In the interval \(0<\lambda<\infty\), there exists a unique \(\lambda^{\ast}=\lambda^{\ast}(u)\) such that
$$ \biggl.\frac{d}{d\lambda}J(\lambda u)\biggr|_{\lambda=\lambda^{\ast}}=0. $$ -
(v)
\(J(\lambda u)\) is increasing on \(0<\lambda<\lambda^{\ast}\), decreasing on \(\lambda^{\ast}<\lambda<\infty\) and \(I(\lambda^{\ast})=0\).
-
(vi)
\(I(\lambda u)>0 \) for \(0<\lambda<\lambda^{\ast}\), \(I(\lambda u)<0\) for \(\lambda^{\ast}<\lambda<\infty\) and \(I(\lambda^{\ast})=0\).
Lemma 3.3
Let \(s>\frac{n}{2}\) with \(s\geq1\), \(u\in H^{s}\), and let \(f(u)\) satisfy (A1). Then:
-
(i)
If \(0<\|u\|_{H^{1}}<r_{0}\), then \(I(u)>0\).
-
(ii)
If \(I(u)<0\), then \(\|u\|_{H^{1}}>r_{0}\).
-
(iii)
If \(I(u)=0\) and \(\|u\|_{H^{1}}\neq0\), then \(\|u\|_{H^{1}}\geq r_{0}\), where \(r_{0}=(\frac{1}{a C_{\ast}^{p+1}})^{\frac{1}{p-1}}\).
Lemma 3.4
Let \(s>\frac{n}{2}\) with \(s\geq1\) and \(f(u)\) satisfy (A1). We have
Lemma 3.5
Let \(s>\frac{n}{2}\) with \(s\geq1\) and \(f(u)\) satisfy (A1). Assume \(u\in H^{s}\) and \(I(u)<0\). Then
Proof of Theorem 1.3
We only prove the invariance of \(W_{1}\) since the proof for the invariance of \(V_{1}\) is similar. Let \(u(t,x)\) be any weak solution of problem (5) with \(\phi\in W_{1}\), T be the maximal existence time of \(u(t,x)\). Next we prove that \(u(t,x)\in W_{1}\) for \(0 < t < T\). Arguing by contradiction we assume there is a \(t_{1}\in (0,T)\) such that \(u(t_{1})\notin W_{1}\). By the continuity of \(I(u(t))\) with respect to t, there exists a \(t_{0}\in (0,T)\) such that \(u(t_{0})\in \partial W_{1}\). From the definition of \(W_{1}\) and (i) of Lemma 3.3 we have \(R_{0}\subset W_{1}\), \(R_{0}=\{u\in H^{1}\mid \|u\|_{H^{1}}< r_{0}\}\). Hence we know \(0\notin \partial W_{1}\). From \(u(t_{0})\in \partial W_{1}\), it holds that \(I(u(t_{0})) =0\) and \(\|u(t_{0})\|_{H^{1}}\neq 0\). The definition of d yields \(J (u(t_{0}))\geq d\), which contradicts
The proof of Theorem 1.3 is complete. □
From Theorem 1.3, we can prove the following corollaries.
Corollary 3.6
Let s, \(f(u)\), ϕ, ψ and \(E(0)\) be the same as those in Theorem 1.3. Then:
-
(i)
All solutions of problem (5) belong to \(W_{1}\), provided that \(\phi\in W_{2}\).
-
(ii)
All solutions of problem (5) belong to \(V_{1}\), provided that \(\phi\in V_{2}\).
Corollary 3.7
Let \(s>\frac{n}{2}\) with \(s\geq1\), and let \(f(u)\) satisfy (A2) with \([p]\geq s\) or (A3), \(\phi\in H^{s}\), \(\psi\in H^{s}\) and \((-\Delta )^{-\frac{1}{2}}\psi\in L^{2}\). Assume that \(E(0)<0\) or \(E(0)=0\), \(\phi\neq0\). Then all the solutions of problem (5) belong to \(V_{1}\).
4 Global existence of solutions
In this section, we prove the global existence of a solution for problem (5).
Lemma 4.1
Let \(s>\frac{n}{2}\) with \(s\geq1\) and \(f(u)\) satisfy (A2) with \([p]\geq s\) or \(\phi\in H^{s}\), \(\psi\in H^{s} \), and \((-\Delta)^{-\frac{1}{2}}\psi\in C^{1}([0,T_{m}), L^{2})\). Assume that \(E(0)< d\) and \(\phi\in W_{2}\). Then, for the local solution u given in Corollary 2.7, one has
Proof
Let u be the unique local solution of problem (5) given in Corollary 2.7. Then \(u\in C^{2}([0, T_{m}); H^{s})\), \((-\Delta )^{-\frac{1}{2}} u_{t}\in C^{1}([0, T_{m}),L^{2})\) satisfying (7) and
From Theorem 1.3, we get \(u\in W_{2}\) and \(I(u)\geq0\) for \(0\leq t\leq T_{m}\). Hence, (19) gives rise to
Thus, we obtain (18). □
Proof of Theorem 1.4
It follows from Corollary 2.7 that problem (5) admits a unique local solution \(u\in C^{2}([0,T_{m}); H^{2})\), with \((-\Delta)^{-\frac{1}{2}}u_{t}\in C^{1}([0,T_{m});L^{2})\) satisfying (7), where \(T_{m}\) is the maximal existence time of u.
Next, we prove that \(T_{m}=+\infty\). Using Lemma 4.1 one derives (19). Since \(u\in C^{2}([0,T_{m}); H^{2})\) satisfies (5), we have
and
Multiplying (22) by \(u_{t}\in C^{1}([0,T_{m}), H^{2})\) and integrating on \(R^{n}\), we obtain
For \(n=3\), we get
For \(n=3\), we have \(H^{1}\hookrightarrow L^{2}\) for \(2\leq q\leq6\), \(|f'(u)|=A|u|^{p-1}\). From \(\frac{7}{3}\leq p\leq5\), we have \(\frac {4}{3}\leq p-1\leq4\) and \(2\leq\frac{3}{2}(p-1)\leq6\). Hence, we have \(\|f'(u)\|_{\frac{3}{2}}\leq C(p)\) for \(0\leq t< T_{m}\). From (23) and (24), we get
For \(n=1 \mbox{ or } 2\), we have
and (25). Let
Using (25) yields
and
From (26), we obtain \(T_{m}=+\infty\). If the conclusion \(T_{m}=+\infty\) is false, then \(T_{m}<\infty\). By (26), we get
which contradicts (16). □
Proof of Theorem 1.5
It follows from Corollary 2.7 that problem (5) admits a unique local solution \(u\in C^{2}([0,T_{m}]; H^{3})\) and \((-\Delta)^{-\frac{1}{2}}u_{t}\in C^{1}([0, T_{m}); L^{2})\).
Multiplying (22) by \(-\Delta u_{t}\), we obtain
From (23) and (27), for \(0\leq t< T_{m}\), we get
and
Let
Using (28) and the estimates above, we obtain
and
Thus \(T_{m}=+\infty\). □
Proof of Theorem 1.6
From Corollary 2.7, it follows that problem (5) admits a unique local solution \(u\in C^{2}([0,T_{m}]; H^{4})\), with \((-\Delta)^{-\frac{1}{2}}u_{t}\in C^{1}([0, T_{m}); L^{2})\), where \(T_{m}\) is the maximal existence time of u.
Multiplying (22) by \(-\Delta^{2} u_{t}\) and integrating on \(R^{n}\), we have
Let
Then, from (31) and the estimates above, we obtain
and
from which one derives \(T_{m}=+\infty\). □
5 Finite-time blow-up of the solution
In this section, we study the finite-time blow-up of the solution for problem (5).
Lemma 5.1
Under the assumptions of Corollary 2.7, we have
provided that \((-\Delta)^{-\frac{1}{2}}u_{0}\in L^{2}\).
Proof
From \((-\Delta )^{-\frac{1}{2}}u\in C^{2}([0, T_{m}); L^{2})\) and
we obtain the result. □
Proof of Theorem 1.7
Let \(u\in C^{2}([0, T_{m}); H^{s})\) and \((-\Delta)^{-\frac{1}{2}}u_{t}\in C^{1}([0, T_{m}); L^{2})\) be the unique local solution of problem (5). Then, by Lemma 5.1, we have \((-\Delta)^{-\frac{1}{2}}u\in C^{2}([0, T_{m});L^{2})\), where \(T_{m}\) is the maximal existence time of u. Suppose that \(T_{m}=+\infty\). Then \(u\in C^{2}([0, \infty); H^{s})\) with \((-\Delta)^{-\frac{1}{2}}u \in C^{2}([0,\infty ); L^{2})\) and \((-\Delta)^{-\frac{1}{2}}u_{t}\in C^{1}([0,\infty); L^{2})\). Let
then
Using the energy equality (7), we obtain
from which one derives
and
Substituting (35) into (33), we obtain
On the other hand, from (33), we get
Using \(I(u_{0})<0\) and Theorem 1.3, we get \(u\in V_{2}\) and \(I(u)<0\) for \(0\leq t\leq\infty\). Hence, by Lemma 3.3, we obtain \(H(t)>0\) for \(0\leq t<\infty\). Using Lemma 3.5, we have \((p-1)\|u\| ^{2}_{H^{1}}>2(p+1)d\). Thus, we get
On the other hand, from (36), we obtain
Hence there exists a \(t_{0}\geq0\) such that \(H'(t)>0\), from which, together with \(H(t_{0})>0\) and (37), one derives that there exists a \(T_{1}>0\) such that
which contradicts \(u\in C^{2}([0,\infty); H^{s})\), \((-\Delta )^{-\frac {1}{2}}u\in C^{2}([0, \infty);L^{2})\). Finally, from \(T_{m}<\infty\) and Corollary 2.4, we obtain
□
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The author would like to thank the editor and the reviewers for their constructive suggestions and helpful comments. The work was partially supported by the National Natural Science Foundation of China (grant number 11101069).
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Wang, Y. Existence and nonexistence of solutions for a generalized Boussinesq equation. Bound Value Probl 2015, 1 (2015). https://doi.org/10.1186/s13661-014-0259-3
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Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-014-0259-3