Scholarly article on topic 'Existence and nonexistence of solutions for a generalized Boussinesq equation'

Existence and nonexistence of solutions for a generalized Boussinesq equation Academic research paper on "Mathematics"

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Academic research paper on topic "Existence and nonexistence of solutions for a generalized Boussinesq equation"

Wang Boundary Value Problems (2015) 2015:1 DOI 10.1186/s13661-014-0259-3

0 Boundary Value Problems

a SpringerOpen Journal

RESEARCH

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Existence and nonexistence of solutions for a generalized Boussinesq equation

Ying Wang*

Correspondence: nadine_1979@163.com Schoolof MathematicalSciences, University of Electronic Science and Technology of China, Chengdu, 611731,China

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. MSC: 35Q20; 76B15

Keywords: Boussinesq equation; blow-up; global solution; nonexistence

ft Spri

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

utt = -auxxxx + Uxx + ß( u2) xx, (1)

ringer

where u(t, x) is an elevation of the free surface of fluid, and the constant coefficients a and p 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

utt = uxx + uxxtt + uxxxx — kUxxt + 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:

utt = uxx + uxxtt + ¡¡uxxxx — uxxxxtt +f (u) xx, (3)

where t, x, ¡i e R and u(t, x) e 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

© 2014 Wang; licensee Springer. 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 originalworkis properly credited.

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:

utt - uxx + uxxxx + uxxxxtt = 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 «-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:

{uu - Autt + A2utt = -A2u + Au + Af (u), x e Rn, t >0, u(x, 0) = <(x), ut (x, 0) = i (x), x e Rn,

where f (u) satisfies one of the following three assumptions: (Ai) f (u) = ±a|u|p or - a|u|p-1u, a >0,p > 1, (A2)

f (u) = ±a\u\p, a > 0,p >1,p = 2k,k = 1,2,... or f (u) = -a\u\p-1u, a > 0, p > 1, p =2k + 1, k = 1, 2, ...,

!f (u) = ±a\u\2k, a >0,p >1,k = 1,2,... or f (u) = -a\u\2k+1u, k =1,2,....

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« of problem (5) can be written as

u(t,x) = (dtS(t)<)(x) +(S(t)i)(x)+ i r(t- rf(u(r))dT. (6)

Here r(t) = S(t)(1 - A + A2)-1 A and

(3tS(t)<)(x) = -1-/' ex cosfJM+EUmm,

V ' (2n)n Jr« W1+ £ |2+ |£ |4 j

1 ()i () (2n )njRn |2 + £ |V |£ ^(?)?

where <(£) = F(<)(£) = fRn e~l(x,M<(x) dx is the Fourier transform of <(x).

Throughout this paper: Lp denotes the usual Lebesgue space on Rn with norm || • |L, Hs denotes the usual Sobolev space

on Rn with norm || u||hs = II (I - A)2 u|| = || (1 + |2)2 u||

and |£| = Vf12 + £ + ••• + .

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 > | andf e Cm with m > s being an integer. Then, for any $ e Hs and f e Hs, the Cauchyproblem (5) has a unique local solution u e C:([0, T],Hs). Moreover, if Tm is the maximal existence time ofu, and

0m<rJlu(t)lHs + IIut(t)L] < œ

then Tm = œ.

Theorem 1.2 Let s > | andf e Cm with m > s being an integer. Assume that $ e Hs(Rn), f e Hs(Rn ), and (-A)-i $ e L2, F ($) e L1, F (u) = fQf (x ) dx. Then, for the local solution u, we have u e C2([0, T);Hs), (-A)-2Ut e C1([0, Tm),L2), satisfying

E(t) = 1 N|(-A)-2 ut |2 + ||V ut ||2 + ||V u\\2 + || u M2 + \\ut II2! + f F (u) dx

= £(0), Vt e (0, Tm). (7)

In order to use the potential well method, for s > | (s > 1) and u e Xs(T), we define

J(u) = 1 ||uy^i + / F(u) dx,

I(u) = \\u\\2Hi ^ / uf (u)dx,

d = inf /(u), ^ = {u e H 1\I(u) = 0, \\u\\Hi =0},

Wi= {u e H1 \I(u) > 0, J(u) <4 U {0}, Vi = {u e H1\I(u) < 0,J(u) < d}

W2 = {u e H 1\I(u)> 0} U{0}, V2 = {u e H 1\I(u) < 0}.

From u e C1([0, T];Hs), we get u e C1([0, T];Lm) and u e C1([0, T];Lq) for all 2 < q < to. Hence, J(u), I(u), d, W1, and V1 are all well defined. Now, we give the following results for problem (5).

Theorem 1.3 Lets > n with s > 1, andf (u) satisfy (A2) with p] > sor (A3). Assume that $ e Hs, ^ e Hs, and (-A)-i ^ e L2, £(0) < d. Then ¿oth W2 and V2 are invariant under the flow of problem (5).

Theorem 1.4 Let n < 3 andf (u) satisfy (A1), where 2 < p < to for n = 1,2; 3 < p < 5 for n = 3, $ e H2, f e H2, and (-A)-2 $ e L2. Assume £(0) < dand$ e W2. Then problem (5) admits a unique global solution u e C2([0, to),H2), with (-A)-2 ut e C1([0, to),L2) and u e W1 for 0 < t < to.

Theorem 1.5 Let n < 3 and letf (u) satisfy (Ai), where 3 < p < to for n = 1,2; 3 < p < 5 for n = 3,0 e W3, f e W3, and (-A)-? 0 e L2. Assume that E(0) < dand0 e W2. Then problem (5) admits a unique global solution u e C2([0, to), W3) with (-A)-2 ut e C1([0, to), L2) and u e ITfor 0 < t < to.

Theorem 1.6 Let n < 3 andf (u) satisfy (A1), where 4 < p < to for n = 1,2; 4 < p < 5 for n = 3,0 e W4, f e W4, and (-A)-2 0 e L2. Assume that £(0) < dand0 e W2. Then problem (5) admitsa unique global solution u e C2([0, to), W4) with (-A)-2 ut e C1([0, to), L2) and u e ITfor 0 < t < to.

Theorem 1.7 Let s > f with s > 1, and f (u) satisfy (A2) with p] > 3 or 0, f e Ws, (-A)-2 0, (—A)-2 f e L2. Assume thatE(0) < d andI(0) < 0. Then the solution of problem (5) blows up in finite time, i.e., the maximal existence time Tm ofu is finite, and

lim sup(||u(t)||Ws + ||u(t)|W^ = to.

t^ 1 m

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 dtS(t), S(t) and r(t) defined in Section 1, we have

dtS№\\HS <\\<p\\w, V0 eHs, (8)

dtSm\\HS < 2(1 + t)\\f \\h, Vf eHs, (9)

dttS(t)4> \\Hs , e Hs, (0)

r(t)f \\H < V2\f \hs-2, Vf e Hs-2, (1)

dtr(tf \\H <\f \\hs-2, Vf e Hs-2. (2)

Proof We only need to prove (9) and (11), since the proofs of the other inequalities are similar. Using the Plancherel theorem, we have

ih ,fc,2v 1+ I £ 12+ I £ I 4 • i t | £ | ^1+ | £ |2 ^7^2^

|9ts(t)f |ws£ |) T+oT+mn ^1+| £|2+| £ | 4)f (£ )| d£

<y < (1+| £ | 2)st2|f (£)|2 d£

+vi£>1(1+|£'2)s If <£"2 d£

< t2 f (1 + |£|2)s|f (£)|2d£ + 4/ (1 + |£|2)s|f (£)|2d£

j|£|<1 j|£|>1

< 4(1 + tf\\f \\H,

Vi + \f\2 + \f \4

i + \f \2 + \f \4 If I4 \2(i + \f \2) (i + \f \2 + \f \4)21

V.M-H2 fti ,tl2\' ■ 2Î t\f W i+\f \2 (t)f "hs = jr{ 1 + lf\)sin( TfTpTp

-if (f )|2 df

< 2 J (1+\ £ \ 2)s-2|/(f)|2 d£ = 2|f ||Hs-2. Therefore (9) and (11) hold. This completes the proof of the lemma. □

Lemma 2.2 ([i7]) Let g e Cm(R), where m > 0 is an integer.

(i) If 0 < s < m and u e Hs(Rn) n Lœ(Rn), then g(u) e Hs(Rn) and

||g(u)||Hs < C(MTOHuHhs. (13)

(ii) If s < m and u, v e Hs(Rn) n LTO(Rn), then

^(u) — g(v)|Hs < K(||u|to, ||v|to|u - v|hs. (14)

In particular, if u, v e Hs for some s > |, then u and v e LTO, (13) and (14) hold. Proof of Theorem 1.1 Let s >

Xs(T) = C1([0, T];Hs),

HuHxs(T) = max {|u(t)|Hs + |ut(t)|Hs} 0<t<Tl" "H " IIH '

Bu = (dtS(t)$)(x) +(S(t)f )(x)+ i r(t- r)f (u(t)) dx,

Ar(T )={u e Xs(T )\|u|xs(T)< R}. Similarly to the proofs in [15,16], we see that for sufficiently small T, Bu : Ar ^ Ar

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, ifTm < to, we have

lim sup(|u(t)|Hs + |ut(t)|H^ = +to.

Corollary 2.4 Let s > | andf (u) satisfy (A2) or (A3). Then, for any $ e Hs and f e Hs, problem (5) admits a unique local solution u e C1([0, Tm),Hs1), where Tm is the maximal existence time ofu. Moreover, either Tm = +to or Tm < to and

lim sup(|u(t)|Hs + |ut(t)|H^ = +TO.

t^ 1 m

Lemma 2.5 Assume s > f e Cm(R), 0 e Ws, and f e Ws. Then for the local solution u e C1([0, Tm), Ws) given in Theorem 1.1, we have utt e C([0, Tm), Ws).

Proof Using the Fourier transformation, we have

if!2 -

m2(i + in2)

Utt _ —-—-77T7 U +

1 + imi2 + imi4 1 + imi2 + imi

-f (u).

|£|2(1+|£|2)<(1+|£|2 + |£|4), |£|2 < (1 + |£|2 + |£|4),

which together with (15) yields

\\utth = iiutt\\hs < \\U\\hs +

< (1 + C\\u\\œ)\\u\\Hs < C(T).

Furthermore, using

\\ Utt (t + At) - Utt (t)\\H = \\Utt (t + At)-Utt (t)\\ < C(T)\U(t + At)-

^ 0 as At ^ 0,

we obtain uu e C([0, Tm);Hs).

Lemma 2.6 Assume s > §, f e Cm(R), 0 e Ws, f e Ws, and (-A)-? 0 e L2. Then for the local solution u e C1([0, Tm), Ws) given in Theorem 1.1, we have (-A)-2 ut e C1([0, Tm),L2).

Proof First for the local solution u given in Theorem 1.1, we obtain (-A)-2 utt e C([0, Tm), Ws+1).

From (15), we get

U _ imi2(1 + imi2) U _ mUtt _ imi(1 + imi2 + imi4)U + imi(1 + imi2 + imi4)

-z(u),

| |(1 + | |2)

1+| |2 + | |4

f (1 + mi)

= i (1 + imi2)s

s+1 mi2(1 + mi2)2

(1 + | |2 + | |4)2 (1 + | |2)4

|u (m)|2 dm |u (m)|2 dm

(1 + imi2 + mi4)2 ci (1 + imi2)s|u(m)|2dm _ cmH,

1+| |2 + | |

-f (u)

f (1 + mi)s

= f (1 + imi)

(1 + | |2 + | |4)2

s+1 (1 + mi2)

(1 + | |2 + | |4)

f (u)|2 dm f (u)|2 dm

< cf (l+|Ç|2)s|/(u)|2dÇ = C\\f(u)\\

< C(||u||№)Huilas.

Furthermore, we get ||(-A) 2utt(t + At)-(-A) 2utt(t)||Hs+1 ^ 0 as At ^ 0. Hence, we have

(-A)-2 utt e C([0, Tm),Hs+1).

(-A)-2 ut e L2

(-A)-2 ut = (-A)-2 f + i (-A)-2 u„ dr, Jo

we get

(-A)-2ut e C1 ([0, Tm),L2). □

Proof of Theorem 1.2 Using (5), it follows by straightforward calculation that

E'(t) = ((-A)-2 utt ,(-A)-2 u^ + (utt, ut) + (V utt, V ut) + (ux, V ut) + (u, ut) + f (u), ut) = ((-A)'1 utt + utt - A utt - Au + u + f (u), ut)XX = 0,

where (•, •) denotes the inner product of L2 space, <•, -)XtX means the usual duality of X* and X with X = H1. Integrating the above equality with respect to t, we have identity (7). The theorem is proved. □

Corollary 2.7 Let s > | with s > 1 andf (u) satisfy (A2), with p] > s or (A3). Assume that $ e Hs, f e Hs, and (-A)-2 f e L2, problem (5) admits a unique local solution u e C2([0, Tm),Hs), with (-A)-2ut e C1([0, Tm),L2) satisfying (7), where Tm is the maximal existence time ofu. Moreover, either Tm = +to orTm < to and

lim sup(||u(t)|Hs + || ut(t)|H^ = +to. (6)

Proof Since $ e Hs and s > 5, we have $ e Lto. Hence, $ e Lq for all 2 < q <to. From \F($)\ = C|$|p+1, 2 <p + 1 < to. We obtain F($) e L1. □

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 Lets > 2 with s > 1 andletf (u) satisfy (A1), u e Hs andg(X) = -X jR„uf (Xu) dx. Assume R uf (u) dx < 0. Then:

(i) g(X) is increasing on 0 < X < to.

(ii) limx^0g(X) = 0, limx g (X) = +to.

Lemma 3.2 Lets > 2 with s > 1, u e Hs, and letf (u) satisfy (A1), u =0. We have:

(i) lima^0J(Xu) = 0.

(ii) I(Xu) = XdXJ(Xu), VX >0. Furthermore, if fR„ uf (u) dx < 0, then:

(iii) limA^

+TO J (Xu) = -TO.

(iv) In the interval 0 < X < to, there exists a unique X* = X*(u) such that

dfxJ (ku)

(v) J (Xu) is increasing on 0 < X < X*, decreasing on X* < X < to and I{X*) = 0.

(vi) I(Xu) > 0 for 0 < X < X*, I (Xu) < 0for X* < X < to and I(X*) = 0.

Lemma 3.3 Let s > n with s > 1, u e Hs, and letf (u) satisfy (Ai). Then:

(i) If 0 < ||u||Hi < r0, then I(u) > 0.

(ii) If I(u) < 0, then ||u|Mi > r0.

(iii) If I(u) = 0 and ||u||Hi =0, then ||u||Hi > r0, where r0 = (^p+r)p-ii.

Lemma 3.4 Let s > n with s > i andf (u) satisfy (Ai). We have

j t p-i 2 p-i / i A*-1 d > d0 = -7-- r2 =

2(p + 1) 0 2(p + 1)V aCP+\ Lemma 3.5 Lets > 2 with s > 1 andf (u) satisfy (A1). Assume u e Hs and I (u) < 0. Then p -1

d < -7--r ||u|H1. (7)

Proof of Theorem 1.3 We only prove the invariance of W\ since the proof for the invariance of V1 is similar. Let u(t, x) be any weak solution of problem (5) with $ e W1, T be the maximal existence time of u(t, x). Next we prove that u(t, x) e W1 for 0 < t < T. Arguing by contradiction we assume there is a t1 e (0, T) such that u(t1) e WÎl. By the continuity of I(u(t)) with respect to t, there exists a t0 e (0, T) such that u(t0) e d Wl. From the definition of Wl and (i) of Lemma 3.3 we have R0 c Wl, R0 = {u e H1 | \\u\\Hi < r0}. Hence we know 0 e d Wl. From u(t0) e d Wl, it holds that I(u(t0)) = 0 and \\u(t0)\Hi =0. The definition of d yields J(u(t0)) > d, which contradicts

i[||(-A)-îut|2 + \\Vut\\2 + \ut\\2] + J(u) <£(0) < d. The proof of Theorem 1.3 is complete. □

From Theorem 1.3, we can prove the following corollaries.

Corollary 3.6 Lets,f (u), $, ty andE(0) be the same as those in Theorem 1.3. Then:

(i) All solutions of problem (5) belong to Wl, provided that $ e W2.

(ii) All solutions of problem (5) belong to V1, provided that $ e V2.

Corollary 3.7 Let s > | with s > 1, and letf (u) satisfy (A2) with [p] > sor (A3), $ e Hs, ty e Hs and (—A)—2ty e L2. Assume thatE(O) < 0 orE(O) = 0, $ = 0. Then all the solutions of problem (5) belong to V1.

4 Global existence of solutions

In this section, we prove the global existence of a solution for problem (5).

Lemma 4.1 Let s > « with s > 1 andf (u) satisfy (A2) with p] > sor $ e №, ty e Hs, and (—A)-2 ty e C1([0, Tm),L2). Assume thatE(0) < d and $ e W2. Then,for the local solution u given in Corollary 2.7, one has

11 m 2 m m 2 4p

II u (t) IIH1 + II u(t) || „1 <-- d. (8)

11 "n " "n p -1

Proof Let u be the unique local solution of problem (5) given in Corollary 2.7. Then u e C2([0, Tm);Hs), (-A)-2ut e C1([0, Tm),L2) satisfying (7) and

1II(—A)-22utI2 + 1 ||Vut||2 + 1 \\utH2 + -f-1 \\u\\2H1 + -^I(u)=E(0) < d. (19) 2" 11 2 2 2(p +1) H p + 1

From Theorem 1.3, we get u e W2 and I(u) > 0 for 0 < t < Tm. Hence, (19) gives rise to

HuH < 2(p +1) d, 0 < t < Tm, (0) H p — 1

2 II(—A)—2utI2+ 2\VutH2 + i \\ut\2< d, 0 < t < Tm. (21)

Thus, we obtain (18). □

Proof of Theorem 1.4 It follows from Corollary 2.7 that problem (5) admits a unique local solution u e C2([0, Tm);H2), with (—A)—2 ut e C1([0, Tm);L2) satisfying (7), where Tm is the maximal existence time of u.

Next, we prove that Tm = +cc. Using Lemma 4.1 one derives (19). Since u e C2([0, Tm); H2) satisfies (5), we have

utt — Autt + A2utt + A2u — Au = Af (u) in C([0, Tm),H—2)

utt + u — Autt + A2utt + A2u — Au = Af (u) +u, 0 < t < Tm. (22)

Multiplying (22) by ut e C1([0, Tm),H2) and integrating on Rn, we obtain

2 dt ["ut\\2 + \\Vut H2 + H Au H2 + HuH2 + \\V u\2 + H AuH2] = —(f'(u)V u, V ut) + (u, ut ). (23)

For n = 3, we get

—(f'(u)Vu, Vut) < I^'(u)13 HVuHeHVut^2. (4)

For n = 3, we have H1 L2 for 2 < q < 6, f'(u)\ = A\u\p 1. From 7 < p < 5, we have 3 <p -1 < 4 and 2 < |(p -1) < 6. Hence, we have |f'(u)|| 3 < C(p) for 0 < t < Tm. From (23) and (24), we get

2 d [H^y2 + UVut "2 + "Aut ||2 + ||u||2 + ||V u||2 + || Au||2]

< C(||«t||2 + ||Vut||2 + ||Aut||2 + ||u||2 + ||Vu||2 + || Au||2). (25)

For n = 1 or 2, we have

-(f (u)Vu, Vut) < If'(u)||3||Vu||3||Vut||3 < C||u||h2 ||«t||h

< C(|ut||2 + ||Vut|2 + ||Aut|2 + ||u|2 + ||Vu|2 + ||Au|2)

and (25). Let

E1(t) = 2 (||ut |2 + ||V ut ||2 + || Aut |2 + ||u||2 + ||V u|2 + || Au|2). Using (25) yields

E1(t) = E1(0) + C i E1(r) dr, 0 < t < Tm

E1(0) < E0eCt, 0 < t < Tm. (6)

From (26), we obtain Tm = +to. If the conclusion Tm = +to is false, then Tm < to. By (26), we get

E1(t) < E1(0)eCTm, for 0 < t < Tm, which contradicts (16). □

Proof of Theorem 1.5 It follows from Corollary 2.7 that problem (5) admits a unique local solution u e C2([0, Tm];H3) and (-A)-2ut e C1([0, Tm);L2). Multiplying (22) by-Aut, we obtain

2 d [||V ut |2 + |V 3ut||2 + ||Aut |2 + ||V u||2 + || V3 u ||2 + ||Au||2]

= -(f'(u)V u, V 3ut) + (V u, V ut), 0 < t < Tm. (27)

From (23) and (27), for 0 < t < Tm, we get

2 d [||ut ||2 + 2||Vut ||2 + ||Aut||2 + ||V3ut ||2

+ ||V 3u|2+ 11 u |2 + ||V 3u||2 + 2|| Au|2] = (u, ut) + (V u, V ut) - (f(u)V u, V 3ut) + (f(u)V u, V ut) (28)

(u, ut) + (V u, V ut) < 2 (HuH2 + Hut H2 + HV uH2 + HV ut H2), —f (u)V u, V ut) < C(Hut H2 + HV ut H2 + H Au H2 + HuH2 +2HVuH2 +2HAuH2), f (u)Vu,V3ut) < I^'(u)I4HVu\41V3utI < CHuHh2Iv3«tI

< C(HuH2H2 + IV3utI2)

< C(Hu\2 + HVu\2 + HAuH2+ IV3utI3).

E2(t) = 2 (Hut H2 + 2HV ut H2 + HAut H2 + IIV 3ut I2 + HuH2 + 2HV uH2 + 2HAuH2+ IV 3uI2).

Using (28) and the estimates above, we obtain

d ft —E2(t) < CE2(t), E2(t) < E2(0) + C E2(t) dT, 0 < t < Tm dt J0

E2(t) < E2(0)eCt, 0 < t < Tm. (9)

Thus Tm = +cc. □

Proof of Theorem 1.6 From Corollary 2.7, it follows that problem (5) admits a unique local solution u e C2([0, Tm];H4), with (—A)—2ut e C1([0, Tm);L2), where Tm is the maximal existence time of u. Multiplying (22) by — A2ut and integrating on Rn, we have

2 dt [HAut H2 + IIV 3utI2 + II A2 ut 12 + HAuH2 + IIV 3uI2 + I A2uI2] = (Af (u), A2ut ) + (Au, Aut ). (30)

From (23) and (30), we obtain

2 dt [Hut H2 + 2HAut H2+ IV3utI2+ II A2ut I2 + HuH2 + HV uH2 + 2HAuH2 + 2IIV 3u 12 + II A2u 12] = (u, ut ) + (Au, Aut ) — Af (u), A2ut) + (Au, Aut ), 0 < t < Tm, (31)

(u,ut) + (Au, Aut) < 2(HuH2 + HutH2 + HAuH2 + HAutH2),

—f'(u)Vu, Vut) < CHuHh2 HutHh2

< C(HuH2 + HV uH2 + HAuH2 + Hut H2 + HAut H2),

(Af (u), A2ut) < ! Af (u)| ! A2ut II

< C(\\u\\tf2 + I A2ut|2)

< C(\u\2 + \\Vu\2 + \Au\2 + I A2ut|2).

E-i(t) = 1 (\ut\2 + 2\\Aut\2 + | V u |2 + \u\2 + \\V u\

+ 21| Au |2 + ||V3 u|2 + || A2 u|2). Then, from (31) and the estimates above, we obtain d ft

—E3(t) < CE3(t), E3(t) < E3(0) + C / E3(r) dr, 0 < t < Tm

E3(t) < E3(0)eCt, 0 < t < Tm, (2)

from which one derives Tm = +to. □

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

(-A)-2 u e C2([0, Tm);L2) provided that (-A)-2 u0 e L2. Proof From (-A)-2u e C2([0, Tm);L2) and

(-A)-2 u = (-A)-2 u0 + / (-A)-2ur dr, 0

we obtain the result. □

Proof of Theorem 1.7 Let u e C2([0, Tm); Hs) and (-A)-2 ut e C1([0, Tm); L2) be the unique local solution of problem (5). Then, by Lemma 5.1, we have (-A)-2u e C2([0, Tm);L2), where Tm is the maximal existence time of u. Suppose that Tm = +to. Then u e C2([0,to);Hs) with (-A)-2u e C2([0, to);L2) and (-A)-2Ut e C1([0, to);L2). Let

H(t)= |(-A)-2u|2 + \u\2+ \Vu\2, 0 < t < to,

H'(t) = 2((-A)-2 ut, (-A)-2 u) + 2(V ut, V u) + 2(u, ut ), (33)

H"(t) = 2||(-A)-I Ut ||2 + 2||V Ut II2 + 2||u II2 + 2((-A)-2 utt,(-A)-2 u) + 2(V Utt, Vu) + 2(utt, u) = 21| (—A)-2 ut|2 + 2||V ut |2 + 21| ut |2 + 2((-A)-1utt, u)

- 2(Autt, u) + 2(utt, u) = 21|(—A)-2ut||2 + 2||Vut||2 + 2||ut||2 -2/(u). (34)

Using the energy equality (7), we obtain

11| (—A)-22 ut||2 + 1 ||ut |2 + 1 ||V ut |2 + 1 ||u|Hi + /" F(u) dx = £(0), 2 2 2 2 J Rn

from which one derives

1 ||(-A)-2 ut||2 + 1 ||ut ||2 + 1 ||V ut ||2 + -f-1 ||u|Hi + -^-I(u)=£(0) 2" 11 2 2 2(p +1) H p + 1

-2I(u) = (p + 1)[||(-A)-2 ut|2 + ||V ut ||2 + ||ut ||2]

+ (p -1)||u|Hi-2(p + 1)£(0). (35)

Substituting (35) into (33), we obtain

H"(t) = (p + 3)[||(-A)-2ut ||2 + ||Vut||2 + ||ut||2] + (p -1)11u|H1 - 2(p + 1)£(0). (36)

On the other hand, from (33), we get

(H'(t))2 = 4((-A)-2 ut ,(-A)-2 u) + (V ut, Vu) + (u, ut )2

= 4[((-A)-2 ut ,(-A)-2 u)2 + (V ut, V u)2 + (u, ut )2 + 2(-A)-2 ut, (-A)-2 u) (V ut, V u) + 2(-A)-2 ut, (-A)-2 u ( u, ut) + 2(V ut, V u)(u, ut)] < 4[||(-A)-2ut||21|(—A)-2u||2 + ||Vut||2||Vu|2 + ||ut||21u|2

+ ||(-A)-2 ut ||2|Vu|2 + ||(-A)-2 u||2 ||V ut ||2+ ||(-A)-2 u||2 ||u|2 + ||(-A)-2 u||2||ut |2 + ||V ut ||21 u |2 + ||V u||2 ||ut |2] = 4H (t)(|| (-A)-2 ut|2 + ||V ut ||2 + ||ut |2). (37)

From (36) and (37), we get

H(t)H"(t) - ^(H'(t))2 > H(t)((p -1) ||u|H1 - 2(p + 1)£(0))

> H(t)((p -1) 11u|H1 - 2(p + 1)d).

Using I(u0) < 0 and Theorem 1.3, we get u e V2 and I(u) < 0 for 0 < t <to. Hence, by

Lemma 3.3, we obtain H(t) > 0 for 0 < t < to. Using Lemma 3.5, we have (p - 1)\\u\H1 >

2(p + 1)d. Thus, we get

H(t)H"(t) - p+3 (H'(t))2 >0, 0 < t < to. (38)

On the other hand, from (36), we obtain

H"(t) > (p -1)||u||H1-2(p + 1)E(0)

= (p - 1)||u|H1 - 2(p + 1)d + 2(p + 1)(d -E(0))

> 2(p + 1)(d -E(0)) = S0, 0 < t < to.

Hence there exists a t0 > 0 such that H' (t) > 0, from which, together with H(t0) > 0 and (37), one derives that there exists a T1 > 0 such that

lim H(t) = to,

which contradicts u e C2([0, to);Hs), (-A)-2 u e C2([0, to);L2). Finally, from Tm < to and Corollary 2.4, we obtain

lim sup( |u(t)|Hs + |ut(t)||Hs) = +TO. □

Competing interests

The author declares to have no competing interests. Author?s contributions

The author declares to have read and approved the final manuscript. Acknowledgements

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).

Received: 10 September 2014 Accepted: 2 December 2014 Published online: 10 January 2015

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