JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 197672 10.1155/2012/197672 197672 Research Article On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System Mi Yongsheng 1, 2 Mu Chunlai 1 Tao Weian 2 Hartung Ferenc 1 College of Mathematics and Statistics Chongqing University Chongqing 400044 China cqu.edu.cn 2 College of Mathematics and Computer Sciences Yangtze Normal University Fuling, Chongqing 408100 China yznu.cn 2012 2 10 2012 2012 16 05 2012 24 07 2012 30 07 2012 2012 Copyright © 2012 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: (1.1)mt+umx+2mux+ρρ¯x+λm=0,t>0,xR,ρt+(ρu)x+λρ=0,t>0,xR,m(0,x)=m0(x),xR,ρ(0,x)=ρ0(x),xR,m(t,x+1)=m(t,x),t0,xR,ρ(t,x+1)=ρ(t,x),t0,xR, where m=(1-x2)u, ρ=(1-x2)(ρ¯-ρ¯0), and λ is a nonnegative dissipative parameter.

The Camassa-Holm equation  has been recently extended to a two-component integrable system (CH2) (1.2)mt+umx+2mux=ρρx,t>0,xR,ρt+(ρu)x=0,t>0,xR, with m=u-uxx, which is a model for wave motion on shallow water, where u(t,x) describes the horizontal velocity of the fluid, and ρ(t,x) is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover, u and ρ satisfy the boundary conditions: u0 and ρ1 as |x|. The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions . Moreover, it is connected with the time-dependent Schrödinger spectral problem . Popowicz  observes that the system is related to the bosonic sector of an N=2 supersymmetric extension of the classical Camassa-Holm equation. Equation (1.2) with ρ0 becomes the Camassa-Holm equation, which has global conservative solutions  and dissipative solutions .

Since the system was derived physically by Constantin and Ivanov  in the context of shallow water theory (also by Chen et al. in  and Falqui et al. in ), many researchers have paid extensive attention to it. In , Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to (1.2) on the line. In , Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of (1.2) on the line. Later, Guan and Yin  obtain a new global existence result for strong solutions to (1.2) and get several blow-up results, which improve the recent results in . Recently, they study the global existence of weak solutions to (1.2) . In , Henry studies the infinite propagation speed for (1.2). Gui and Liu  establish the local well-posedness for (1.2) in a range of the Besov spaces, they also derive a wave breaking mechanism for strong solutions. Mustafa  gives a simple proof of existence for the smooth travelling waves for (1.2). Hu and Yin [14, 15] study the blow-up phenomena and the global existence of (1.2) on the circle.

Recently, the CH2 system was generalized into the following modified two-component Camassa-Holm (MCH2) system: (1.3)mt+umx+2mux=-gρρ¯x,t>0,xR,ρt+(ρu)x=0,t>0,xR, where m=(1-x2)u, ρ=(1-x2)(ρ¯-ρ¯0), u denotes the velocity field, ρ¯0 is taken to be a constant, and g is 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  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 . 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-x2)(ρ¯-ρ¯0), but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law: (1.4)R(u2+ux2+ρ2+ρx2)dx, which play a crucial role in the study of (1.3). Noting that for the CH2 system, we cannot obtain the conservation of H1 norm.

In general, it is quite difficult to avoid energy dissipation mechanisms in a real world. Ghidaglia  studies the long time behaviour of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin  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, (1.1) on the circle (periodic case) has not been studied yet. The aim of this paper is to study the blow-up phenomena of the strong solutions to (1.1). We find that the behavior of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system (1.1) is similar to that of the modified two-component Camassa- Holm system (1.3), such as the local well-posedness and the blow-up scenario. In addition, we also find that the blow-up rate of (1.1) is not affected by the weakly dissipative term, but the occurrence of blow-up of (1.1) is 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 of (1.1).

2. Local Well-Posedness

In this section, by applying Kato’s semigroup theory , we can obtain the local well-posedness for the Cauchy problem of (1.1) in Hs×Hs, s>3/2, with with S=R/Z (the circle of unit length).

First, we introduce some notations. All spaces of functions are assumed to be over S; for simplicity, we drop S in our notation for function spaces if there is no ambiguity. If A is an unbounded operator, we denote by D(A) the domain of A. [A;B] denotes the commutator of two linear operators A and B. ·X denotes the norm of Banach space X. We denote the norm and the inner product of Hs; sR+, by ·s and (·,·)s, respectively.

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

Consider the following abstract quasilinear evolution equation: (2.1)dudt+A(u)=f(u),t>0,u(0)=u0.

Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X and let Q:YX be a topological isomorphism. L(Y,X) denotes the space of all bounded linear operator from Y to X (and we write L(X), if X=Y).

Theorem 2.1 (see [<xref ref-type="bibr" rid="B28">30</xref>]).

Assume that

A(y)L(Y,X) for yX with (2.2)(A(y)-A(z))wXμ1y-zXwY,y,z,wY,

and A(y)G(X,1,β) uniformly on bounded sets in Y.

QA(y)Q-1=A(y)+B(y), where B(y)L(X) is bounded, uniformly on bounded sets in Y. Moreover, (2.3)(B(y)-B(z))wXμ2y-zYwX,y,z,Y,wX.

f:YY and extends also to a map from X into X,f is bounded on bounded sets in Y and (2.4)f(y)-f(z)Yμ3y-zY,y,zY,f(y)-f(z)Xμ3y-zX,y,zY,

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 maximal T>0 depending only on u0Y and a unique solution u to (2.1) such that (2.5)u=u(·,u0)C([0,T];Y)C1([0,T];X).

Moreover, the map u0u(·,u0) is continuous from Y to C([0,T];Y)C1([0,T];X).

We now provide the framework in which we will reformulate system (1.1). With m=u-uxx, ρ=γ-γxx, and γ=ρ¯-ρ¯0, we can rewrite (1.1) as follows: (2.6)mt+umx+2mux+ργx+λm=0,t>0,xR,ρt+(ρu)x+λρ=0,t>0,xR,m(0,x)=u0(x)-u0,xx(x),xR,ρ(0,x)=γ0(x)-γ0,xx(x),xR,m(t,x+1)=m(t,x),t0,xR,ρ(t,x+1)=ρ(t,x),t0,xR.

Note that if p(x)=cosh(x-[x]-(1/2))/2sinh(1/2),xR is the kernel of (1-x2)-1, where [x] stands for the integer part of xR, then (1-x2)-1f=p*f for all fL2(S),p*m=u, and p*ρ=γ. Here we denote by * the convolution. Using this identity, we can rewrite (2.6) as follows: (2.7)ut+uux=-xp*(u2+12ux2+12γ2-12γx2)-λu,t>0,xR,γt+uγx=-p*((uxγx)x+uxγ)-λγ,t>0,xR,u(0,x)=u0(x),xR,γ(0,x)=γ0(x),xR,u(t,x+1)=u(t,x),t0,xR,γ(t,x+1)=γ(t,x),t0,xR, or we can write it in the following equivalent form: (2.8)ut+uux=-x(1-x2)-1(u2+12ux2+12γ2-12γx2)-λu,t>0,xR,γt+uγx=-x(1-x2)-1(uxγx)-(1-x2)-1uxγ-λγ,t>0,xR,u(0,x)=u0(x),xR,γ(0,x)=γ0(x),xR,u(t,x+1)=u(t,x),t0,xR,γ(t,x+1)=γ(t,x),t0,xR.

Theorem 2.2.

Given z0=z(x,0)=(u0,γ0)Hs×Hss>3/2, then there exist a maximal T=T(z0)>0 and a unique solution z=(u,γ) to (1.1) or (2.7) such that (2.9)z=z(·,z0)C([0,T);Hs×Hs)C1([0,T);Hs-1×Hs-1).

Moreover, the solution depends continuously on the initial data, that is, the mapping z0z(·,z0):Hs×HsC([0,T);Hs×Hs)C1([0,T);Hs-1×Hs-1) is continuous and the maximal time of existence T>0 can be chosen to be independent of s.

The remainder of this section is devoted to the proof of Theorem 2.2.

Let z=(uγ), A(z)=(ux,00,ux) and (2.10)f(z)=(-x(1-x2)-1(u2+12ux2+12γ2-12γx2)-λu-x(1-x2)-1(uxγx)-(1-x2)-1uxγ-λγ.).

Set Y=Hs×Hs, X=Hs-1×Hs-1, Λ=(1-x2)1/2 and Q=(Λ00Λ). Obviously, Q is an isomorphism of Hs×Hs onto Hs-1×Hs-1. In order to prove Theorem 2.2 by applying Theorem 2.1, we only need to verify A(z) and f(z) which satisfy the conditions (i)–(iii).

We break the argument into several lemmas.

Lemma 2.3.

The operator A(z)=(ux,00,ux), with zHs×Hs, s>3/2, belongs to G(L2×L2,1,β).

Lemma 2.4.

The operator A(z)=(ux,00,ux), with zHs×Hs, s>3/2, belongs to G(Hs-1×Hs-1,1,β).

Lemma 2.5.

A ( z ) = ( u x , 0 0 , u x ) , with zHs×Hs, s>3/2. The operator A(z)L(Hs×Hs,Hs-1×Hs-1). Moreover, (2.11)(A(y)-A(z))wHs-1×Hs-1μ1y-zHs×HswHs×Hs,y,z,w×Hs.

Lemma 2.6.

The operator B(z)=[Q,A(z)]Q-1 with zHs×Hs, s>3/2. Then B(z)L(Hs-1×Hs-1) and (2.12)(B(y)-B(z))wHs-1×Hs-1μ2y-zHs×HswHs-1×Hs-1, for y,zHs×Hs and wHs-1×Hs-1.

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

Hence, according to Kato’s theorem (Theorem 2.1), in order to prove Theorem 2.2, we only need to verify condition (iii), that is, we need to prove the following lemma.

Lemma 2.7.

Let zHs×Hs, s>3/2 and (2.13)f(z)=(-x(1-x2)-1(u2+12ux2+12γ2-12γx2)-λu-x(1-x2)-1(uxγx)-(1-x2)-1uxγ-λγ).

Then f is bounded on bounded sets in Hs×Hs and satisfies

f(y)-f(z)Hs×Hsμ3y-zHs×Hs,y,zHs×Hs,

f(y)-f(z)Hs-1×Hs-1μ4y-zHs-1×Hs-1,y,zHs×Hs.

Proof.

Let y,zHs×Hs, s>3/2. Since Hs-1 is a Banach algebra, it follows that (2.14)f(y)-f(z)Hs×Hs-x(1-x2)-1((y12-u2)+12(y1x2-ux2)+12(y22-γ2)-12(y2x2-γx2))-λ(y1-u)Hs+-x(1-x2)-1(y1xy2x-uxγx)-(1-x2)-1(y1xy2-uxγ)-λ(y2-γ)Hs(y1-u)(y1+u)Hs-1+12(y1x-ux)(y1x+ux)Hs-1+12(y2-γ)(y2+γ)Hs-1+12(y2x-γx)(y2x+γx)Hs-1+λy1-uHs+ux(y2x-γx)Hs-1+(y1x-ux)y2xHs-1+ux(y2-γ)Hs-2+(y1x-ux)y2Hs-2+λy2-γc(y1-uHs-1y1+uHs-1+12y1-uHsy1+uHs+12y2-γHs-1y2+γHs-1)+12y2-γHsy2+γHs-1+λy1-uHs+uHsy2-γHs+y1-uHsy2Hs+λy2-γHs+uHs-1y2-γHs-2+y1-uHs-1y2Hs-2c(yHs×Hs+zHs×Hs+λ)y-zHs×Hs.

This proves (a). Taking y=0 in the above inequality, we obtain that f is bounded on bounded set in Hs×Hs.

Next, we prove (b). Note that Hs-1 is a Banach algebra. Then, we have (2.15)f(y)-f(z)Hs-1×Hs-1-x(1-x2)-1((y12-u2)+12(y1x2-ux2)+12(y22-γ2)-12(y2x2-γx2))-λ(y1-u)Hs-1+-x(1-x2)-1(y1xy2x-uxγx)-(1-x2)-1(y1xy2-uxγ)-λ(y2-γ)Hs-1(y1-u)(y1+u)Hs-2+12(y1x-ux)(y1x+ux)Hs-2+12(y2-γ)(y2+γ)Hs-2+12(y2x-γx)(y2x+γx)Hs-2+λy1-uHs-1+ux(y2x-γx)Hs-2+(y1x-ux)y2xHs-2+ux(y2-γ)Hs-3+(y1x-ux)y2Hs-3+λy2-γc(y1-uHs-2y1+uHs-1+12y1-uHs-1y1+uHs-1+12y2-γHs-2y2+γHs-2)+12y2-γHs-1y2+γHs-2+λy1-uHs-1+uHsy2-γHs-1+y1-uHs-1y2Hs-1+λy2-γHs-1+uHs-2y2-γHs-3+y1-uHs-2y2Hs-3c(yHs×Hs-1+zHs-1×Hs-1+λ)y-zHs-1×Hs-1.

This proves (b) and completes the proof of the Lemma 2.7.

Proof of Theorem <xref ref-type="statement" rid="thm2.2">2.2</xref>.

Combining Theorem 2.1 and Lemmas 2.32.7, we can get the statement of Theorem 2.2.

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 to (2.7).

Lemma 3.1.

Let z0=(u0,γ0)Hs×Hs, s>3/2, and let T be the maximal existence time of the solution z=(u,γ) to (2.7) with the initial data z0. Then for all t[0,T), we have (3.1)u(t,·)H12+γ(t,·)H12=e-2λt(u0H12+γ0H12).

Proof.

Denote (3.2)f(u,γ)=u2+12ux2+12γ2-12γx2,g=g(u,γ)=(uxγx)x+uxγ.

In view of the identity -x2p*f=f-p*f, we can obtain from (2.7), (3.3)utx=-ux2-uuxx+f-p*f,γtx=-uxγx-uγxx-xp*g.

Therefore, an integration by parts yields (3.4)12ddt(uH12+γH12)=R(uut+uxutx+γγt+γxγtx)dx=Ru(-uux-x2p*f-λu)+ux(-ux2-uuxx+f-p*f-λux)  +γ(-uγx-p*g-λγ)+γx(-uγx-uγxx-xp*g-λγx)=R[-12ux3+ux(u2+12ux2+12γ2-12γx2)-uγγx-γ(uxxγx+uxγ)  -uγxγx2-uγxγxx-λ(u2+ux2+γ2+γx2){12}]dx=-λR(u2+ux2+γ2+γx2)dx.

Thus, the statement of the conservation law follows.

Lemma 3.2 (see [<xref ref-type="bibr" rid="B38">31</xref>]).

( i ) For every fH1(S), we have (3.5)maxx[0,1]f2(x)e+12(e-1)fH12, where the constant (e+1)/2(e-1) is sharp.

For every fH3(S), we have (3.6)maxx[0,1]f2(x)cfH12,

with the best possible constant c lying within the range (1,13/12]. Moreover, the best constant c is (e+1)/2(e-1).

So, if zH3×H3, then by Lemmas 3.1 and 3.2, we have (3.7)u(t,·)L2+γ(t,·)L2e+12(e-1)uH12+e+12(e-1)γH12=e+12(e-1)(u0H12+γ0H12)=e+12(e-1)z0H1×H12, for all t[0,T).

Theorem 3.3.

Let z0=(u,γ)Hs×Hs, s>3/2 be given and assume that T is the maximal existence time of the corresponding solution z=(u,γ) to (2.7) with initial data z0, if there exists M>0 such that (3.8)ux(t,·)L+γx(t,·)LM,t[0,T), then the Hs×Hs norm of z(t,·) does not blow-up on [0,T).

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

Consider the following differential equation equation: (3.9)dq(x,t)dt=u(q(x,t),t),t[0,T),q(0,t)=x,xR, where u denotes the first component of the solution z to (2.7). Applying classical results in the theory of ordinary differential equations, one can obtain the following result on q which is crucial in the proof of blow-up scenario.

Lemma 3.4 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

Let u0C1([0,T);Hs-1), s>3/2, and T be the maximal existence time of the corresponding solution u(t,x) to (3.7). Then (3.7) has a unique solution qC1([0,T)×R,R). Moreover, the map q(t,·) is an increasing diffeomorphism of R with (3.10)qx(x,t)=exp(0tux(q(x,s),s)ds)>0,qx(x,0)=1,xR,0t<T.

The following result is proved only with regard to r=3, since we can obtain the same conclusion for the general case r>3/2 by using Theorem 2.1 and a simple density argument.

We now present a precise blow-up scenario for strong solutions to (2.6).

Theorem 3.5.

Let y0=(u0,γ0)Hs×Hs, s>3/2, and let T be the maximal existence of the corresponding solution z=(u,γ) to (2.7). Then the solution blows up in finite time if and only if (3.11)liminftT,xRux(t,x)=-orlimsuptT{γx(t,·)L}=+.

Proof.

Multiplying the first equation in (2.6) by m=u-uxx and integrating by parts, we obtain (3.12)ddtSm2dx=2Smmtdx=2Sm(-umx-2mux-ργx)dx-2λSm2dx=-3Sm2uxdx-2Smργxdx-2λSm2dx.

Repeating the same procedure to the second equation in (2.6) we get (3.13)ddtSρ2dx=-Sρ2ux-2λSρ2dx.

A combination of (3.7) and (3.9) yields (3.14)ddtS(m2+ρ2)dx=-3Sm2uxdx-2Smργxdx-Sρ2ux-2λS(m2+ρ2)dx.

Differentiating the first equation in (2.6) with respect to x, multiplying by mx=ux-uxxx, then integrating over S, we obtain (3.15)ddtSmx2dx=-5Smx2uxdx+2Sm2uxdx-2Smxρxγxdx-2Smxργxxdx-2λSmx2dx.

Similarly, (3.16)ddtSρx2dx=-3Sρx2uxdx+Sρ2uxxxdx-2λSρx2dx.

A combination of (3.12)–(3.16) yields (3.17)ddtS(m2+ρ2+mx2+ρx2)dx=-Sm2uxdx-5Smx2uxdx-2Smργxdx-2Smxρxγxdx-2λS(m2+ρ2)dx-2Smxργxxdx-Sρ2uxdx-3Sρx2uxdx+Sρ2uxxxdx-2λS(mx2+ρx2)dx=-Sm2uxdx-5Smx2uxdx-Sρ2uxdx-3Sρx2uxdx-2λS(m2+ρ2)dx+Sρ2uxxxdx-2Smργxdx-2Smxρxγxdx-2Smxργxxdx-2λS(mx2+ρx2)dx.

Assume that there exists M1>0 and M2>0 such that ux(t,x)-M1 and γx(t,·)LM2 for all (t,x)[0,T)×R, then it follows from Lemma 2.4 that (3.18)ρ(t,·)LeM1Tρ0(·)L.

Therefore, (3.19)ddtS(m2+ρ2+mx2+ρx2)dx(5M1)S(m2+ρ2+mx2+ρx2)dx+(M2+eM1Tρ0(·)L)S(m2+ρ2+mx2+ρx2+uxxx2+γxx2)dx(5M1)S(m2+ρ2+mx2+ρx2)dx+2(M2+eM1Tρ0(·)L)S(m2+ρ2+mx2+ρx2)dx(5M1+2(M2+eM1Tρ0(·)L))S(m2+ρ2+mx2+ρx2)dx.

The above discussion shows that if there exist M1>0 and M2>0 such that ux(t,x)-M1 and γx(t,·)M2 for all (t,x)[0,T)×S, then there exist two positive constants K and k such that the following estimate holds (3.20)u(t,·)Hs2+γ(t,·)Hs2Kekt,t[0,T).

This inequality, Sobolev’s embedding theorem and Theorem 3.3 guarantee that the solution does not blow-up in finite time.

On the other hand, we see that if (3.21)liminftT,xRux(t,x)=-orlimsuptT{γx(t,·)L}=+, then by Sobolev’s embedding theorem, the solution will blow-up in finite time. This completes the proof of the theorem.

Lemma 3.6 (see [<xref ref-type="bibr" rid="B3">32</xref>]).

Let T>0 and vC1([0,T);H2). Then for every t[0,T), there exists at least one point ξR with (3.22)ζ(t)=infxR[vx(t,x)]=vx(t,ξ(t)).

The function ζ(t) is absolutely continuous on (0,T) with (3.23)dζdt=vtx(t,ξ(t)),a.e.,on(0,T).

Theorem 3.7.

Let z0=(u0,γ0)Hs×Hs, s>3/2,z=(u,γ) be the corresponding solution to (2.7) with initial data z0 and satisfies γx(t,x)LM, for all (t,x)[0,T)×S, T be the maximal existence time of the solution. Then we have (3.24)limtT(infxRux(t,x)(T-t))=-2.

Proof.

Applying Theorems 2.1 and a simple density argument, we only need to show that the above theorem holds for some s>3/2. Here, we assume s=3 to prove the above theorem.

Define now (3.25)g(t)=infxsux(t,x),t[0,T), and let ξS be a point where this minimum is attained. Clearly, uxx(t,ξ(t))=0 since u(t,·)H3C2(S). Differentiating the first equation of (2.7) with respect to x, in view of x2p*f=p*f-f, we have (3.26)utx+uuxx=-12ux2+u2+12γ2-12γx2-p*(u2+12ux2+12γ2-12γx2)-λux.

Evaluating (3.26) at ξ(t) and using Lemma 3.6, we obtain (3.27)ddtg(t)+12g2(t)+λg(t)=u2(t,ξ(t))+12γ2(t,ξ(t))-12γx2(t,ξ(t))-[p*f](t,ξ(t)), where f=u2+(1/2)ux2+(1/2)γ2-(1/2)γx2. By Lemma 3.1 and Young’s inequality, we have for all t[0,T) that (3.28)p*fLGLu2+12ux2+12γ2-12γx2L1cosh(1/2)2sinh(1/2)(uH12+γH12)=cosh(1/2)2sinh(1/2)zH1×H12cosh(1/2)2sinh(1/2)z0H1×H12.

This relation together with (3.7) and γx(t,x)LM implies that there is a constant K>0 such that (3.29)|g(t)+12g(t)+λg(t)|K, where K depends only on u0H1 and γ0H1. It follows that (3.30)-K-12λ2g(t)+12(g(t)+λ)2K+12λ2a.e.,on(0,T).

Choose ϵ(0,1/2). Since liminftT(y(t)+λ)=- by Theorem 3.5, there is some t0(0,T) with g(t0)+λ<0 and (g(t0)+λ)2>K+(1/2)λ2/ϵ. Let us first prove that (3.31)(g(t)+λ)2>1ϵ(K+12λ2),t[t0,T).

Since g is locally Lipschitz, there is some δ>0 such that (3.32)(g(t)+λ)2>1ϵ(K+12λ2),t(t0,t0+δ).

Note that g is locally Lipschitz (it belongs to Wloc1,(s) by Lemma 3.6) and therefore absolutely continuous. Integrating the previous relation on (t0,t0+δ) yields that (3.33)g(t0+δ)+λg(t0)+λ<0.

It follows from the above inequality that (3.34)(g(t0+δ)+λ)2(g(t0)+λ)2>1ϵ(K+12λ2).

The obtained contradiction completes the proof of the relation (3.31). By (3.30)-(3.31), we infer (3.35)12-ϵ-g(t)(m+λ)212+ϵ,a.e.on(0,T).

For T(t0,T), integrating (3.35) on (t,T) to get (3.36)(12-ϵ)(T-t)-1g(t)+λ(12+ϵ)(T-t),t(t0,T).

Since g(t)+λ<0 on [t0,T), it follows that (3.37)1(1/2)+ϵ-(g(t)+λ)(T-t)1(1/2)+ϵ,t(t0,T).

By the arbitrariness of ϵ(0,1/2), the statement of the theorem follows.

4. Blow-Up

In this section, we discuss the blow-up phenomena of (2.7) and prove that there exist strong solutions to (2.7) which do not exist globally in time.

Theorem 4.1.

Let z0=(u0,γ0)Hs×Hs, s>3/2 and T be the maximal existence time of the solution z=(u,γ) to (2.7) with the initial data z0. If there exists some x0S such that (4.1)u0(x0)<-λ-λ2+(e+1e-1+cosh(1/2)2sinh(1/2))z0H1×H12, then the existence time T is finite and the slope of u tends to negative infinity as t goes to T while u remains uniformly bounded on [0,T).

Proof.

As mentioned earlier, here we only need to show that the above theorem holds for s=3. Differentiating the first equation of (2.7) with respect to x, in view of x2p*f=p*f-f, we have (4.2)utx+uuxx=-12ux2+u2+12γ2-12γx2-p*(u2+12ux2+12γ2-12γx2)-λux.

Define now (4.3)g(t)=minxS[ux(t,x)],t[0,T), and let ξ(t)S be a point where this minimum is attained. It follows that (4.4)g(t)=ux(t,ξ(t)).

Clearly uxx(t,ξ(t))=0 since u(t,·)H3(S)C2(S). Evaluating (4.2) at ξ(t), we obtain (4.5)utx(t,ξ(t))+12ux2(t,ξ(t))+λux(t,ξ(t))=u2(t,ξ(t))+12γ2(t,ξ(t))-12γx2(t,ξ(t))-p*(u2+12ux2+12γ2-12γx2)(t,ξ(t))u2(t,ξ(t))+12γ2(t,ξ(t))+12p*γx2(t,ξ(t))e+12(e-1)z0H1×H12+cosh(1/2)4sinh(1/2)γx2L1(e+12(e-1)+cosh(1/2)4sinh(1/2))z0H1×H12, here, we used Lemma 3.2 and (4.6)p*γx2LpLγx2L1=cosh(1/2)2sinh(1/2)γx2L1.

Inequality (4.5) and Lemma 3.4 imply (4.7)ddtg(t)+12g2(t)+λg(t)(e+12(e-1)+cosh(1/2)4sinh(1/2))z0H1×H12, that is, (4.8)ddtg(t)-12g2(t)-λg(t)+(e+12(e-1)+cosh(1/2)4sinh(1/2))z0H1×H12,

Take (4.9)K=e+12(e-1)+cosh(1/2)4sinh(1/2)z0H1×H1.

It then follows that (4.10)g(t)-12g2(t)-λg+K2=-12(g(t)+λ+λ2+2K2)(g(t)+λ-λ2+2K2).

Note that if g(0)=u0(ξ(0))u(x0)-λ-λ2+2K2, then g(t)-λ-λ2+2K2, for all t[0,T). Therefore, we can solve the above inequality to obtain (4.11)g(0)+λ+λ2+2K2g(0)+λ-λ2+2K2eλ2+2K2t-12λ2+2K2g(t)+λ-λ2+2K20.

Due to 0<(g(0)+λ+λ2+2K2)/(g(0)+λ-λ2+2K2)<1, then there exists T, and 0<T<(1/λ2+2K2)ln((g(0)+λ+λ2+2K2)/(g(0)+λ-λ2+2K2)), such that limtTg(t)=-. This completes the proof of the theorem.

Theorem 4.2.

Let z0=(u0,γ0)Hs×Hs, s>3/2,z=(u,γ) be the corresponding solution to (2.7) with initial data z0 and satisfies γx(t,x)LM, for all (t,x)[0,T)×S, T be the maximal existence time of the solution. If z0 satisfies the following condition: (4.12)Su0x3dx<-3λz0H1×H12-9λ2z0H1×H14-2K2z0H1×H12, where K=((9(e+1)/4(e-1))+(3cosh(1/2)/4sinh(1/2)))z0H1×H14. Then the corresponding solution to (2.7) blows up in finite time.

Proof.

In view of (4.2), we obtain (4.13)ddtSux3=3Sux2uxtdx=3Sux2(-uuxx-12ux2+u2+12γ2-12γx2-p*(u2+12ux2+12γ2-12γx2)-λux)dx=3S(-uux2uxx+ux2u2-12ux4+12ux2γ2-12ux2γx2-ux2p*(u2+12ux2+12γ2)  +12ux2p*γx2-λux2)dx3S(-uux2uxx+ux2u2-12ux4+12ux2γ2+12ux2p*γx2-λux3)dx=-12Sux4dx+3Sux2u2dx+32Suxγ2dx+32Sux2p*γx2dx-3λSux3dx.

Note that (4.14)p*γx2LGLγx2L1=cosh(1/2)2sinh(1/2)γx2L1,Sux2u2dxu2LSux2dxe+12(e-1)z0H1×H14,Sux2γ2dxγ2LSux2dxe+12(e-1)z0H1×H14,Sux2p*γx2dxcosh(1/2)2sinh(1/2)γx2L1Sux2dxcosh(1/2)2sinh(1/2)z0H1×H14.

Thus, (4.15)ddtSux3dx-12Sux4-3λSux3dx+(9(e+1)4(e-1)+3cosh(1/2)4sinh(1/2))z0H1×H14.

Using the following inequality: (4.16)|Sux3dx|(Sux4dx)1/2(Sux2dx)1/2(Sux4dx)1/2z0H1×H1. and letting (4.17)g(t)=Sux3dx, we obtain (4.18)ddtg(t)-12z0H1×H12g2(t)-3λg(t)+(9(e+1)4(e-1)+3cosh(1/2)4sinh(1/2))z0H1×H14.

Taking (4.19)K=(9(e+1)4(e-1)+3cosh(1/2)4sinh(1/2))z0H1×H14, we get (4.20)ddtg(t)-12z0H1×H12(g(t)+3λz0H1×H12+9λ2z0H1×H14+2K2z0H1×H12)×(g(t)+3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H12).

Note that if (4.21)g(0)<-3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H12, then (4.22)g(t)<-3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H12, for all t[0,T). From the above inequality, we obtain (4.23)g(0)+3λz0H1×H12+9λ2z0H1×H14+2K2z0H1×H12g(0)+3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H12e9λ2z0H1×H14+2K2z0H1×H12t-129λ2z0H1×H14+2K2z0H1×H12g(t)+3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H120.

Since 0<(g(0)+3λz0H1×H12+9λ2z0H1×H14+2K2z0H1×H12)/(g(0)+3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H12)<1 then there exists (4.24)0<T19λ2z0H1×H14+2K2z0H1×H12×ln(g(0)+3λz0H1×H12+9λ2z0H1×H14+2K2z0H1×H12g(0)+3λz0H1×H12-9λ2z0H1×H14+2K2z0H1×H12). such that limtTg(t)=-. On the other hand, (4.25)|Sux3dx|uxLSux2uxLu2LH1uxLz0H1×H12.

Applying Theorem 3.5, the solution z blows up in finite time.

Acknowledgments

This work was partially supported by NSF of China (11071266), Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, and the Educational Science Foundation of Chongqing China (KJ121302).

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