DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 163070 10.1155/2013/163070 163070 Research Article Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation Wang Ke-chuang Xia Yonghui 1 Department of Basic Zhejiang Dongfang Vocational Technical College Wenzhou, Zhejiang 325011 China 2013 4 4 2013 2013 09 02 2013 10 03 2013 2013 Copyright © 2013 Ke-chuang Wang. 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.

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipation. We establish a sufficient condition to guarantee global-in-time solutions, then present persistence property for the Cauchy problem, and describe the asymptotic behavior of solutions for compactly supported initial data.

1. Introduction

Dullin et al.  derived a new equation describing the unidirectional propagation of surface waves in a shallow water regime:(1)ut-α2uxxt+c0ux+3uux+γuxxx=α2(2uxuxx+uuxxx),x,t>0, where the constants α2 and γ/c0 are squares of length scales and the constant c0>0 is the critical shallow water wave speed for undisturbed water at rest at spatial infinity. Since this equation is derived by Dullin, Gottwald, and Holm, in what follows, we call this new integrable shallow water equation (1) DGH equation.

If α=0, (1) becomes the well-known KdV equation, whose solutions are global as long as the initial data is square integrable. This is proved by Bourgain . If γ=0 and α=1, (1) reduces to the Camassa-Holm equation which was derived physically by Camassa and Holm in  by approximating directly the Hamiltonian for Euler's equations in the shallow water regime, where u(x,t) represents the free surface above a flat bottom. The properties about the well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works , and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in .

It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities: (2)E(u)=12(u2+α2ux2)dx,F(u)=12(u3+α3uux2+c0u2-γux2)dx.

Recently, in , local well-posedness of strong solutions to (1) was established by applying Kato's theory , and some sufficient conditions were found to guarantee finite time blow-up phenomenon. Moreover, Zhou  found the best constants for two convolution problems on the unit circle via variational method and applied the best constants on (1) to give some blow-up criteria. Later, Zhou and Guo improved the results and got some new criteria for wave breaking .

In general, it is quite difficult to avoid energy dissipation mechanism in the real world. Ghidaglia  studied the long time behavior of solutions to the weakly dissipative KdV equation as a finite dimensional dynamic system. Moreover, some results on blow-up criteria and the global existence condition for the weakly dissipative Camassa-Holm equation are presented in , and very related work can be found in [21, 22]. In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation (3)ut-α2uxxt+c0ux+3uux+γuxxx+λ(1-α2x2)u=α2(2uxuxx+uuxxx), where x, t>0,  λ(1-α2x2)u is the weakly dissipative term and λ is a positive dissipation parameter. Set Q=(1-α2x2)1/2, then the operator Q-2 can be expressed by (4)Q-2f=G*f=G(x-y)f(y)dy, for all fL2() with G(x)=(1/2α)e-|x|/α. With this in hand, we can rewrite (3) as a quasilinear equation of hyperbolic type (5)ut+(u-γα2)ux+xG*(u2+α22ux2+(c0+γα2)u)+λu=0. It is the dissipative term that causes the previous conserved quantities E(u) and F(u) to be no longer conserved for (3), and this model could also be regarded as a model of a type of a certain rate-dependent continuum material called a compressible second grade fluid . Our consideration is based on this fact. Furthermore, we will show how the dissipation term affects the behavior of solutions in our forthcoming paper. As a whole, the current dissipation model is of great importance mathematically and physically, and it is worthy of being considered. In what follows, we assume that c0+γ/α2=0 and α>0 just for simplicity. Since u(x,t) is bounded by its H1-norm, a general case with c0+γ/α20 does not change our results essentially, but it would lead to unnecessary technical complications. So the above equation is reduced to a simpler form as follows: (6)ut+(u+c0)ux+xG*F(u)+λu=0,x,t>0, where (7)F(u)=u2+α22ux2.

The rest of this paper is organized as follows. In Section 2, we list the local well-posedness theorem for (6) with initial datum u0Hs,  s>3/2 and collect some auxiliary results. In Section 3, we establish the condition for global existence in view of the initial potential. Persistence properties of the strong solutions are explored in Section 4. Finally, in Section 5, we give a detailed description of the corresponding solution with compactly supported initial data.

2. Preliminaries

In this section, we make some preparations for our consideration. Firstly, the local well-posedness of the Cauchy problem of (6) with initial data u0Hs with s>3/2 can be obtained by applying Kato's theorem . More precisely, we have the following local well-posedness result.

Theorem 1.

Given that u0(x)Hs, s>3/2, there exist T=T(λ,u0Hs)>0 and a unique solution u to (6), such that (8)u=u(·,u0)C([0,T);Hs)C1([0,T);Hs-1). Moreover, the solution depends continuously on the initial data; that is, the mapping u0u(·,u0):HsC([0,T);Hs)C1([0,T);Hs-1) is continuous, and the maximal time of existence T>0 is independent of s.

Proof.

Set A(u)=(u+c0)x,  f(u)=-x(1-α2x2)-1F(u)-λu,  Y=Hs,X=Hs-1,   s>3/2, and Q=(1-α2x2)1/2. Applying Kato's theory for abstract quasilinear evolution equation of hyperbolic type, we can obtain the local well-posedness of (6) in Hs, s>3/2 and uC([0,T);Hs)C1([0,T);Hs-1).

The maximal value of T in Theorem 1 is called the lifespan of the solution in general. If T<, that is limsuptTu(·,t)Hs=, we say that the solution blows up in finite time, otherwise, the solution exists globally in time. Next, we show that the solution blows up if and only if its first-order derivative blows up.

Lemma 2.

Given that u0Hs,   s>3/2, the solution u=u(·,u0) of (3) blows up in finite time T<+ if and only if (9)liminftT{infx[ux(x,t)]}=-.

Proof.

We first assume that u0Hs for some s,   s4. Equation (6) can be written into the following form in terms of y=(1-α2x2)u(10)yt+(yu)x+12(u2-α2ux2)x+c0yx+λy=0,yt+(yu)x+12(u20-α2ux2)x+x,t>0. Multiplying (10) by y=(1-α2x2)u, and integrating by parts, we have (11)ddty2dx=2yytdx=-3uxy2dx-2λy2dx. Differentiating (10) with respect to the spatial variable x, then multiplying by yx=(1-α2x2)ux, and integrating by parts again, we obtain (12)ddtyx2dx=2yxyxtdx=-5uxyx2dx+2α2uxy2dx-2λyx2dx. Summarizing (11) and (12), we obtain (13)ddt((y2+yx2)dx)=-(3-2α2)uxy2dx-5uxyx2dx-2λ((y2+yx2)dx). If ux is bounded from below on [0,T), for example, ux-C, C is a positive constant, then we get by (13) and Gronwall's inequality the following: (14)yH12exp{(KC-2λ)t}y0H12, where K=max{5,(3-2/α2)}. Therefore, the H3-norm of the solution to (10) does not blow up in finite time. Furthermore, similar argument shows that the Hk-norm with k4 does not blow up either in finite time. Consequently, this theorem can be proved by Theorem 1 and simple density argument for all s>3/2.

Lemma 3.

Let u0Hα1, then as long as the solution u(x,t) given by Theorem 1 exists, for any t[0,T), one has (15)uHα12=exp(-2λt)u0Hα12, where the norm is defined as (16)uHα12=(u2+α2ux2)dx.

Proof.

Multiplying both sides of (10) by u and integrating by parts on , we get (17)uytdx+(yu)xudx+12(u2-α2ux2)xudx+c0yxudx+λyudx=0. Note that (18)(yu)xudx+12(u2-α2ux2)xudx=0,c0yxudx=0. Then, we have (19)u(ut-α2uxxt)dx+λ(u2-α2uuxx)dx=0. Hence, (20)uutdx-α2uuxxtdx+λu2dx-λα2uuxxdx=0. Thus, we easily get (21)(uut+α2uxuxt)dx+λ(u2+α2ux2)dx=0, and, therefore, (22)ddtuHα12+2λuHα12=0. By integration from 0 to t, we get (23)uHα12=exp(-2λt)u0Hα12,      for  any  t[0,T). Hence, the lemma is proved.

We also need to introduce the standard particle trajectory method for later use. Consider now the following initial value problem as follows: (24)qt=u(t,q)+c0,t[0,T),q(0,x)=x,x, where uC1([0,T),Hs-1) is the solution to (6) with initial data u0Hs, (s>3/2) and T>0 is the maximal time of existence. By direct computation, we have (25)qtx(t,x)=ux(t,q(t,x))qx(t,x). Then, (26)qx(t,x)=exp(0tux(τ,q(τ,x))dτ)>0,t>0,  x, which means that q(t,·): is a diffeomorphism of the line for every t[0,T). Consequently, the L-norm of any function v(t,·) is preserved under the family of the diffeomorphism q(t,·), that is, (27)v(t,·)L=v(t,q(t,·))L,t[0,T). Similarly, (28)infxv(t,x)=infxv(t,q(t,x)),  t[0,T),supxv(t,x)=supxv(t,q(t,x)),  t[0,T). Moreover, one can verify the following important identity for the strong solution in its lifespan: (29)ddt(y(q(x,t),t)qx2(x,t))=-λy(q(x,t),t)qx2(x,t). We get that (30)y(q(x,t),t)qx2(x,t)=y0(x)exp(-λt), where y(x,t) is defined by y(x,t)=(1-α2x2)u(x,t), for t0 in its lifespan.

From the expression of u(x,t) in terms of y(x,t), for all t[0,T), x, we can rewrite u(x,t) and ux(x,t) as follows: (31)u(x,t)=12αe-x/α-xeξ/αy(ξ,t)dξ+12αex/αxe-ξ/αy(ξ,t)dξ, from which we get that (32)ux(x,t)=-12α2e-x/α-xeξ/αy(ξ,t)dξ+12α2ex/αxe-ξ/αy(ξ,t)dξ.

3. Global Existence

It is shown that it is the sign of initial potential not the size of it that can guarantee the global existence of strong solutions.

Theorem 4.

Assume that u0Hs,  s>3/2, and  y0=u0-α2u0xx satisfies (33)y0(x)0,x(-,x0),y0(x)0,x(x0,), for some point x0. Then, the solution u(x,t) to (6) exists globally in time.

Proof.

From the hypothesis and (30), we obtain that y(x,t)0, q(x0,t)x<; y(x,t)0, -<xq(x0,t). According to (31) and (32), one can get that when x>x0, (34)u(q(x,t),t)+αux(q(x,t),t)=1αeq(x,t)/αq(x,t)e-ξ/αy(ξ,t)dξ0, it follows that (35)-αux(q(x,t),t)u(q(x,t),t)uLexp(-λt)2αu0Hα112αu0Hα1, that is, ux(x,t) is bounded below. Similarly, when x<x0, (36)u(q(x,t),t)-αux(q(x,t),t)=1αe-q(x,t)/α-q(x,t)eξ/αy(ξ,t)dξ0, so -αux(q(x,t),t)-u(q(x,t),t). We also get the bounded below result as above. Therefore, the theorem is proved by Lemma 2.

Corollary 5.

Assume that u0Hs,  s>3/2, and  y0=u0-α2u0xx is of one sign, then the corresponding solution u(x,t) to (6) exists globally.

In fact, if x0 is regarded as ±, we prove this corollary immediately from Theorem 4.

4. Persistence Properties

In this section, we will investigate the following property for the strong solutions to (6) in L-space which behave algebraically at infinity as their initial profiles do. The main idea comes from the recent work of Himonas and his collaborators .

Theorem 6.

Assume that for some T>0 and s>3/2,   uC([0,T];Hs) is a strong solution of the initial value problem associated to (6), and that u0(x)=u(x,0) satisfies (37)|u0(x)|,|u0x(x)|~O(x-θ/α)x, for some θ(0,1) and α1. Then, (38)|u(x,t)|,|ux(x,t)|~O(x-θ/α)x, uniformly in the time interval [0,T].

Proof.

The proof is organized as follows. Firstly, we will estimate u(x,t)L and ux(x,t)L. Then, we apply the weight function to obtain the desired result. In the following proof, we denote some constants by c; they may be different from instance to instance, changing even within the same line.

Multiplying (6) by u2n-1 with n+, then integrating both sides with respect to x variable, we can get (39)u2n-1utdx+u2n-1(u+c0)uxdx+u2n-1xG*F(u)dx=-λu2ndx. The first term of the above identity is (40)u2n-1utdx=12nddtu(t)L2n2n=u(t)L2n2n-1ddtu(t)L2n, and the estimates of the second term is (41)u2n-1uuxdxux(t)Lu(t)L2n2n,c0u2n-1uxdx=c0(u2n2n)xdx=0. In view of Hölder's inequality, we can obtain the following estimate for the third term in (39) (42)|u2n-1xG*F(u)dx|u(t)L2n2n-1xG*F(u)L2n. For the last term (43)|u2n-1λudx|λu(t)L2n2n, putting all the inequalities above into (39) yields (44)ddtu(t)L2n(ux(t)L2n+λ)u(t)L2n+xG*F(u)L2n. Using the Sobolev embedding theorem, there exists a constant (45)M=supt[0,T]u(x,t)Hs, such that we have by applying Gronwall's inequality (46)u(t)L2nceMt(u(0)L2n+0txG*F(u)L2ndτ). For any fL1()L(), we know that (47)limqfLq=fL. Taking the limits in (46) (notice that GL1 and F(u)L1L) from (47), we get (48)u(t)LceMt(u(0)L+0txG*F(u)Ldτ). Then, differentiating (6) with respect to variable x produces the following equation: (49)uxt+uuxx+c0uxx+ux2+x2G*F(u)+λux=0. Again, multiplying (49) by ux2n-1 with n+, integrating the result in x variable, and considering the second term and the third term in the above identity with integration by parts, one gets (50)uuxxux2n-1dx=u(ux2n-12n)xdx=-12nuxux2ndx,c0uxxux2n-1dx=c0(ux2n-12n)xdx=0, so, we have (51)uxtux2n-1dx-12nuxux2ndx+ux2n+1dx=-ux2n-1x2G*F(u)dx-λux2n-1uxdx. Similarly, the following inequality holds (52)ddtux(t)L2n(2ux(t)L+λ)ux(t)L2n+x2G*F(u)(t)L2n, and therefore as before, we obtain (53)ux(t)L2nce2Mt(ux(0)L2n+0tx2G*F(u)L2ndτ). Taking the limits in(53), we obtain(54)ux(t)Lce2Mt(ux(0)L+0tx2G*F(u)Ldτ).

Next, we will introduce the weight function to get our desired result. This function φN(x) with N+ is independent of t as the following: (55)φN(x)={1,x1,xθ/α,x(1,N),Nθ/α,xN. From (6) and (49), we get the following two equations: (56)φNut+φNuux+φNc0ux+φNxG*F(u)+λφNu=0,φNuxt+φNuuxx+φNc0uxx+φNux2+φNx2G*F(u)+λφNux=0. We need some tricks to deal with the following term as in : (57)(φN)2n-1u2n-1φNuxdx=(φNu)2n-1[(uφN)x-u(φN)x]dx=(φNu)2n-1d(φNu)-(φNu)2n-1u(φN)xdx(φNu)2ndx, where we have used the fact 0φN(x)φN(x), a.e. x. Similar technique is used for the term (φN)2n-1ux2n-1φNuxxdx. Hence, as in the weightless case, we get the following inequality in view of (48) and (54) as follows: (58)u(t)φNL+ux(t)φNLce2Mt(u(0)φNL+ux(0)φNL)+ce2Mt×(0t(φNxG*F(u)L+φNx2G*F(u)L)dτ). On the other hand, a simple calculation shows that there exists C>0, depending only on α and θ such that for any N+, (59)φN(x)e-|x-y|/α1φN(y)dyC. Therefore, for any appropriate function g, one obtains that (60)|φNxG*g2(x)|=|12αφN(x)e-|x-y|/αg2(y)dy|12αφN(x)e-|x-y|/α1φN(y)φN(y)g(y)g(y)dy12α(φN(x)e-|x-y|/α1φN(y)dy)gφNLgLCαgφNLgL, and similarly, |φNx2G*g2(x)|(C/α)gφNLgL. Using the same method, we can estimate the following two terms (61)|φNG*g(x)|CαgφNL,|φNxG*g(x)|CαgφNL. Therefore, it follows that there exists a constant C1(M,T,α,λ)>0 such that (62)u(t)φNL+ux(t)φNLC1(u(0)φNL+ux(0)φNL)+C10t((u(τ)L+ux(τ)L)·(φNu(τ)L+φNux(τ)L))dτC1(0tu(0)φNL+ux(0)φNL+0t(φNu(τ)L+φNux(τ)L)dτ). Hence, the following inequality is obtained for any N+ and any t[0,T]: (63)u(t)φNL+ux(t)φNLC1(u(0)φNL+ux(0)φNL)C1(u(0)max(1,xθ/α)L+ux(0)max(1,xθ/α)L). Finally, taking the limit as N goes to infinity in the above inequality, we can find that for any t[0,T], (64)(|u(x,t)xθ/α|+|ux(x,t)xθ/α|)C1(u(0)max(1,xθ/α)L+ux(0)max(1,xθ/α)L), which completes the proof of Theorem 6.

5. Asymptotic Description

The following result is to give a detailed description on the corresponding strong solution u(x,t) in its lifespan with u0(x) being compactly supported.

Theorem 7.

Assume that the initial datum 0u0(x)Hs with s>5/2 is compactly supported in [a,c], then the corresponding solution u(x,t)C([0,T);Hs) to (6) has the following property: for any t(0,T), (65)u(x,t)=L(t)e-x/α      as  x>q(c,t),u(x,t)=l(t)ex/α      as  x<q(a,t), where q(x,t) is defined by (24) and T is its lifespan. Furthermore, L(t) and l(t) denote continuous nonvanishing functions, with L(t)>0 and l(t)<0 for t(0,T). Moreover, L(t) is a strictly increasing function, while l(t) is strictly decreasing.

Remark 8.

This is an interesting phenomenon for our model; it implies that the strong solution does not have compact x-support for any t>0 in its lifespan anymore, although the corresponding u0(x) is compactly supported. No matter that the initial profile u0(x) is (no matter it is positive or negative), for any t>0 in its lifespan, the nontrivial solution u(x;t) is always positive at infinity and negative at negative infinity. Moreover, we found that the dissipative coefficient does not affect this behavior.

Proof.

First, since u0(x) has a compact support, so does y0(x)=(1-α2x2)u0(x). Equation (30) tells us that y=(1-α2x2)u(x,t)=((1-α2x2)u0(q-1(x,t))exp(-λt))/(xq-1((x,t),t))2 is compactly supported in [q(a,t),q(c,t)] in its lifespan. Hence, the following functions are well defined (66)E(t)=eξ/αy(ξ,t)dξ,F(t)=e-ξ/αy(ξ,t)dξ, with (67)E(0)=eξ/αy0(ξ)dξ=eξ/αu0(ξ)dξ-α2eξ/αu0xx(ξ)dξ=0. And F(0)=0 by integration by parts.

Then, for x>q(c,t), we have (68)u(x,t)=12αe-|x|/α*y(x,t)=12αe-x/αq(a,t)q(c,t)eξ/αy(ξ,t)dξ=12αe-x/αE(t), where (66) is used.

Similarly, when x<q(a,t), we get (69)u(x,t)=12αe-|x|/α*y(x,t)=12αex/αq(a,t)q(c,t)e-ξ/αy(ξ,t)dξ=12αex/αF(t). Because y(x,t) has a compact support in x in the interval [q(a,t),q(c,t)] for any t[0,T], we get y(x,t)=u(x,t)-α2uxx(x,t)=0, for x>q(c,t) or x<q(a,t). Hence, as consequences of (68) and (69), we have (70)u(x,t)=-αux(x,t)=α2uxx(x,t)=12αe-x/αE(t),as  x>q(c,t),u(x,t)=αux(x,t)=α2uxx(x,t)=12αex/αF(t),as  x<q(a,t). On the other hand, (71)dE(t)dt=eξ/αyt(ξ,t)dx.

Substituting the identity (10) into dE(t)/dt, we obtain (72)dE(t)dt=-eξ/α[(yu)x+12(u2-α2ux2)x+c0yx+λy]dξ=1αeξ/αyudξ+12αeξ/α(u2-α2ux2)dξ+c0αeξ/αydξ+α2eξ/αλuxxdξ-eξ/αλudξ=32αeξ/αu2dξ+α2eξ/αux2dξ+eξ/αuuxdξ=eξ/α(1αu2+α2ux2)dξ>0, where we used (70). Therefore, in the lifespan of the solution, we have that E(t) is an increasing function with E(0)=0; thus, it follows that E(t)>0 for t(0,T]; that is, (73)E(t)=0teξ/α(1αu2+α2ux2)(ξ,τ)dξdτ>0. By similar argument, one can verify that the following identity for F(t) is true: (74)F(t)=-0te-ξ/α(1αu2+α2ux2)(ξ,τ)dξdτ<0. In order to finish the proof, it is sufficient to let L(t)=(1/2α)E(t), and to let l(t)=(1/2α)F(t), respectively.

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