# On the Blow-Up of Solutions of a Weakly

## Full text

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

## Dissipative Modified Two-Component PeriodicCamassa-Holm System

1, 2

1

### and Weian Tao

2

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

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:

mtumx2muxρρxλm0, t >0, x∈R, ρt

ρu

xλρ0, t >0, x∈R, m0, x m0x, xR,

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

ρt, x1 ρt, x, t≥0, x∈R,

1.1

wherem 1−2xu,ρ 1−2xρ−ρ0, andλis a nonnegative dissipative parameter.

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

mtumx2muxρρx, t >0, x∈R, ρt

ρu

x0, t >0, x∈R, 1.2

withmu−uxx, 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:

mtumx2mux−gρρx, t >0, x∈R, ρt

ρu

x0, t >0, x∈R, 1.3

wherem 1−2xu,ρ 1−2xρ−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

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

R

u2u2xρ2ρ2x

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 of1.1inHs×Hs,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 ofHs;sR, 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:YXbe a topological isomorphism.LY, Xdenotes the space of all bounded linear operator fromY toXand we writeLX, ifXY.

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Theorem 2.1see30. Assume that iAyLY, XforyXwith

A y

Az w

Xμ1yz

XwY, y, z, wY, 2.2 andAyGX,1, βuniformly on bounded sets inY.

iiQAyQ−1 Ay By, whereByLXis bounded, uniformly on bounded sets in Y. Moreover,

B y

Bz w

Xμ2yz

YwX, y, z,Y, wX. 2.3 iiif :YY and extends also to a map fromXintoX, f is bounded on bounded sets inY

and

f y

fz

Yμ3yz

Y, y, zY, f

y

fz

Xμ3yz

X, y, zY, 2.4

where, μ1, μ2, and μ3 depend only on max{yX,zX} and μ4 depends only on max{yY,zY}. If the above conditions (i), (ii), and (iii) hold, given u0Y, there is a maximalT >0 depending only onu0Y and a unique solution u to2.1such that

uu·, u0C0, T;YC10, T;X. 2.5

Moreover, the mapu0u·, u0is continuous fromYtoC0, T;YC10, T;X.

We now provide the framework in which we will reformulate system1.1. Withm uuxx,ργγxx, andγρρ0, we can rewrite1.1as follows:

mtumx2muxργxλm0, t >0, x∈R, ρt

ρu

xλρ0, t >0, x∈R, m0, x u0x−u0,xxx, xR, ρ0, x γ0x−γ0,xxx, xR, 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,xRis the kernel of1−2x−1, where xstands for the integer part ofxR, then1−2x−1f pffor allfL2S,pmu,

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

utuux−∂xp

u21 2u2x1

2γ2−1

2γx2λu, t >0, x∈R, γtx−p∗

uxγx

xuxγ

λγ, t >0, x∈R, u0, x u0x, xR,

γ0, x γ0x, xR, 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

1−2x−1 u21

2u2x1 2γ2−1

2γx2λu, t >0, x∈R, γtx−∂x

1−2x−1 uxγx

1−2x−1

uxγλγ, t >0, x∈R, u0, x u0x, xR,

γ0, x γ0x, xR, 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, γ0Hs×Hss > 3/2, then there exist a maximal T Tz0>0 and a unique solutionz u, γto1.1or2.7such that

zz·, z0C0, T;Hs×HsC1

0, T;Hs−1×Hs−1

. 2.9

Moreover, the solution depends continuously on the initial data, that is, the mappingz0z·, z0: Hs×HsC0, T;Hs×HsC10, T;Hs−1×Hs−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.

Letzu

γ

,Az

u∂x, 0 0, u∂x

and

fz

⎝−∂x

1−2x−1 u2 1

2u2x1 2γ2−1

2γx2λu

−∂x

1−2x−1 uxγx

1−2x−1

uxγλγ.

. 2.10

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SetY Hs ×Hs,X Hs−1×Hs−1,Λ 1−2x1/2 and Q Λ0

0 Λ

. Obviously, Q is an isomorphism of Hs ×Hs onto Hs−1 ×Hs−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

, withzHs×Hs,s >3/2, belongs toGL2×L2,1, β.

Lemma 2.4. The operatorAz

u∂x, 0 0, u∂x

, withzHs×Hs,s > 3/2, belongs toGHs−1× Hs−1,1, β.

Lemma 2.5. Az

u∂x, 0 0, u∂x

, withzHs×Hs,s >3/2. The operatorAzLHs×Hs, Hs−1× Hs−1. Moreover,

A y

Az w

Hs−1×Hs−1μ1yz

Hs×HswHs×Hs, y, z, w∈ ×Hs. 2.11

Lemma 2.6. The operatorBz Q, AzQ−1withzHs×Hs,s >3/2. ThenBzLHs−1× Hs−1and

B y

Bz w

Hs−1×Hs−1μ2yz

Hs×HswHs−1×Hs−1, 2.12

fory, zHs×HsandwHs−1×Hs−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. LetzHs×Hs,s >3/2 and

fz

⎜⎜

⎜⎜

−∂x

1−2x−1 u2 1

2u2x1 2γ2−1

2γx2λu

−∂x

1−2x−1 uxγx

1−2x−1

uxγλγ

⎟⎟

⎟⎟

. 2.13

Thenfis bounded on bounded sets inHs×Hsand satisfies

afy−fzHs×Hsμ3y−zHs×Hs, y, zHs×Hs, bfy−fzHs−1×Hs−1μ4y−zHs−1×Hs−1, y, zHs×Hs.

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Proof. Lety, zHs×Hs,s >3/2. SinceHs−1is a Banach algebra, it follows that f

y

fz

Hs×Hs

≤ −∂x

1−2x−1

y21u2 1

2

y21xu2x 1

2

y22γ2

−1 2

y2x2γx2λ

y1u

Hs

−∂x

1−2x−1

y1xy2xuxγx

1−2x−1

y1xy2uxγ

λ

y2γ

Hs

y1u

y1u

Hs−11

2y1xux

y1xux

Hs−1 1

2y2γ

y2γ

Hs−1

1

2y2xγx

y2xγx

Hs−1λy1u

Hsux

y2xγx

Hs−1

y1xux y2x

Hs−1ux

y2γ

Hs−2y1xux y2

Hs−2λy2γ

cy1u

Hs−1y1u

Hs−11

2y1u

Hsy1u

Hs1

2y2γ

Hs−1y2γ

Hs−1

1

2y2γ

Hsy2γ

Hs−1λy1u

HsuHsy2γ

Hsy1u

Hsy2

Hs

λy2γ

HsuHs−1y2γ

Hs−2y1u

Hs−1y2

Hs−2

cy

Hs×HszHs×Hsλyz

Hs×Hs.

2.14

This provesa. Takingy0 in the above inequality, we obtain thatfis bounded on bounded set inHs×Hs.

Next, we proveb. Note thatHs−1is a Banach algebra. Then, we have f

y

fz

Hs−1×Hs−1

≤ −∂x

1−2x−1

y21u2 1

2

y21xu2x 1

2

y22γ2

−1 2

y2x2γx2 −λ

y1u

Hs−1

−∂x

1−2x−1

y1xy2xuxγx

1−2x−1

y1xy2uxγ

λ

y2γ

Hs−1

y1u

y1u

Hs−21

2y1xux

y1xux

Hs−2 1

2y2γ

y2γ

Hs−2

1

2y2xγx

y2xγx

Hs−2λy1u

Hs−1ux

y2xγx

Hs−2

y1xux y2x

Hs−2ux

y2γ

Hs−3y1xux y2

Hs−3λy2γ

cy1u

Hs−2y1u

Hs−11

2y1u

Hs−1y1u

Hs−11

2y2γ

Hs−2y2γ

Hs−2

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1

2y2γ

Hs−1y2γ

Hs−2λy1u

Hs−1uHsy2γ

Hs−1

y1u

Hs−1y2

Hs−1λy2γ

Hs−1uHs−2y2γ

Hs−3y1u

Hs−2y2

Hs−3

cy

Hs×Hs−1zHs−1×Hs−1λyz

Hs−1×Hs−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, γ0Hs×Hs,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,·2H1γt,·2

H1e−2λt

u02H1γ02

H1

. 3.1

Proof. Denote

f u, γ

u21 2u2x1

2γ2−1

2γx2, g g u, γ

uxγx

xuxγ. 3.2

In view of the identity−∂2xpffpf, we can obtain from2.7,

utx−u2xuuxxfpf, γtx−uxγxxxxpg. 3.3

Therefore, an integration by parts yields

1 2

d dt

u2H1γ2

H1

R

uutuxutxγγtγxγtx

dx

R

u

−uux2xpfλu ux

−u2xuuxxfpfλux γ

−uγxpgλγ γx

−uγxxxxpgλγx

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R

−1 2u3xux

u2 1

2u2x1 2γ2−1

2γx2uγγxγ

uxxγxuxγ

−uγxγx2xγxxλ

u2u2xγ2γx2 dx −λ

R

u2u2xγ2γx2 dx.

3.4

Thus, the statement of the conservation law follows.

Lemma 3.2see31. iFor everyfH1S, we have

x∈0,1maxf2x≤ e1 2e−1f2

H1, 3.5

where the constante1/2e−1is sharp.

iiFor everyfH3S, 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, ifzH3×H3, then by Lemmas3.1and3.2, we have ut,·2Lγt,·2

Le1

2e−1u2H1 e1 2e−1γ2

H1

e1 2e−1

u02H1γ02

H1

e1

2e−1z02H1×H1,

3.7

for allt∈0, T.

Theorem 3.3. Letz0 u, γ ∈ Hs×Hs,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,·Lγxt,·

LM, t∈0, T, 3.8

then theHs×Hsnorm ofzt,·does not blow-up on0, T.

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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, xR,

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;Hs−1,s >3/2, andTbe the maximal existence time of the corresponding solutionut, xto3.7. Then3.7has a unique solutionqC10, 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, γ0Hs×Hs,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 uuxx and integrating by parts, we obtain

d dt

S

m2dx2

S

mmtdx2

S

m

−umx−2muxργx

dx−2λ

S

m2dx −3

S

m2uxdx−2

S

mργxdx−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

m2uxdx−2

S

mργxdx

S

ρ2ux−2λ

S

m2ρ2

dx. 3.14

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

d dt

S

m2xdx−5

S

m2xuxdx2

S

m2uxdx−2

S

mxρxγxdx

−2

S

mxργxxdx−2λ

S

m2xdx.

3.15

Similarly,

d dt

S

ρ2xdx−3

S

ρ2xuxdx

S

ρ2uxxxdx−2λ

S

ρ2xdx. 3.16

A combination of3.12–3.16yields d

dt

S

m2ρ2m2xρ2x dx

S

m2uxdx−5

S

m2xuxdx−2

S

mργxdx−2

S

mxρxγxdx−2λ

S

m2ρ2 dx

−2

S

mxργxxdx

S

ρ2uxdx−3

S

ρ2xuxdx

S

ρ2uxxxdx−2λ

S

m2xρx2 dx

S

m2uxdx−5

S

m2xuxdx

S

ρ2uxdx−3

S

ρ2xuxdx−2λ

S

m2ρ2 dx

S

ρ2uxxxdx−2

S

mργxdx−2

S

mxρxγxdx−2

S

mxργxxdx−2λ

S

m2xρx2 dx.

3.17 Assume that there existsM1 >0 andM2 >0 such thatuxt, x≥ −M1andγxt,·LM2

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

LeM1Tρ0·

L. 3.18

Therefore, d dt

S

m2ρ2m2xρ2x dx

≤5M1

S

m2ρ2m2xρ2x dx

M2eM1Tρ0·

L S

m2ρ2m2xρ2xu2xxxγxx2 dx

≤5M1

S

m2ρ2m2xρ2x dx2

M2eM1Tρ0·

L S

m2ρ2m2xρ2x dx

5M12

M2eM1Tρ0·

L S

m2ρ2m2xρ2x dx.

3.19

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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,·2Hsγt,·2

HsKekt, 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∈Ruxt, x −∞ or lim sup

t→T

γxt,·

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 andvC10, T;H2. Then for everyt∈0, T, there exists at least one pointξRwith

ζt:inf

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

dt vtxt, ξt, a.e., on0, T. 3.23 Theorem 3.7. Letz0 u0, γ0Hs×Hs,s > 3/2, z u, γbe the corresponding solution to 2.7with initial dataz0and satisfiesγxt, xLM, for allt, x∈0, T×S,T be the maximal existence time of the solution. Then we have

t→limT

x∈Rinfuxt, 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 letξSbe a point where this minimum is attained. Clearly,uxxt, ξt 0 sinceut,·∈ H3C2S. Diﬀerentiating the first equation of2.7with respect tox, in view of∂2xpf pff, we have

utxuuxx−1

2u2xu21 2γ2−1

2γx2p

u2 1 2u2x1

2γ2−1

2γx2λux. 3.26

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

dtgt 1

2g2t λgt u2t, ξt 1

2γ2t, ξt−1

2γx2t, ξt− pf

t, ξt, 3.27 wheref u2 1/2u2x 1/2γ2−1/2γx2. By Lemma3.1and Young’s inequality, we have for allt∈0, Tthat

pf

LGL

u2 1 2u2x1

2γ2−1 2γx2

L1

≤ cosh1/2 2 sinh1/2

u2H1γ2

H1

cosh1/2

2 sinh1/2z2H1×H1≤ cosh1/2

2 sinh1/2z02H1×H1.

3.28

This relation together with3.7andγxt, xLMimplies that there is a constantK >0 such that

gt 1

2gt λgt

K, 3.29

whereKdepends only onu0H1andγ0H1. It follows that

−K−1

2λ2gt 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 toWloc1,∞sby Lemma3.6and therefore absolutely continuous. Integrating the previous relation ont0, t0δyields that

gt0δ λgt0 λ <0. 3.33

It follows from the above inequality that gt0δ λ2

gt0 λ2

> 1 K1

2λ2 . 3.34

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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, γ0Hs×Hs,s > 3/2 andT be the maximal existence time of the solutionz u, γto2.7with the initial dataz0. If there exists somex0Ssuch that

u0x0<−λ−

λ2 e1

e−1 cosh1/2

2 sinh1/2 z02H1×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∂2xpfpff, we have

utxuuxx−1

2u2xu21 2γ2−1

2γx2p

u2 1 2u2x1

2γ2−1

2γx2λux. 4.2 Define now

gt:min

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

gt uxt, ξt. 4.4

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

utxt, ξt 1

2u2xt, ξt λuxt, ξt u2t, ξt 1

2γ2t, ξt− 1

2γx2t, ξt

p

u21 2u2x1

2γ2−1

2γx2 t, ξt

u2t, ξt 1

2γ2t, ξt 1

2pγx2t, ξt

e1

2e−1z02H1×H1 cosh1/2 4 sinh1/2γx2

L1

e1

2e−1 cosh1/2

4 sinh1/2 z02H1×H1,

4.5

here, we used Lemma3.2and p∗γx2

Lp

L

γx2

L1 cosh1/2 2 sinh1/2γx2

L1. 4.6

Inequality4.5and Lemma 3.4 imply

d

dtgt 1

2g2t λgt

e1

2e−1 cosh1/2

4 sinh1/2 z02H1×H1, 4.7

that is,

d

dtgt≤ −1

2g2t−λgt

e1

2e−1 cosh1/2

4 sinh1/2 z02H1×H1, 4.8

Take

K:

e1

2e−1 cosh1/2

4 sinh1/2z0H1×H1. 4.9

It then follows that

gt≤ −1

2g2t−λgK2 −1

2

gt λ

λ22K2

gt λ

λ22K2 .

4.10

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