existence and nonexistence of solutions for a generalized boussinesq equation

The existence and uniqueness for the local solution and global solution of the problem are established under certain conditions.. MSC: 35Q20; 76B15 Keywords: Boussinesq equation; blow-up

R E S E A R C H Open Access

Existence and nonexistence of solutions for a

generalized Boussinesq equation

Ying Wang*

* Correspondence:

nadine_1979@163.com

School of Mathematical Sciences,

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

1 Introduction

In , the Boussinesq equation was derived by Boussinesq [] to describe the propa-gation 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 + β

u

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 () has been discussed in [–] In [–], 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 [] investigated the following Boussinesq model:

u tt = u xx + u xxtt + μu xxxx – u xxxxtt + f (u) xx, ()

problem For a degenerate case, Schneider and Eugene [] have proved that the long wave limit can be described approximately by two decoupled Kawahara equations In [, ], 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 Attribu-tion License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribuAttribu-tion, and reproducAttribu-tion in any

of solutions and nonlinear scattering for small amplitude solutions to the Cauchy problem

of () In [, ], 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 () are proved by Wang and Xu [] Wang and Wang [] also proved the global existence

and asymptotic behavior of the solution in n-dimensional Sobolev spaces Recently, Xu et

al.[, ] proved the global existence and finite-time blow-up of the solutions for () by means of the family of potential wells The results in [] improve the results obtained by Wang and Xu []

This work considers the Cauchy problem for the following equation:



u tt – u tt + u tt = –u + u + f (u), x ∈ R n , t > ,

where f (u) satisfies one of the following three assumptions:

(A) f (u) = ±a|u| p or – a|u| p– u, a > , p > ,

(A)



f (u) = ±a|u| p, a > , p > , p = k, k = , , or

f (u) = –a |u| p– u, a > , p > , p = k + , k = , , ,

(A)



f (u) = ±a|u| k, a > , p > , k = , , or

f (u) = –a|u| k+ u, k= , ,

In this paper, we discuss problem () in high dimensional space To our knowledge, there have been few results on the global existence of a solution to problem () In [], Wang and Xue only proved the global existence and finite-time blow-up of the solution to () in one space dimension Though the arguments and methods used in this paper are similar

to those in [], the first equation of problem () is different from () and ()

By the Fourier transform and Duhamel?s principle, the solutionu of problem () can be

written as

u (t, x) =

∂ t S (t)φ

(x) +

S (t)ψ

(x) +

 t



 (t – τ )f

u (τ )

Here (t) = S(t)( –  + )–and



∂ t S (t)φ

(π ) n



R n e ixξcos

 +|ξ|t



 +|ξ|+|ξ|



ˆφ(ξ)dξ,



∂ t S (t)ψ

(π ) n



R n

e ixξsin

 +|ξ|t



 +|ξ|+|ξ|



 +|ξ|+|ξ|

|ξ| +|ξ| ˆψ(ξ)dξ,

where ˆφ (ξ ) = F(φ)(ξ ) =

R n e –i(x,ξ ) φ (x) dx is the Fourier transform of φ(x).

H s denotes the usual Sobolev space on R nwith normu H s=(I – ) s

u  = ( + |ξ|)s ˆu

and|ξ| =ξ+ ξ+· · · + ξ

First, by using the contraction mapping theorem, we obtain the following existence and uniqueness of the local solution to problem ()

Theorem . Let s>n and f ∈ C m with m ≥ s being an integer Then, for any φ ∈ H s and

ψ ∈ H s , the Cauchy problem () has a unique local solution u ∈ C([, T], H s ) Moreover, if

T m is the maximal existence time of u , and

max

≤t<Tm

u (t)

H s+ u t (t)

H s

<∞,

then T m=∞

Theorem . Let s>n and f ∈ C m with m ≥ s being an integer Assume that φ ∈ H s (R n),

ψ ∈ H s (R n ), and (–)–φ ∈ L, F(φ) ∈ L, F(u) = t

f (τ ) dτ Then, for the local solution u,

we have u ∈ C([, T); H s ), (–)–u t ∈ C([, T m ), L), satisfying

E (t) = 

–u t 

+∇u t+∇u+u+u t

+



R n

F (u) dx

In order to use the potential well method, for s > n (s ≥ ) and u ∈ X s (T), we define

J (u) = 

u

H+



R n

F (u) dx,

I (u) = u

H+



R n uf (u) dx,

d= inf

u∈N J (u), N= u ∈ H|I(u) = , u H= ,

W= u ∈ H|I(u) > , J(u) < d∪ {},

V= u ∈ H|I(u) < , J(u) < d

and

W= u ∈ H|I(u) > ∪ {},

V= u ∈ H|I(u) < 

From u ∈ C([, T]; H s ), we get u ∈ C([, T]; L∞) and u ∈ C([, T]; L q) for all ≤ q < ∞ Hence, J(u), I(u), d, W, and Vare all well defined Now, we give the following results for problem ()

Theorem . Let s>n with s ≥ , and f (u) satisfy (A) with [p] ≥ s or (A) Assume that

φ ∈ H s , ψ ∈ H s , and (–)–ψ ∈ L, E() < d Then both Wand V are invariant under the flow of problem()

Theorem . Let n ≤  and f (u) satisfy (A), where  ≤ p < ∞ for n = , ;

 ≤ p ≤  for

n = , φ ∈ H, ψ ∈ H, and (–)–φ ∈ L Assume E() < d and φ ∈ W Then problem ()

admits a unique global solution u ∈ C([,∞), H), with (–)–u t ∈ C([,∞), L) and

u ∈ W for≤ t < ∞.

Theorem . Let n ≤  and let f (u) satisfy (A), where  ≤ p < ∞ for n = , ;  ≤ p ≤  for

n = , φ ∈ H, ψ ∈ H, and (–)–φ ∈ L Assume that E() < d and φ ∈ W Then problem () admits a unique global solution u ∈ C([,∞), H) with (–)–u t ∈ C([,∞), L) and

u ∈ W for  ≤ t < ∞.

Theorem . Let n ≤  and f (u) satisfy (A), where  ≤ p < ∞ for n = , ;  ≤ p ≤  for

n = , φ ∈ H, ψ ∈ H, and (–)–φ ∈ L Assume that E() < d and φ ∈ W Then problem () admits a unique global solution u ∈ C([,∞), H) with (–)–u t ∈ C([,∞), L) and

u ∈ W for  ≤ t < ∞.

Theorem . Let s> n

 with s ≥ , and f (u) satisfy (A) with [p] ≥  or φ, ψ ∈ H s,

(–)–φ , (–)–ψ ∈ L Assume that E() < d and I(φ) <  Then the solution of problem () blows up in finite time, i.e., the maximal existence time T m of u is finite , and

lim

t→T m

sup u (t)

H s+ u t (t)

H s



The remainder of this paper is organized as follows In Section , Theorems . and . are proved In Section , we give some preliminary lemmas and the proof of Theorem . The proofs of Theorems ., . and . are given in Section  Finally, Section  is devoted

to the proof of Theorem .

2 Existence of local solutions

In this section, we consider the local existence and uniqueness of solutions to problem ()

Lemma . For the operators ∂ t S (t), S(t) and (t) defined in Section , we have

∂ t S (t)φ

∂ t S (t)ψ

∂ tt S (t)φ

 (t)f

∂ t  (t)f

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

∂ t S (t)ψ 

H s=



R



 +|ξ|s +|ξ|+|ξ|

|ξ|( +|ξ|) sin



t |ξ| +|ξ|



 +|ξ|+|ξ|

 ˆψ(ξ)

dξ

≤



|ξ|≤



 +|ξ|s

t ˆψ (ξ )

dξ

+



|ξ|>



 +|ξ|s( +|ξ|+|ξ|)

|ξ|( +|ξ|)  ˆψ (ξ )

dξ

≤ t



|ξ|≤



 +|ξ|s ˆψ (ξ )

dξ+ 



|ξ|>



 +|ξ|s ˆψ (ξ )

dξ

≤ ( + t)ψ

s,

 (t)f 

H s=



R



 +|ξ|s

sin



t |ξ| +|ξ|



 +|ξ|+|ξ|



× +|ξ||ξ|+|ξ| ( +|ξ|)

|ξ|

( +|ξ|+|ξ|)ˆf (ξ )

dξ

≤ 



R



 +|ξ|s–ˆf (ξ )

dξ= f 

H s–

Lemma .([]) Let g ∈ C m (R), where m ≥  is an integer.

(i) If  ≤ s ≤ m and u ∈ H s (R n)∩ L∞(R n), then g(u)∈ H s (R n)and

g (u)

H s ≤ Cu∞

(ii) If s ≤ m and u, v ∈ H s (R n)∩ L∞(R n), then

g (u) – g(v)

H s ≤ Ku∞,v∞

In particular, if u, v ∈ H s for some s > n, then u and v ∈ L∞, () and () hold

Proof of Theorem . Let s > n,

X s (T) = C

[, T]; H s

,

u X s (T)= max

≤t≤T u (t)

H s+ u t (t)

H s

 and

Bu=

∂ t S (t)φ

(x) +

S (t)ψ

(x) +

 t



 (t – τ )f

u (τ )

dτ,

A R (T) = u ∈ X s (T)|u X s (T) < R

Similarly to the proofs in [, ], we see that for sufficiently small T ,

Bu : A R → A R

is a contract mapping Hence by the contracting-mapping principle we obtain the result

Corollary . Under the assumption of Theorem ., if T m<∞, we have

lim

t→T m

sup u (t)

H s+ u t (t)

H s



= +∞

Corollary . Let s> n and f (u) satisfy (A) or (A) Then, for any φ ∈ H s and ψ ∈ H s,

problem () admits a unique local solution u ∈ C([, T m ), H s), where T m is the maximal existence time of u Moreover, either T m= +∞ or Tm<∞ and

lim

t→T m

sup u (t)

H s+ u t (t)

H s



= +∞

Lemma . Assume s>n, f ∈ C m (R), φ ∈ H s , and ψ ∈ H s Then for the local solution

u ∈ C([, T m ), H s ) given in Theorem ., we have u tt ∈ C([, T m ), H s)

Proof Using the Fourier transformation, we have

ˆu tt= –|ξ|( +|ξ|)

 +|ξ|+|ξ|ˆu + |ξ|

Since

|ξ|

 +|ξ|

<

 +|ξ|+|ξ|

,

|ξ|<

 +|ξ|+|ξ|

, which together with () yields

u ttH s=ˆu ttH s ≤ ˆu H s+ ˆf (u)

H s≤ + Cu∞

u H s ≤ C(T).

Furthermore, using

u tt (t + t) – u tt (t)

H s = ˆu tt (t + t) – ˆu tt (t)

H s

≤ C(T) u (t + t) – u(t)

Lemma . Assume s> n, f ∈ C m (R), φ ∈ H s , ψ ∈ H s , and (–)–φ ∈ L Then for the

local solution u ∈ C([, T m ), H s ) given in Theorem ., we have (–)–u t ∈ C([, T m ), L)

Proof First for the local solution u given in Theorem ., we obtain

(–)–u tt ∈ C[, T m ), H s+

From (), we get



|ξ| ˆu tt= – |ξ|( +|ξ|)

|ξ|( + |ξ|+|ξ|)ˆu + |ξ|

|ξ|( + |ξ|+|ξ|)ˆf(u),

 +|ξ|( + |ξ| |ξ|+|ξ|)ˆu



H s+

=



R n



 +|ξ|s+ |ξ|( +|ξ|)

( +|ξ|+|ξ|)ˆu (ξ )

dξ

≤



R n



 +|ξ|s ( +|ξ|)

( +|ξ|+|ξ|)ˆu (ξ )

dξ

≤ C



R n



 +|ξ|sˆu (ξ )

dξ = C u

H s,

 +|ξ| |ξ|+|ξ|ˆf(u)



H s+

=



R n



( +|ξ|+|ξ|)ˆf (u)

dξ

≤



n



 +|ξ|s+ ( +|ξ|)

( +|ξ|+|ξ|)ˆf (u)

dξ

≤ C



R n



 +|ξ|sˆf (u)

dξ = C ˆf (u) 

H s

≤ Cu H s

u

H s

Furthermore, we get(–)–u tt (t + t) – (–)–u tt (t)H s+→  as t → .

Hence, we have

(–)–u tt ∈ C[, T m ), H s+

Using

(–)–u t ∈ L

and

(–)–u t = (–)–ψ+

 t



(–)–u τ τ dτ,

we get

(–)–u t ∈ C

[, T m ), L

Proof of Theorem. Using (), it follows by straightforward calculation that

E(t) =

(–)–u tt , (–)–u t



+ (u tt , u t) + (∇u tt,∇u t ) + (u x,∇u t ) + (u, u t) +

f (u), u t



=

(–)–u tt + u tt – u tt – u + u + f (u), u t

X∗X= , where (·, ·) denotes the inner product of Lspace,·, ·X∗X means the usual duality of X∗ and X with X = H Integrating the above equality with respect to t, we have identity ().

Corollary . Let s> n with s ≥  and f (u) satisfy (A), with [p] ≥ s or (A) Assume

that φ ∈ H s , ψ ∈ H s , and (–)–ψ ∈ L, problem () admits a unique local solution

u ∈ C([, T m ), H s ), with (–)–u t ∈ C([, T m ), L) satisfying (), where T m is the max-imal existence time of u Moreover, either T m= +∞ or T m<∞ and

lim

t→T m

sup u (t)

H s+ u t (t)

H s



Proof Since φ ∈ H s and s > n, we have φ ∈ L∞ Hence, φ ∈ L q for all ≤ q ≤ ∞ From

3 Preliminary lemmas and invariant sets

In this section, we will prove several lemmas which are related with the potential well for problem () By arguments similar to those in [], we obtain the following lemmas

Lemma . Let s>nwith s ≥  and let f (u) satisfy (A), u ∈ H s and g (λ) = –λ

R n uf (λu) dx.

Assume

n uf (u) dx <  Then:

(i) g(λ) is increasing on  < λ <∞.

(ii) limλ→g (λ) = , lim λ→+∞g (λ) = +∞.

Lemma . Let s>n with s ≥ , u ∈ H s , and let f (u) satisfy (A), u =  We have:

(i) limλ→ J (λu) = .

(ii) I(λu) = λ dλ d J (λu), ∀λ > .

Furthermore , if

R n uf (u) dx < , then:

(iii) limλ→+∞ J (λu) = –∞

(iv) In the interval  < λ < ∞, there exists a unique λ∗= λ∗(u) such that

d

dλ J (λu)





λ=λ∗= 

(v) J(λu) is increasing on  < λ < λ∗, decreasing on λ∗< λ < ∞ and I(λ∗) = 

(vi) I(λu) >  for  < λ < λ∗, I(λu) <  for λ∗< λ < ∞ and I(λ∗) = 

Lemma . Let s>n

 with s ≥ , u ∈ H s , and let f (u) satisfy (A) Then:

(i) If  < u H< r, then I(u) > .

(ii) If I(u) < , then u H> r

(iii) If I(u) =  and u H= , then u H≥ r, where r= ( 

aC p∗+)p–

Lemma . Let s>n

 with s ≥  and f (u) satisfy (A) We have

d ≥ d= p– 

(p + ) r



= p– 

(p + )





aC p+∗

p–

Lemma . Let s>n with s ≥  and f (u) satisfy (A) Assume u ∈ H s and I (u) <  Then

d< p– 

(p + ) u

Proof of Theorem . We only prove the invariance of Wsince the proof for the invariance

maximal existence time of u(t, x) Next we prove that u(t, x) ∈ Wfor  < t < T Arguing

by contradiction we assume there is a t∈ (, T) such that u(t) /∈ W By the continuity of

I (u(t)) with respect to t, there exists a t∈ (, T) such that u(t)∈ ∂W From the definition

of Wand (i) of Lemma . we have R⊂ W, R={u ∈ H| u H< r} Hence we know

 /∈ ∂W From u(t)∈ ∂W, it holds that I(u(t)) =  andu(t)H=  The definition of

d yields J(u(t))≥ d, which contradicts



–u t 

+∇u t+u t

+ J(u) ≤ E() < d.

From Theorem ., we can prove the following corollaries

Corollary . Let s , f (u), φ, ψ and E() be the same as those in Theorem . Then:

(i) All solutions of problem () belong to W, provided that φ ∈ W

(ii) All solutions of problem () belong to V , provided that φ ∈ V

Corollary . Let s>n with s ≥ , and let f (u) satisfy (A) with [p] ≥ s or (A), φ ∈ H s,

ψ ∈ H s and (–)–ψ ∈ L Assume that E() <  or E() = , φ =  Then all the solutions

of problem () belong to V

4 Global existence of solutions

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

Lemma . Let s>n with s ≥  and f (u) satisfy (A) with [p] ≥ s or φ ∈ H s , ψ ∈ H s , and (–)–ψ ∈ C([, T m ), L) Assume that E() < d and φ ∈ W Then, for the local solution

u given in Corollary ., one has

u t (t) 

H+ u (t) 

H< p

Proof Let u be the unique local solution of problem () given in Corollary . Then u∈

C([, T m ); H s ), (–)–u t ∈ C([, T m ), L) satisfying () and



 (–)–u t 

∇u t+

u t+ p– 

(p + ) u

H+ 

p+ I (u) = E() < d. ()

From Theorem ., we get u ∈ Wand I(u) ≥  for  ≤ t ≤ T m Hence, () gives rise to

u

H<(p + )



 (–)–u t 

∇u t+

Proof of Theorem. It follows from Corollary . that problem () admits a unique local

solution u ∈ C([, T m ); H), with (–)–u t ∈ C([, T m ); L) satisfying (), where T m is

the maximal existence time of u.

Next, we prove that T m= +∞ Using Lemma . one derives () Since u ∈ C([, T m);

H) satisfies (), we have

u tt – u tt + u tt + u – u = f (u) in C

[, T m ), H– and

Multiplying () by u t ∈ C([, T m ), H) and integrating on R n, we obtain





d

dt u t+∇u t+u t+u+∇u+u

= –

f(u)∇u, ∇u t



For n = , we get

–

f(u)∇u, ∇u t



For n = , we have H → L for ≤ q ≤ , |f(u)| = A|u| p– From  ≤ p ≤ , we have



(p – ) ≤  Hence, we have f(u) ≤ C(p) for  ≤ t < T m From () and (), we get





d

dt u t+∇u t+u t+u+∇u+u

≤ Cu t+∇u t+u t+u+∇u+u

For n =  or , we have

–

f(u)∇u, ∇u t



≤ f(u)

∇u∇u t≤ Cu Hu tH

≤ Cu t+∇u t+u t+u+∇u+u and () Let

E(t) =





u t+∇u t+u t+u+∇u+u

Using () yields

E(t) = E() + C

 t



E(τ ) dτ , ≤ t < T m

and

From (), we obtain T m= +∞ If the conclusion T m= +∞ is false, then T m<∞ By (),

we get

E(t) ≤ E()e CT m, for ≤ t < T m,

Proof of Theorem. It follows from Corollary . that problem () admits a unique local

solution u ∈ C([, T m ]; H) and (–)–u t ∈ C([, T m ); L)





d

dt ∇u t+ ∇u t 

+u t+∇u+ ∇u 

+u

= –

f(u)∇u, ∇u t



From () and (), for ≤ t < T m, we get





d

dt u t+ ∇u t+u t+ ∇u t 

+ ∇u 

+u+ ∇u 

= (u, u) + (∇u, ∇u) –

f(u)∇u, ∇u

+