On the Blow-Up of Solutions of a Weakly

Journal of Applied Mathematics Volume 2012, Article ID 197672,20pages doi:10.1155/2012/197672

Research Article

On the Blow-Up of Solutions of a Weakly

Dissipative Modified Two-Component Periodic Camassa-Holm System

Yongsheng Mi,

Chunlai Mu,

and Weian Tao

1College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

2College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling, Chongqing 408100, China

Correspondence should be addressed to Yongsheng Mi,miyongshen@163.com Received 16 May 2012; Revised 24 July 2012; Accepted 30 July 2012

Academic Editor: Ferenc Hartung

Copyrightq2012 Yongsheng Mi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa- Holm equation. We first establish the local well-posedness result. Then we derive the precise blow- up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow- up results for strong solutions to the system.

1. Introduction

In this paper, we consider the Cauchy problem of the following weakly dissipative modified two-component Camassa-Holm system:

m_tum_x2mu_xρρ_xλm0, t >0, x∈R, ρt

ρu

xλρ0, t >0, x∈R, m0, x m0x, x∈R,

ρ0, x ρ₀x, x∈R, mt, x1 mt, x, t≥0, x∈R,

ρt, x1 ρt, x, t≥0, x∈R,

1.1

wherem 1−∂²_xu,ρ 1−∂²_xρ−ρ₀, andλis a nonnegative dissipative parameter.

The Camassa-Holm equation 1 has been recently extended to a two-component integrable systemCH2

m_tum_x2mu_xρρ_x, t >0, x∈R, ρt

ρu

x0, t >0, x∈R, 1.2

withmu−u_xx, which is a model for wave motion on shallow water, whereut, xdescribes the horizontal velocity of the fluid, andρt, xis in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover,uandρsatisfy the boundary conditions: u → 0 and ρ → 1 as |x| → ∞. The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions2. Moreover, it is connected with the time-dependent Schr ¨odinger spectral problem2. Popowicz3observes that the system is related to the bosonic sector of anN 2 supersymmetric extension of the classical Camassa-Holm equation. Equation1.2 withρ≡0 becomes the Camassa-Holm equation, which has global conservative solutions4 and dissipative solutions5.

Since the system was derived physically by Constantin and Ivanov6in the context of shallow water theoryalso by Chen et al. in2and Falqui et al. in7, many researchers have paid extensive attention to it. In 8, Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to 1.2on the line. In6, Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of1.2 on the line. Later, Guan and Yin9obtain a new global existence result for strong solutions to1.2and get several blow-up results, which improve the recent results in6. Recently, they study the global existence of weak solutions to1.2 10. In11, Henry studies the infinite propagation speed for1.2. Gui and Liu12 establish the local well-posedness for1.2in a range of the Besov spaces, they also derive a wave breaking mechanism for strong solutions. Mustafa13gives a simple proof of existence for the smooth travelling waves for1.2. Hu and Yin14,15study the blow-up phenomena and the global existence of1.2on the circle.

Recently, the CH2 system was generalized into the following modified two-component Camassa-HolmMCH2system:

m_tum_x2mu_x−gρρ_x, t >0, x∈R, ρ_t

ρu

x0, t >0, x∈R, 1.3

wherem 1−∂²_xu,ρ 1−∂²_xρ−ρ₀,udenotes the velocity field,ρ₀ is taken to be a constant, andgis the downward constant acceleration of gravity in applications to shallow water waves. This MCH2 system admits peaked solutions in the velocity and average density, we refer this to 16 for details. There, the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density ρ at the point of vertical slope. Some other recent work can be found in 17–25. We find that the MCH2 system is expressed in terms of an averaged or filtered density ρ in analogy to the relation between momentum and velocity by setting

ρ 1−∂²_xρ−ρ₀, but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law:

R

u²u²_xρ²ρ²_x

dx, 1.4

which play a crucial role in the study of1.3. Noting that for the CH2 system, we cannot obtain the conservation ofH¹norm.

In general, it is quite difficult to avoid energy dissipation mechanisms in a real world.

Ghidaglia26studies the long time behaviour of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin 27 study the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation. In 28,29, Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system on the circle and on the line. However,1.1on the circleperiodic casehas not been studied yet. The aim of this paper is to study the blow- up phenomena of the strong solutions to1.1. We find that the behavior of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system1.1is similar to that of the modified two-component Camassa- Holm system1.3, such as the local well- posedness and the blow-up scenario. In addition, we also find that the blow-up rate of1.1is not affected by the weakly dissipative term, but the occurrence of blow-up of1.1is affected by the dissipative parameter.

This paper is organized as follows: In Section 2, we establish local well-posedness of the Cauchy problem associated with 1.1. In Section 3, we derive precise the blow- up scenario of strong solution and the blow-up rate. In Section 4, we discuss the blow-up phenomena of1.1.

2. Local Well-Posedness

In this section, by applying Kato’s semigroup theory 30, we can obtain the local well- posedness for the Cauchy problem of1.1inH^s×H^s,s > 3/2, with withS R/Zthe circle of unit length.

First, we introduce some notations. All spaces of functions are assumed to be overS;

for simplicity, we dropSin our notation for function spaces if there is no ambiguity. IfAis an unbounded operator, we denote byDAthe domain ofA.A;Bdenotes the commutator of two linear operatorsAandB. · _X denotes the norm of Banach spaceX. We denote the norm and the inner product ofH^s;s∈R, by · _sand·,·_s, respectively.

For convenience, we state here Kato’s theorem in the form suitable for our purpose.

Consider the following abstract quasilinear evolution equation:

du

dt Au fu, t >0, u0 u₀. 2.1

LetX andY be Hilbert spaces such thatY is continuously and densely embedded inXand letQ:Y → Xbe a topological isomorphism.LY, Xdenotes the space of all bounded linear operator fromY toXand we writeLX, ifXY.

Theorem 2.1see30. Assume that iAy∈LY, Xfory∈Xwith

A y

−Az w

X ≤μ₁y−z

Xw_Y, y, z, w∈Y, 2.2 andAy∈GX,1, βuniformly on bounded sets inY.

iiQAyQ⁻¹ Ay By, whereBy∈LXis bounded, uniformly on bounded sets in Y. Moreover,

B y

−Bz w

X≤μ₂y−z

Yw_X, y, z,∈Y, w∈X. 2.3 iiif :Y → Y and extends also to a map fromXintoX, f is bounded on bounded sets inY

and

f y

−fz

Y ≤μ₃y−z

Y, y, z∈Y, f

y

−fz

X ≤μ3y−z

X, y, z∈Y, 2.4

where, μ₁, μ₂, and μ₃ depend only on max{y_X,z_X} and μ₄ depends only on max{y_Y,z_Y}. If the above conditions (i), (ii), and (iii) hold, given u₀ ∈ Y, there is a maximalT >0 depending only onu0_Y and a unique solution u to2.1such that

uu·, u0∈C0, T;Y∩C¹0, T;X. 2.5

Moreover, the mapu₀ → u·, u0is continuous fromYtoC0, T;Y∩C¹0, T;X.

We now provide the framework in which we will reformulate system1.1. Withm u−u_xx,ργ−γ_xx, andγρ−ρ₀, we can rewrite1.1as follows:

mtumx2muxργxλm0, t >0, x∈R, ρ_t

ρu

xλρ0, t >0, x∈R, m0, x u₀x−u_0,xxx, x∈R, ρ0, x γ0x−γ0,xxx, x∈R, mt, x1 mt, x, t≥0, x∈R,

ρt, x1 ρt, x, t≥0, x∈R.

2.6

Note that ifpx:coshx−x−1/2/2 sinh1/2,x∈Ris the kernel of1−∂²_x⁻¹, where xstands for the integer part ofx∈R, then1−∂²_x⁻¹f p∗ffor allf ∈L²S,p∗mu,

andp∗ργ. Here we denote by∗the convolution. Using this identity, we can rewrite2.6 as follows:

u_tuu_x−∂xp∗

u²1 2u²_x1

2γ²−1

2γ_x² −λu, t >0, x∈R, γ_tuγ_x−p∗

u_xγ_x

xu_xγ

−λγ, t >0, x∈R, u0, x u₀x, x∈R,

γ0, x γ0x, x∈R, ut, x1 ut, x, t≥0, x∈R, γt, x1 γt, x, t≥0, x∈R,

2.7

or we can write it in the following equivalent form:

utuux−∂x

1−∂²_x−1 u²1

2u²_x1 2γ²−1

2γ_x² −λu, t >0, x∈R, γ_tuγ_x−∂x

1−∂²_x−1 u_xγ_x

−

1−∂²_x−1

u_xγ−λγ, t >0, x∈R, u0, x u₀x, x∈R,

γ0, x γ0x, x∈R, ut, x1 ut, x, t≥0, x∈R, γt, x1 γt, x, t≥0, x∈R.

2.8

Theorem 2.2. Given z0 zx,0 u0, γ0 ∈ H^s×H^ss > 3/2, then there exist a maximal T Tz0>0 and a unique solutionz u, γto1.1or2.7such that

zz·, z0∈C0, T;H^s×H^s∩C¹

0, T;H^s−1×H^s−1

. 2.9

Moreover, the solution depends continuously on the initial data, that is, the mappingz0 → z·, z0: H^s×H^s → C0, T;H^s×H^s∩C¹0, T;H^s−1×H^s−1is continuous and the maximal time of existenceT >0 can be chosen to be independent ofs.

The remainder of this section is devoted to the proof of Theorem2.2.

Letz_u

γ

,Az

u∂x, 0 0, u∂x

and

fz

⎛

⎝−∂x

1−∂²_x−1 u² 1

2u²_x1 2γ²−1

2γ_x² −λu

−∂x

1−∂²_x−1 uxγx

−

1−∂²_x−1

uxγ−λγ.

⎞

⎠. 2.10

SetY H^s ×H^s,X H^s−1×H^s−1,Λ 1−∂²_x^1/2 and Q _Λ0

0 Λ

. Obviously, Q is an isomorphism of H^s ×H^s onto H^s−1 ×H^s−1. In order to prove Theorem 2.2 by applying Theorem2.1, we only need to verifyAzandfzwhich satisfy the conditionsi–iii.

We break the argument into several lemmas.

Lemma 2.3. The operatorAz

u∂x, 0 0, u∂x

, withz∈H^s×H^s,s >3/2, belongs toGL²×L²,1, β.

Lemma 2.4. The operatorAz

u∂x, 0 0, u∂x

, withz ∈ H^s×H^s,s > 3/2, belongs toGH^s−1× H^s−1,1, β.

Lemma 2.5. Az

u∂x, 0 0, u∂x

, withz∈H^s×H^s,s >3/2. The operatorAz∈LH^s×H^s, H^s−1× H^s−1. Moreover,

A y

−Az w

H^s−1×H^s−1 ≤μ₁y−z

H^s×H^sw_Hs×H^s, y, z, w∈ ×H^s. 2.11

Lemma 2.6. The operatorBz Q, AzQ⁻¹withz∈H^s×H^s,s >3/2. ThenBz∈LH^s−1× H^s−1and

B y

−Bz w

H^s−1×H^s−1 ≤μ₂y−z

H^s×H^sw_Hs−1×H^s−1, 2.12

fory, z∈H^s×H^sandw∈H^s−1×H^s−1.

The proof of the above five lemmas can be done similarly as in8, therefore we omit it here.

Hence, according to Kato’s theoremTheorem2.1, in order to prove Theorem2.2, we only need to verify conditioniii, that is, we need to prove the following lemma.

Lemma 2.7. Letz∈H^s×H^s,s >3/2 and

fz

⎛

⎜⎜

⎜⎜

⎝

−∂x

1−∂²_x−1 u² 1

2u²_x1 2γ²−1

2γ_x² −λu

−∂x

1−∂²_x−1 uxγx

−

1−∂²_x−1

uxγ−λγ

⎞

⎟⎟

⎟⎟

⎠. 2.13

Thenfis bounded on bounded sets inH^s×H^sand satisfies

afy−fz_Hs×H^s≤μ₃y−z_Hs×H^s, y, z∈H^s×H^s, bfy−fz_Hs−1×H^s−1≤μ4y−z_Hs−1×H^s−1, y, z∈H^s×H^s.

Proof. Lety, z∈H^s×H^s,s >3/2. SinceH^s−1is a Banach algebra, it follows that f

y

−fz

H^s×H^s

≤ −∂x

1−∂²_x−1

y²₁−u² 1

2

y²_1x−u²_x 1

2

y²₂−γ²

−1 2

y_2x² −γ_x² −λ

y₁−u

H^s

−∂x

1−∂²_x−1

y_1xy_2x−u_xγ_x

−

1−∂²_x−1

y_1xy₂−u_xγ

−λ

y₂−γ

H^s

≤y₁−u

y₁u

H^s−11

2y_1x−u_x

y_1xu_x

H^s−1 1

2y₂−γ

y₂γ

H^s−1

1

2y2x−γx

y2xγx

H^s−1λy1−u

H^sux

y2x−γx

H^s−1

y_1x−u_x y_2x

H^s−1u_x

y₂−γ

H^s−2y_1x−u_x y₂

H^s−2λy₂−γ

≤cy₁−u

H^s−1y₁u

H^s−11

2y₁−u

H^sy₁u

H^s1

2y₂−γ

H^s−1y₂γ

H^s−1

1

2y2−γ

H^sy2γ

H^s−1λy1−u

H^su_H^sy2−γ

H^sy1−u

H^sy2

H^s

λy₂−γ

H^su_Hs−1y₂−γ

H^s−2y₁−u

H^s−1y₂

H^s−2

≤cy

H^s×H^sz_H^s_×H^sλy−z

H^s×H^s.

2.14

This provesa. Takingy0 in the above inequality, we obtain thatfis bounded on bounded set inH^s×H^s.

Next, we proveb. Note thatH^s−1is a Banach algebra. Then, we have f

y

−fz

H^s−1×H^s−1

≤ −∂x

1−∂²_x−1

y²₁−u² 1

2

y²_1x−u²_x 1

2

y²₂−γ²

−1 2

y_2x² −γ_x² −λ

y₁−u

H^s−1

−∂x

1−∂²_x−1

y_1xy_2x−u_xγ_x

−

1−∂²_x−1

y_1xy₂−u_xγ

−λ

y₂−γ

H^s−1

≤y₁−u

y₁u

H^s−21

2y_1x−u_x

y_1xu_x

H^s−2 1

2y₂−γ

y₂γ

H^s−2

1

2y_2x−γ_x

y_2xγ_x

H^s−2λy₁−u

H^s−1u_x

y_2x−γ_x

H^s−2

y_1x−u_x y_2x

H^s−2u_x

y₂−γ

H^s−3y_1x−u_x y₂

H^s−3λy₂−γ

≤cy₁−u

H^s−2y₁u

H^s−11

2y₁−u

H^s−1y₁u

H^s−11

2y₂−γ

H^s−2y₂γ

H^s−2

1

2y₂−γ

H^s−1y₂γ

H^s−2λy₁−u

H^s−1u_Hsy₂−γ

H^s−1

y₁−u

H^s−1y₂

H^s−1λy₂−γ

H^s−1u_Hs−2y₂−γ

H^s−3y₁−u

H^s−2y₂

H^s−3

≤cy

H^s×H^s−1z_Hs−1×H^s−1λy−z

H^s−1×H^s−1.

2.15

This provesband completes the proof of the Lemma2.7.

Proof of Theorem2.2. Combining Theorem2.1and Lemmas2.3–2.7, we can get the statement of Theorem2.2.

3. The Precise Blow-Up Scenario and Blow-Up Rate

In this section, we present the precise blow-up scenario and the blow-up rate for strong solutions to2.7.

Lemma 3.1. Letz₀ u0, γ₀∈H^s×H^s,s >3/2, and letT be the maximal existence time of the solutionz u, γto2.7with the initial dataz₀. Then for allt∈0, T, we have

ut,·²_H1γt,·²

H¹e^−2λt

u0²_H1γ0²

H¹

. 3.1

Proof. Denote

f u, γ

u²1 2u²_x1

2γ²−1

2γ_x², g g u, γ

u_xγ_x

xu_xγ. 3.2

In view of the identity−∂²_xp∗ff−p∗f, we can obtain from2.7,

utx−u²_x−uuxxf−p∗f, γtx−uxγx−uγxx−∂xp∗g. 3.3

Therefore, an integration by parts yields

1 2

d dt

u²_H1γ²

H¹

R

uutuxutxγγtγxγtx

dx

R

u

−uux−∂²_xp∗f−λu u_x

−u²_x−uu_xxf−p∗f−λu_x γ

−uγx−p∗g−λγ γ_x

−uγx−uγ_xx−∂_xp∗g−λγ_x

R

−1 2u³_xux

u² 1

2u²_x1 2γ²−1

2γ_x² −uγγx−γ

uxxγxuxγ

−uγxγ_x²−uγxγxx−λ

u²u²_xγ²γ_x² dx −λ

R

u²u²_xγ²γ_x² dx.

3.4

Thus, the statement of the conservation law follows.

Lemma 3.2see31. iFor everyf ∈H¹S, we have

x∈0,1maxf²x≤ e1 2e−1f²

H¹, 3.5

where the constante1/2e−1is sharp.

iiFor everyf∈H³S, we have

x∈0,1max f²x≤cf²

H¹, 3.6

with the best possible constantclying within the range1,13/12. Moreover, the best constantcis e1/2e−1.

So, ifz∈H³×H³, then by Lemmas3.1and3.2, we have ut,·²_L∞γt,·²

L^∞ ≤ e1

2e−1u²_H1 e1 2e−1γ²

H¹

e1 2e−1

u0²_H1γ₀²

H¹

e1

2e−1z0²_H1×H¹,

3.7

for allt∈0, T.

Theorem 3.3. Letz0 u, γ ∈ H^s×H^s,s > 3/2 be given and assume thatT is the maximal existence time of the corresponding solutionz u, γ to2.7with initial dataz₀, if there exists M >0 such that

uxt,·_L∞γxt,·

L^∞ ≤M, t∈0, T, 3.8

then theH^s×H^snorm ofzt,·does not blow-up on0, T.

The proof of the theorem is similar to the proof of Theorem 3.1 in20, we omit it here.

Consider the following differential equation equation:

dqx, t dt u

qx, t, t

, t∈0, T, q0, t x, x∈R,

3.9

whereudenotes the first component of the solutionzto2.7. Applying classical results in the theory of ordinary differential equations, one can obtain the following result onqwhich is crucial in the proof of blow-up scenario.

Lemma 3.4see8. Letu₀ ∈ ∩C¹0, T;H^s−1,s >3/2, andTbe the maximal existence time of the corresponding solutionut, xto3.7. Then3.7has a unique solutionq∈C¹0, T×R, R.

Moreover, the mapqt,·is an increasing diffeomorphism ofRwith

qxx, t exp _t

0

ux

qx, s, s ds

>0, qxx,0 1, x∈R, 0≤t < T. 3.10

The following result is proved only with regard to r 3, since we can obtain the same conclusion for the general caser >3/2 by using Theorem2.1and a simple density argument.

We now present a precise blow-up scenario for strong solutions to2.6.

Theorem 3.5. Lety₀ u0, γ₀ ∈ H^s×H^s,s > 3/2, and letT be the maximal existence of the corresponding solutionz u, γto2.7. Then the solution blows up in finite time if and only if

lim inf

t→T,x∈Ruxt, x −∞ or lim sup

t→T

γxt,·

L^∞

∞. 3.11

Proof. Multiplying the first equation in 2.6 bym u−u_xx and integrating by parts, we obtain

d dt

S

m²dx2

S

mmtdx2

S

m

−umx−2mux−ργx

dx−2λ

S

m²dx −3

S

m²u_xdx−2

S

mργ_xdx−2λ

S

m²dx.

3.12

Repeating the same procedure to the second equation in2.6we get d

dt

S

ρ²dx−

S

ρ²ux−2λ

S

ρ²dx. 3.13

A combination of3.7and3.9yields d

dt

S

m²ρ²

dx−3

S

m²u_xdx−2

S

mργ_xdx−

S

ρ²u_x−2λ

S

m²ρ²

dx. 3.14

Differentiating the first equation in2.6with respect tox, multiplying bymx ux−uxxx, then integrating overS, we obtain

d dt

S

m²_xdx−5

S

m²_xu_xdx2

S

m²u_xdx−2

S

m_xρ_xγ_xdx

−2

S

mxργxxdx−2λ

S

m²_xdx.

3.15

Similarly,

d dt

S

ρ²_xdx−3

S

ρ²_xuxdx

S

ρ²uxxxdx−2λ

S

ρ²_xdx. 3.16

A combination of3.12–3.16yields d

dt

S

m²ρ²m²_xρ²_x dx −

S

m²uxdx−5

S

m²_xuxdx−2

S

mργxdx−2

S

mxρxγxdx−2λ

S

m²ρ² dx

−2

S

m_xργ_xxdx−

S

ρ²u_xdx−3

S

ρ²_xu_xdx

S

ρ²u_xxxdx−2λ

S

m²_xρ_x² dx −

S

m²u_xdx−5

S

m²_xu_xdx−

S

ρ²u_xdx−3

S

ρ²_xu_xdx−2λ

S

m²ρ² dx

S

ρ²u_xxxdx−2

S

mργ_xdx−2

S

m_xρ_xγ_xdx−2

S

m_xργ_xxdx−2λ

S

m²_xρ_x² dx.

3.17 Assume that there existsM1 >0 andM2 >0 such thatuxt, x≥ −M1andγxt,·_L∞ ≤M2

for allt, x∈0, T×R, then it follows from Lemma2.4that ρt,·

L^∞ ≤e^M1Tρ₀·

L^∞. 3.18

Therefore, d dt

S

m²ρ²m²_xρ²_x dx

≤5M1

S

m²ρ²m²_xρ²_x dx

M2e^M1Tρ0·

L^∞ S

m²ρ²m²_xρ²_xu²_xxxγ_xx² dx

≤5M1

S

m²ρ²m²_xρ²_x dx2

M2e^M1Tρ0·

L^∞ S

m²ρ²m²_xρ²_x dx

≤

5M₁2

M₂e^M1Tρ₀·

L^∞ S

m²ρ²m²_xρ²_x dx.

3.19

The above discussion shows that if there existM1 >0 andM2 >0 such thatuxt, x≥ −M1

andγxt,· ≤ M₂for allt, x∈0, T×S, then there exist two positive constantsKandk such that the following estimate holds

ut,·²_Hsγt,·²

H^s ≤Ke^kt, t∈0, T. 3.20 This inequality, Sobolev’s embedding theorem and Theorem 3.3guarantee that the solution does not blow-up in finite time.

On the other hand, we see that if lim inf

t→T,x∈Ru_xt, x −∞ or lim sup

t→T

γ_xt,·

L^∞

∞, 3.21

then by Sobolev’s embedding theorem, the solution will blow-up in finite time. This completes the proof of the theorem.

Lemma 3.6see32. LetT >0 andv∈C¹0, T;H². Then for everyt∈0, T, there exists at least one pointξ∈Rwith

ζt:inf

x∈Rvxt, x v_xt, ξt. 3.22 The functionζtis absolutely continuous on0, Twith

dζ

dt v_txt, ξt, a.e., on0, T. 3.23 Theorem 3.7. Letz₀ u0, γ₀ ∈H^s×H^s,s > 3/2, z u, γbe the corresponding solution to 2.7with initial dataz0and satisfiesγxt, x_L∞ ≤M, for allt, x∈0, T×S,T be the maximal existence time of the solution. Then we have

t→limT

x∈Rinfu_xt, xT−t −2. 3.24

Proof. Applying Theorems2.1and a simple density argument, we only need to show that the above theorem holds for somes >3/2. Here, we assumes3 to prove the above theorem.

Define now

gt inf

x∈suxt, x, t∈0, T, 3.25 and letξ∈Sbe a point where this minimum is attained. Clearly,uxxt, ξt 0 sinceut,·∈ H³ ⊂C²S. Differentiating the first equation of2.7with respect tox, in view of∂²_xp∗f p∗f−f, we have

u_txuu_xx−1

2u²_xu²1 2γ²−1

2γ_x²−p∗

u² 1 2u²_x1

2γ²−1

2γ_x² −λu_x. 3.26

Evaluating3.26atξtand using Lemma3.6, we obtain d

dtgt 1

2g²t λgt u²t, ξt 1

2γ²t, ξt−1

2γ_x²t, ξt− p∗f

t, ξt, 3.27 wheref u² 1/2u²_x 1/2γ²−1/2γ_x². By Lemma3.1and Young’s inequality, we have for allt∈0, Tthat

p∗f

L^∞≤ G_L∞

u² 1 2u²_x1

2γ²−1 2γ_x²

L¹

≤ cosh1/2 2 sinh1/2

u²_H1γ²

H¹

cosh1/2

2 sinh1/2z²_H1×H¹≤ cosh1/2

2 sinh1/2z0²_H1×H¹.

3.28

This relation together with3.7andγxt, x_L∞ ≤Mimplies that there is a constantK >0 such that

gt 1

2gt λgt

≤K, 3.29

whereKdepends only onu0_H1andγ0_H1. It follows that

−K−1

2λ² ≤gt 1 2

gt λ₂

≤K1

2λ² a.e., on0, T. 3.30 Choose ∈0,1/2. Since lim inft→Tyt λ −∞by Theorem3.5, there is somet0∈0, T withgt0 λ <0 andgt0 λ²> K 1/2λ²/ . Let us first prove that

gt λ2

> 1 K1

2λ² , t∈t0, T. 3.31

Sincegis locally Lipschitz, there is someδ >0 such that gt λ2

> 1 K1

2λ² , t∈t0, t₀δ. 3.32 Note thatg is locally Lipschitzit belongs toW_loc^1,∞sby Lemma3.6and therefore absolutely continuous. Integrating the previous relation ont0, t₀δyields that

gt0δ λ≤gt0 λ <0. 3.33

It follows from the above inequality that gt0δ λ2≥

gt0 λ2

> 1 K1

2λ² . 3.34

The obtained contradiction completes the proof of the relation3.31. By 3.30-3.31, we infer

1

2 − ≤ − gt mλ² ≤ 1

2 , a.e.on0, T. 3.35

ForT ∈t0, T, integrating3.35ont, Tto get 1

2 − T−t≤ − 1 gt λ ≤

1

2 T−t, t∈t0, T. 3.36 Sincegt λ <0 ont0, T, it follows that

1

1/2 ≤ −

gt λ

T−t≤ 1

1/2 , t∈t0, T. 3.37 By the arbitrariness of ∈0,1/2, the statement of the theorem follows.

4. Blow-Up

In this section, we discuss the blow-up phenomena of2.7and prove that there exist strong solutions to2.7which do not exist globally in time.

Theorem 4.1. Letz0 u0, γ0 ∈ H^s×H^s,s > 3/2 andT be the maximal existence time of the solutionz u, γto2.7with the initial dataz₀. If there exists somex₀ ∈Ssuch that

u₀x0<−λ−

λ² e1

e−1 cosh1/2

2 sinh1/2 z0²_H1×H¹, 4.1 then the existence timeT is finite and the slope ofutends to negative infinity astgoes toT whileu remains uniformly bounded on0, T.

Proof. As mentioned earlier, here we only need to show that the above theorem holds for s3. Differentiating the first equation of2.7with respect tox, in view of∂²_xp∗fp∗f−f, we have

u_txuu_xx−1

2u²_xu²1 2γ²−1

2γ_x²−p∗

u² 1 2u²_x1

2γ²−1

2γ_x² −λu_x. 4.2 Define now

gt:min

x∈Suxt, x, t∈0, T, 4.3 and letξt∈Sbe a point where this minimum is attained. It follows that

gt uxt, ξt. 4.4

Clearlyuxxt, ξt 0 sinceut,·∈H³S⊂C²S. Evaluating4.2atξt, we obtain

u_txt, ξt 1

2u²_xt, ξt λu_xt, ξt u²t, ξt 1

2γ²t, ξt− 1

2γ_x²t, ξt

−p∗

u²1 2u²_x1

2γ²−1

2γ_x² t, ξt

≤u²t, ξt 1

2γ²t, ξt 1

2p∗γ_x²t, ξt

≤ e1

2e−1z0²_H1×H¹ cosh1/2 4 sinh1/2γ_x²

L¹

≤

e1

2e−1 cosh1/2

4 sinh1/2 z0²_H1×H¹,

4.5

here, we used Lemma3.2and p∗γ_x²

L^∞≤p

L^∞

γ_x²

L¹ cosh1/2 2 sinh1/2γ_x²

L¹. 4.6

Inequality4.5and Lemma 3.4 imply

d

dtgt 1

2g²t λgt≤

e1

2e−1 cosh1/2

4 sinh1/2 z0²_H1×H¹, 4.7

that is,

d

dtgt≤ −1

2g²t−λgt

e1

2e−1 cosh1/2

4 sinh1/2 z0²_H1×H¹, 4.8

Take

K:

e1

2e−1 cosh1/2

4 sinh1/2z0_H1×H¹. 4.9

It then follows that

gt≤ −1

2g²t−λgK² −1

2

gt λ

λ²2K²

gt λ−

λ²2K² .

4.10