# The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

Authors: Ying Wang; YunXi Guo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that(1-∂x2)u0∈M+(R), u0∈H1(R), and u0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.


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## Body

## 1.  Introduction

The Camassa-Holm equation (C-H equation)(1.1)ut-utxx+3uux-2uxuxx-uuxxx=0,t>0,x∈R,
as a model for wave motion on shallow water, has a bi-Hamiltonian structure and is completely integrable. After the equation was derived by Camassa and Holm [1], a lot of works was devoted to its investigation of dynamical properties. The local well posedness of solution for initial data u0∈Hs(R) with s>3/2 was given by several authors [2–4]. Under certain assumptions, (1.1) has not only global strong solutions and blow-up solutions [2, 5–7] but also global weak solutions in H1(R) (see [8–10]). For other methods to handle the problems related to the Camassa-Holm equation and functional spaces, the reader is referred to [11–14] and the references therein.To study the effect of the weakly dissipative term on the C-H equation, Guo [15] and Wu and Yin [16] discussed the weakly dissipative C-H equation
(1.2)ut-utxx+3uux-2uxuxx-uuxxx+λ(u-uxx)=0,t>0,x∈R.
The global existence of strong solutions and blow-up in finite time were presented in [16] provided that y0=(1-∂x2)u0 changes sign. The sufficient conditions on the infinite propagation speed for (1.2) are offered in [15]. It is found that the local well posedness and the blow-up phenomena of (1.2) are similar to the C-H equation in a finite interval of time. However, there are differences between (1.2) and the C-H equation in their long time behaviors. For example, the global strong solutions of (1.2) decay to zero as time tends to infinite under suitable assumptions, which implies that (1.2) has no traveling wave solutions (see [16]).Degasperis and Procesi [17] derived the equation (D-P equation)
(1.3)ut-utxx+4uux-3uxuxx-uuxxx=0,t>0,x∈R,
as a model for shallow water dynamics. Although the D-P equation (1.3) has a similar form to the C-H equation (1.1), it should be addressed that they are truly different (see [18]). In fact, many researchers have paid their attention to the study of solutions to (1.3). Constantin et al. [19] developed an inverse scattering approach for smooth localized solutions to (1.3). Liu and Yin [20] and Yin [21, 22] investigated the global existence of strong solutions and global weak solutions to (1.3). Henry [23] showed that the smooth solutions to (1.3) have infinite speed of propagation. Coclite and Karlsen [24] obtained global existence results for entropy solutions in L1(R)∩BV(R) and in L2(R)∩L4(R).The weakly dissipative D-P equation(1.4)ut-utxx+4uux-3uxuxx-uuxxx+λ(u-uxx)=0,t>0,x∈R
is investigated by several authors [25–27] to find out the effect of the weakly dissipative term on the D-P equation. The global existence, persistence properties, unique continuation and the infinite propagation speed of the strong solutions to (1.4) are studied in [26]. The blow-up solution modeling wave breaking and the decay of solution were discussed in [27]. The existence and uniqueness of the global weak solution in space Wloc1,∞(R+×R)⋂Lloc∞(R+;H1(R)) were proved (see [25]).In this paper, we will consider the Cauchy problem for the weakly dissipative shallow water wave equation(1.5)ut-utxx+(a+b)uux-auxuxx-buuxxx+λ(u-uxx)=0,t>0,x∈R,u(0,x)=u0(x),x∈R,
where a>0,b>0, and λ≥0 are arbitrary constants, u is the fluid velocity in the x direction, λ(u-uxx) represents the weakly dissipative term. For λ=0, we notice that (1.5) is a special case of the shallow water equation derived by Constantin and Lannes [28].Since (1.5) is a generalization of the Camassa-Holm equation and the Degasperis-Procesi equation, (1.5) loses some important conservation laws that C-H equation and D-P equation possesses. It needs to be pointed out that Lai and Wu [12] studied global existence and blow-up criteria for (1.5) with λ=0. To the best of our knowledge, the dynamical behaviors related to (1.5) with λ=0, such as the global weak solution in space Wloc1,∞(R+×R)⋂Lloc∞(R+;H1(R)), have not been yet investigated. The objective of this paper is to investigate several dynamical properties of solutions to (1.5). More precisely, we firstly use the Kato theorem [29] to establish the local well-posedness for (1.5) with initial value u0∈Hs with s>3/2. Then, we present a precise blow-up scenario for (1.5). Provided that u0∈Hs(R)⋂L1(R) and the potential y0=(1-∂x2)u0 does not change sign, the global existence of the strong solution is shown to be true. Finally, under suitable assumptions, the existence and uniqueness of global weak solution in W1,∞(R+×R)⋂Lloc∞(R+;H1(R)) are proved. Our main ideas to prove the existence and uniqueness of the global weak solution come from those presented in Constantin and Molinet [8] and Yin [22].


## 2.  Notations

The space of all infinitely differentiable functionsϕ(t,x) with compact support in [0,+∞)×R is denoted by C0∞. Let 1≤p<+∞, and let Lp=Lp(R) be the space of all measurable functions h(t,x) such that ∥h∥LPP=∫R|h(t,x)|pdx<∞. We define L∞=L∞(R) with the standard norm ∥h∥L∞=infm(e)=0supx∈R∖e|h(t,x)|. For any real number s, let Hs=Hs(R) denote the Sobolev space with the norm defined by ∥h∥Hs=(∫R(1+|ξ|2)s|ĥ(t,ξ)|2dξ)1/2<∞, where ĥ(t,ξ)=∫Re-ixξh(t,x)dx.We denote by* the convolution. Let ∥·∥X denote the norm of Banach space X and 〈·,·〉 the H1(R), H-1(R) duality bracket. Let M(R) be the space of the Radon measures on R with bounded total variation and M+(R) the subset of positive measures. Finally, we write BV(R) for the space of functions with bounded variation, V(f) being the total variation of f∈BV(R).Note that ifG(x):=(1/2)e-|x|, x∈R. Then, (1-∂x2)-1f=G*f for all f∈L2(R) and G*(u-uxx)=u. Using this identity, we rewrite problem (1.5) in the form
(2.1)ut+buux+∂xG*[a2u2+3b-a2(ux)2]+λu=0,t>0,x∈R,u(0,x)=u0(x),x∈R,
which is equivalent to
(2.2)yt+buyx+ayux+λy=0,t>0,x∈R,y=u-uxx,u(0,x)=u0(x).


## 3.  Preliminaries

Throughout this paper, let{ρn}n≥1 denote the mollifiers
(3.1)ρn(x):=(∫Rρ(ξ)dξ)-1nρ(nx),x∈R,n≥1,
where ρ∈Cc∞(R) is defined by
(3.2)ρ(x):={e1/(x2-1)for|x|<1,0for|x|≥1.
Thus, we get
(3.3)∫Rρn(x)dx=1,ρn≥0,x∈R,n≥1.
Next, we give some useful results.Lemma 3.1 (see [8]).
Letf:R→R be uniformly continuous and bounded. If  μ∈M(R), then
(3.4)[ρn*(fμ)-(ρn*f)(ρn*μ)]n⟶∞⃗0inL1(R).Lemma 3.2 (see [8]).
Letf:R→R be uniformly continuous and bounded. If  g∈L∞(R), then
(3.5)[ρn*(fg)-(ρn*f)(ρn*g)]n⟶∞⃗0inL∞(R).Lemma 3.3 (see [30]).
LetT>0. If  f,g∈L2((0,T);H1(R)) and  df/dt,dg/dt∈L2((0,T);H-1(R)), then  f,g are a.e. equal to a function continuous from  [0,T] into  L2(R) and
(3.6)〈f(t),g(t)〉-〈f(s),g(s)〉=∫st〈d(f(τ))dτ,g(τ)〉dτ+∫st〈d(g(τ))dτ,f(τ)〉dτ
for all s,t∈[0,T].Lemma 3.4 (see [8]).
Assume thatu(t,·)∈W1,1(R) is uniformly bounded in  W1,1(R) for all  t∈R+. Then, for a.e. t∈R+(3.7)ddt∫R|ρn*u|dx=∫R(ρn*ut)sgn⁡(ρn*u)dx,ddt∫R|ρn*ux|dx=∫R(ρn*uxt)sgn⁡(ρn*ux)dx.


## 4.  Global Strong Solution

We firstly present the existence and uniqueness of the local solutions to the problem (2.1).Theorem 4.1.
Letu0∈Hs(R),s>3/2. Then, the problem (2.1) has a unique solution  u, such that
(4.1)u=u(t,x)∈C([0,T);Hs(R))∩C1([0,T);Hs-1(R)),
where T>0 depends on ∥u0∥Hs(R).Proof.
The proof of Theorem4.1 can be finished by using Kato’s semigroup theory (see [29] or [4]). Here, we omit the detailed proof.Theorem 4.2.
Givenu0∈Hs,s>3/2, the solution  u=(·,u0) of problem (2.1) blows up in finite time  T<+∞ if and only if
(4.2)lim⁡t→Tinf⁡{inf⁡x∈R[ux(t,x)]}=-∞.Proof.
Settingy(t,x)=u(t,x)-uxx(t,x), we get
(4.3)‖y‖L22=∫R(u-uxx)2dx=∫R(u2+2ux2+uxx2)dx,
which yields
(4.4)‖u‖H22≤‖y‖L22≤2‖u‖H22.
Using system (2.2), one has
(4.5)ddt∫Ry2(t,x)dx=2∫Ryytdx=-2b∫Ruyyxdx-2a∫Ruxy2dx-2λ∫Ry2dx=-(2a-b)∫Ruxy2dx-2λ∫Ry2dx.
Assume that there is a constant M>0 such that
(4.6)-ux(t,x)≤Mon⁡[0,T)×R.
From (4.5), we get
(4.7)ddt∫Ry2(t,x)dx≤|2a-b|M∫Ry2dx-2λ∫Ry2dx.
Using Gronwall’ inequality, we deduce the ∥u∥H2 is bounded on [0,T). On the other hand,
(4.8)u(t,x)=(1-∂x2)-1y(t,x)=∫RG(x-s)y(s)ds.
Therefore, using (4.4) leads to
(4.9)‖ux‖L∞≤|∫RGx(x-s)y(s)ds|≤‖Gx‖L2‖y‖L2=12‖y‖L2≤‖u‖H2.
It shows that if ∥u∥H2 is bounded, then ∥ux∥L∞ is also bounded. This completes the proof.We consider the differential equation(4.10)qt=bu(t,q),t∈[0,T),x∈R,q(0,x)=x,x∈R,
where u solves (1.5) and T>0.Lemma 4.3.
Letu0∈Hs(R)(s>3);  T is the maximal existence time of the corresponding solution  u to (1.5). Then, system (4.10) has a unique solution  q∈C1([0,T)×R;R). Moreover, the map  q(t,·) is an increasing diffeomorphism of  R with
(4.11)qx(t,x)=exp⁡(∫0tbux(s,q(s,x))ds)>0,∀(t,x)∈[0,T)×R,y(t,q(t,x))qx2(t,x)=y0(x)exp⁡(∫0t[-(a-2b)ux(s,q(s,x))-λ]ds).Proof.
From Theorem4.1, we have u∈C1([0,T);Hs-1(R)) and Hs-1∈C1(R). We conclude that both functions u(t,x) and ux(t,x) are bounded, Lipschitz in space, and C1 in time. Applying the existence and uniqueness theorem of ordinary differential equations implies that system (4.10) has a unique solution q∈C1([0,T)×R,R).
Differentiating (4.10) with respect to x leads to(4.12)ddtqx=bux(t,q)qx,t∈[0,T),b>0,qx(0,x)=1,x∈R,
which yields
(4.13)qx=exp⁡(∫0tbux(s,q(s,x))ds).
For every T′<T, using the Sobolev embedding theorem gives rise to
(4.14)sup⁡(s,x)∈[0,T′)×R|ux(s,x)|<∞.
It is inferred that there exists a constant K0>0 such that qx≥e-K0t for (t,x)∈[0,T)×R.
By computing directly, we derive(4.15)ddt[y(t,q(t,x))qx2(t,x)]=-(a-2b)ux(t,x)yqx2-λyqx2,
which results in(4.16)y(t,q(t,x))qx2(t,x)=y0(x)exp⁡(∫0t[-(a-2b)ux(s,q(s,x))-λ]ds).
The proof of Lemma 4.3 is completed.Theorem 4.4.
Letu0∈L1(R)∩Hs(R),s>3/2, and  (1-∂x2)u0≥0 for all  x∈R (or equivalently  (1-∂x2)u0≤0 for all  x∈R). Then, problem (2.1) has a global strong solution
(4.17)u=u(⋅,u0)∈C([0,∞);Hs(R))∩C1([0,∞);Hs-1(R)).
Moreover, if y(t,·)=u-uxx, then one has for all t∈R+,(i)
y(t,·)≥0, u(t,·)≥0, and |ux(t,·)|≤u(t,·) on R,(ii)
e-λt∥y0∥L1(R)=∥y(t,·)∥L1(R)=∥u(t,·)∥L1(R)=e-λt∥u0∥L1(R) and ∥ux(t,·)∥L∞(R)≤e-λt∥u0∥L1(R),(iii)
∥u∥H12≤∥u0∥H12exp[(|a-2b|/λ)(1-e-λt)∥u0∥L1-2λt].Proof.
Letu0∈Hs,s>3/2, and let T>0 be the maximal existence time of the solution u with initial date u0 (cf. Theorem 4.1). If y0≥0, then Theorem 4.2 ensures that y≥0 for all t∈[0,T).
Note thatu0=G*y0 and y0=(u0-u0,xx)∈L1(R). By Young’s inequality, we get(4.18)‖u0‖L1(R)=‖G*y0(t,⋅)‖L1(R)≤‖G‖L1(R)≤‖G‖L1(R)‖y0(t,⋅)‖L1(R)≤‖y0‖L1(R).
Integrating the first equation of problem (2.1) by parts, we get
(4.19)ddt∫Rudx=-b∫Ruuxdx-∫R∂x(G*[a2u2+3b-a2(ux)2])dx+λ∫Rudx=0.
It follows that
(4.20)∫Rudx=e-λt∫Ru0dx.
Since y=u-uxx, we have
(4.21)∫Rydx=∫Rudx-∫Ruxxdx=∫Rudx=e-λt∫Ru0dx=e-λt(∫Ru0dx-∫Ru0,xxdx)=e-λt∫Ry0dx.
Given t∈[0,T), due to u(t,x)∈Hs,s≥3/2, from Theorem 4.1 and (4.21), we obtain
(4.22)-ux(t,x)+∫-∞xudx=∫-∞xu-uxxdx=∫-∞xydx≤∫-∞∞ydx=e-λt∫Ry0dx=e-λt∫Ru0dx.
Note that u=G*y,  y≥0 on [0,T) and the positivity of G. Thus, we can infer that u≥0 on [0,T). From (4.22) we have
(4.23)ux(t,x)≥-e-λt∫Ru0dx,∀(t,x)∈[0,T)×R.
From Theorem 4.2 and (4.23), we find T=∞. This implies that problem (2.1) has a unique solution
(4.24)u=u(⋅,u0)∈C([0,∞);Hs(R))⋂C1([0,∞);Hs-1(R)),s≥32.
Due to y(t,x)≥0 and u(t,x)≥0 for all t≥0, it shows that
(4.25)u(t,x)=e-x2∫-∞xeξy(ξ)dξ+ex2∫x∞e-ξy(ξ)dξ,ux(t,x)=-e-x2∫-∞xeξy(ξ)dξ+ex2∫x∞e-ξy(ξ)dξ.
From the two identities above, we infer that (ux)2≤u2 on R for all t≥0. This proves (i).
Due toy≥0, we obtain(4.26)ux(t,x)-∫-∞xudx=-∫-∞x(u-uxx)dx=-∫-∞xydx≤0.
From u≥0 and the inequality above, we get
(4.27)ux(t,x)≤∫-∞xudx≤∫-∞∞udx=e-λt∫-∞∞u0dx=e-λt‖u0‖L1(R).
On the other hand, from (4.23), we have that ux(t,x)≥-e-λt∥u0∥L1(R). This proves (ii).
Multiplying the first equation of problem (2.1) by u and integrating by parts, we find
(4.28)12ddt∫R(u2+ux2)dx=-(a+b)∫Ru2uxdx+a∫Ruuxuxxdx+b∫Ru2uxxxdx-λ∫R(u2+ux2)dx=-a-2b2∫Rux3dx-λ∫R(u2+ux2)dx,
which yields(4.29)∫R(u2+ux2)dx=-(a-2b)∫Rux3dx-2λ∫R(u2+ux2)dx≤|a-2b|∫R|ux|3dx-2λ∫R(u2+ux2)dx≤|a-2b|‖ux‖L∞‖ux‖L22-2λ∫R(u2+ux2)dx≤|a-2b|‖ux‖L∞‖u‖H12-2λ∫R(u2+ux2)dx.
From Gronwall’s inequality, one has
(4.30)‖u‖H12≤‖u0‖H12exp⁡[|a-2b|λ(1-e-λt)‖u0‖L1-2λt].
This proves (iii) and completes the proof of the theorem.


## 5.  Global Weak Solution

Theorem 5.1.
Letu0∈H1(R)⋂L1(R) and  y0=(u0-u0xx)∈M+(R). Then  equation (1.5) has a unique solution  u∈W1,∞(R+×R)⋂Lloc∞(R+;H1(R)) with initial data  u(0)=u0 and such that  (u-uxx)∈M+,a.e.∈R+ is uniformly bounded on R.Proof.
We split the proof of Theorem5.1 in two parts.
Letu0∈H1(R) and y0=u0-u0,xx∈M+(R). Note that u0=G*y0. Thus, for φ∈L∞(R), we have(5.1)‖u0‖L1(R)=‖G*y0‖L1(R)=sup⁡‖φ‖L∞(R)≤1∫Rφ(x)(G*y0)(x)dx=sup⁡‖φ‖L∞(R)≤1∫Rφ(x)∫RG(x-ξ)dy0(ξ)dx=sup⁡‖φ‖L∞(R)≤1∫R(G*φ)(ξ)dy0(ξ)=sup⁡‖φ‖L∞(R)≤1‖G*φ‖L∞(R)‖y0‖M(R)≤sup⁡‖φ‖L∞(R)≤1‖G‖L1(R)‖φ‖L∞(R)‖y0‖M(R)=‖y0‖M(R).
Let us define u0n:=ρn*u0∈H∞(R) for n≥1. Obviously, we get
(5.2)u0n⟶u0inH1(R)forn⟶∞,‖u0n‖H1(R)=‖ρn*u0‖H1(R)=‖ρn*u0‖L2+‖ρn*u0′‖L2≤‖u0‖H1(R),∥u0n∥L1(R)=∥ρn*u0∥L1(R)≤‖u0‖L1(R).
Note that, for all n≥1,
(5.3)y0n:=u0n-u0,xxn=ρn*(y0)≥0.
Referring to the proof of (5.1), we have
(5.4)‖y0n‖L1(R)≤‖y0‖M(R),n≥1.
From the Theorem 4.4, we know that there exists a global strong solution
(5.5)un=un(⋅,u0n)∈C([0,∞);Hs(R))∩C1([0,∞);Hs-1(R)),s≥32,
and un(t,x)-uxxn(t,x)≥0 for all (t,x)∈R+×R.
Note that for all(t,x)∈R+×R(5.6)(un)2=∫-∞x2unuxndξ≤∫R[(un)2+(uxn)2]dξ=‖un‖H12.
From Theorem4.4 and (5.2), we obtain
(5.7)‖uxn‖L∞(R)2≤‖un‖L∞(R)2≤‖un‖H1(R)2≤‖u0n‖H12exp⁡[|a-2b|λ(1-e-λt)∥u0n∥L1-2λt]≤‖u0‖H12exp⁡[|a-2b|λ(1-e-λt)∥u0∥L1-2λt].
From the Hölder inequality, Theorem 4.4, and (5.2), for all t≥0 and n≥1, we have
(5.8)∥bun(t)uxn(t)∥L2(R)≤b‖un(t)‖L∞(R)‖uxn(t)‖L2(R)≤b‖un‖H1(R)2≤b‖u0n‖H12exp⁡[|a-2b|λ(1-e-λt)‖u0n‖L1-2λt]≤b‖u0‖H12exp⁡[|a-2b|λ(1-e-λt)‖u0‖L1-2λt].
Using Young’s inequality, we get
(5.9)‖∂xG*[a2(un)2+3b-a2(uxn)2]‖L2(R)≤a2‖∂xG*(un)2‖L2(R)+3b+a2‖∂xG*(uxn)2‖L2(R)≤a2‖∂xG‖L2(R)‖(un)2‖L1(R)+3b+a2‖∂xG‖L2(R)‖(uxn)2‖L1(R)≤3b+a2‖∂xG‖L2(R)‖un‖H1(R)2≤3b+a2‖∂xG‖L2(R)‖u0n‖H12exp⁡[|a-2b|λ(1-e-λt)‖u0n‖L1-2λt]≤3b+a2‖∂xG‖L2(R)‖u0‖H12exp⁡[|a-2b|λ(1-e-λt)‖u0‖L1-2λt],
where ∥∂xG∥L2(R) is bounded, and
(5.10)‖λun‖L2≤λ‖un‖H1≤λ∥u0∥H1exp⁡[|a-2b|2λ(1-e-λt)‖u0‖L1-λt].
Applying (5.8)–(5.10) and problem (2.1), we have
(5.11)∥ddtun∥L2(R)≤(b+3b+a2‖∂xG‖L2(R))∥u0∥H12exp⁡[|a-2b|λ(1-e-λt)‖u0‖L1-2λt]+λ∥u0∥H1exp⁡[|a-2b|2λ(1-e-λt)‖u0‖L1-λt].
For fixed T>0, from (5.7) and (5.11), we deduce
(5.12)∫0T∫R([un(t,x)]2+[uxn(t,x)]2+[utn(t,x)]2)dxdt≤M,
where M is a positive constant depending only on ∥Gx∥L2(R), ∥u0∥H1(R), ∥u0∥L1(R), and T. It follows that the sequence {un}n≥1 is uniformly bounded in the space H1((0,T)×R). Thus, we can extract a subsequence such that
(5.13)unk⇀u,weaklyinH1((0,T)×R)fornk⟶∞,(5.14)unk⟶u,a.e.on⁡(0,T)×Rfornk⟶∞,
for some u∈H1((0,T)×R). From Theorem 4.4 and (5.2), for fixed t∈(0,T), we have that the sequence uxnk(t,·)∈BV(R) satisfies
(5.15)V[uxnk(t,x)]=‖uxxnk(t,⋅)‖L1(R)≤‖unk(t,⋅)‖L1(R)+‖ynk(t,⋅)‖L1(R)≤2e-λt‖u0nk(t,⋅)‖L1(R)≤2e-λt‖u0(t,⋅)‖L1(R)≤2e-λt‖y0(t,⋅)‖M(R),‖uxnk(t,⋅)‖L∞≤e-λt‖u0nk‖L1(R)≤e-λt‖u0‖L1(R)≤e-λt‖y0‖M(R).
Applying Helly’s theorem [31], we infer that there exists a subsequence, denoted again by {uxnk(t,·)}, which converges at every point to some function ν(t,·) of finite variation with
(5.16)V(ν(t,⋅))≤2e-λt‖y0‖M(R).
From (5.14), we get that for almost all t∈(0,T), uxnk(t,·)→ux(t,·) in D′(R), it follows that ν(t,·)=ux(t,·) for a.e. t∈(0,T). Therefore, we have
(5.17)uxnk(t,⋅)⟶ux(t,⋅)a.e.on⁡(0,T)×Rfornk⟶∞,
and, for a.e. t∈(0,T),
(5.18)V[ux(t,⋅)]=‖uxx(t,⋅)‖M(R)=2e-λt‖u0‖L1≤2e-λt‖y0‖M(R).
By Theorem 4.4 and (5.7), we have
(5.19)‖a2(un)2+3b-a2(uxn)2‖L2(R)≤a2‖(un)2‖L2(R)+3b+a2‖(uxn)2‖L2≤a2‖un‖L∞‖un‖L2(R)+3b+a2‖uxn‖L∞‖uxn‖L2≤a2‖un‖H1(R)2+3b+a2‖un‖H1(R)2=(a+3b2)‖un‖H1(R)2≤(a+3b2)‖u0‖H12exp⁡[|a-2b|λ(1-e-λt)x‖u0‖L1-2λt].
Note that for fixed t∈(0,T), the sequence {(a/2)(un)2+((3b-a)/2)(uxn)2}n≥1 is uniformly bounded in L2(R). Therefore, it has a subsequence {(a/2)(unk)2+((3b-a)/2)(uxnk)2}nk≥1, which converges weakly in L2(R). From (5.14), we infer that the weak L2(R)-limit is {(a/2)(u)2+((3b-a)/2)(ux)2}. It follows from Gx∈L2(R) that
(5.20)∂xG*(a2(unk)2+3b-a2(uxnk)2)⟶∂xG*(a2(u)2+3b-a2(ux)2)fornk⟶∞.
From (5.14), (5.17), and(5.20), we have that u solves (2.1) in D′((0,T)×R).
For fixedT>0, note that utnk is uniformly bounded in L2(R) as t∈[0,T) and ∥unk(t)∥H1(R) is uniformly bounded for all t∈[0,T) and n≥1, and we infer that the family t→unk∈H1(R) is weakly equicontinous on [0,T]. An application of the Arzela-Ascoli theorem yields that {unk} has a subsequence, denoted again {unk}, which converges weakly in H1(R), uniformly in t∈[0,T). The limit function is u. T being arbitrary, we have that u is locally and weakly continuous from [0,∞) into H1(R), that is, u∈Cw,loc(R+;H1(R)).
Since, for a.e.t∈R+, unk(t,·)⇀u(t,·) weakly in H1(R), from Theorem 4.4, we get(5.21)‖u(t,⋅)‖L∞(R)≤‖u(t,⋅)‖H1(R)≤lim⁡inf⁡nk→∞‖unk(t,⋅)‖H1(R)≤‖u0‖H1exp⁡[|a-2b|2λ(1-e-λt)‖u0‖L1(R)-λt].
Inequality (5.21) shows that
(5.22)u∈Lloc⁡∞(R+×R)∩Lloc⁡∞(R+;H1(R)).
From Theorem 4.4, (5.1) and (5.2), for t∈R+, we obtain
(5.23)‖uxn(t,⋅)‖L∞≤e-λt‖u0n‖L1(R)≤e-λt‖u0‖L1(R)≤e-λt‖y0‖M(R).
Combining with (5.14), we have
(5.24)ux∈L∞(R+×R).
Next, we will prove that ∫Ru(t,·)dx=e-λt∫Ru(0,·)dx by using a regularization approach.
Sinceu satisfies (2.1) in distribution sense, convoluting (2.1) with ρn, we have that, for a.e.t∈R+,
(5.25)ρn*ut+ρn*(buux)+ρn*∂xp*[a2u2+3b-a2(ux)2]+λρn*u=0.
Integrating the above equation with respect tox on R, we obtain
(5.26)ddt∫Rρn*udx+∫Rρn*(buux)dx+∫Rρn*∂xp*[a2u2+3b-a2(ux)2]dx+λ∫Rρn*udx=0.
Integration by parts gives rise to
(5.27)ddt∫Rρn*udx=-λ∫Rρn*udx,t∈R+,n≥1.
Utilizing Lemma 3.3, we obtain that
(5.28)∫Rρn*u(t,⋅)dx=e-λt∫Rρn*u0dx.
Since
(5.29)lim⁡n→∞‖ρn*u(t,⋅)-u(t,⋅)‖L1(R)=lim⁡n→∞‖ρn*u0-u0‖L1(R)=0.
it follows that, for a.e. t∈R+,
(5.30)∫Ru(t,⋅)dx=lim⁡n→∞∫Rρn*u(t,⋅)dx=lim⁡n→∞e-λt∫Rρn*u0dx=e-λt∫Ru0dx.
Finally, we prove that (u(t,·)-uxx(t,·))∈M+ is uniformly bounded on R and u(t,x)∈W1,∞(R+×R).
Due to(5.31)L1(R)⊂(L∞)*⊂(C0(R))*=M(R),
from (5.18), we get that, for a.e. t∈R+,
(5.32)‖u(t,⋅)-uxx(t,⋅)‖M(R)≤‖u(t,⋅)‖L1(R)+‖uxx(t,⋅)‖M(R)≤e-λt‖u0‖L1(R)+2e-λt‖y0‖M(R)≤3e-λt‖y0‖M(R).
The above inequality implies that, for a.e. t∈R+, (u(t,·)-uxx(t,·))∈M(R) is uniformly bounded on R. For fixed T≥0, applying (5.13) and (5.14), we have
(5.33)[unk(t,⋅)-uxxnk(t,⋅)]⟶[u(t,⋅)-uxx(t,⋅)]inD′(R)forn⟶∞.
Since (unk(t,·)-uxxnk(t,·))≥0 for all (t,x)∈R+×R, we obtain that for a.e. t∈R+, (u(t,·)-uxx(t,·))∈M+(R).
Note thatu(t,x)=G*(u(t,x)-uxx(t,x)). Then we get(5.34)|u(t,x)|=|G*(u(t,x)-uxx(t,x))|≤‖G‖L∞(R)‖u(t,x)-uxx(t,x)‖M(R)≤3e-λt‖y0‖M(R).
Combining with (5.24), it implies that u(t,x)∈W1,∞(R+×R).
This completes the proof of the existence of Theorem5.1.
Next, we present the uniqueness proof of the Theorem5.1.
Letu,v∈W1,∞(R+×R)∩Lloc∞(R+;H1(R)) be two global weak solutions of problem (2.1) with the same initial data u0. Assume that (u(t,·)-uxx(t,·))∈M+(R) and (v(t,·)-vxx(t,·))∈M+(R) are uniformly bounded on R+ and set
(5.35)N:=sup⁡t∈R+{‖u(t,⋅)-uxx(t,⋅)‖M(R)+‖v(t,⋅)-vxx(t,⋅)‖M(R)}.
From assumption, we know thatN<∞. Then, for all (t,x)∈R+×R,
(5.36)|u(t,x)|=|G*(u(t,x)-uxx(t,x))|≤‖G‖L∞(R)‖u(t,x)-uxx(t,x)‖M(R)≤N2,(5.37)|ux(t,x)|=|Gx*(u(t,x)-uxx(t,x))|≤‖Gx‖L∞(R)‖u(t,x)-uxx(t,x)‖M(R)≤N2.
Similarly,
(5.38)|v(t,x)|≤N2,|vx(t,x)|≤N2,(t,x)∈R+×R.
Following the same procedure as in (5.1), we may also get that
(5.39)‖u(t,x)‖L1=‖G*(u(t,x)-uxx(t,x))‖L1(R)≤‖G‖L1(R)‖u(t,x)-uxx(t,x)‖M(R)≤N,‖ux(t,x)‖L1(R)=‖Gx*(u(t,x)-uxx(t,x))‖L1(R)≤‖Gx‖L1(R)‖u(t,x)-uxx(t,x)‖M(R)≤N
and, for all (t,x)∈R+×R,
(5.40)|v(t,x)|≤N,|vx(t,x)|≤N,(t,x)∈R+×R.
We define
(5.41)w(t,x):=u(t,x)-v(t,x),(t,x)∈R+×R.
Convoluting (2.1) for u and v with ρn, we get that for a.e. t∈R+ and all n≥1,
(5.42)ρn*ut+ρn*(buux)+ρn*∂xG*[a2u2+3b-a2(ux)2]+λρn*u=0,(5.43)ρn*vt+ρn*(bvvx)+ρn*∂xG*[a2v2+3b-a2(vx)2]+λρn*v=0.
Subtracting (5.43) from (5.42) and using Lemma 3.4, integration by parts shows that, for a.e. t∈R+ and alln≥1(5.44)ddt∫R|ρn*w|dx=∫R(ρn*wt)sgn⁡(ρn*w)dx=-b∫R(ρn*wux)sgn⁡(ρn*w)dx-b∫R(ρn*vwx)sgn⁡(ρn*w)dx-a2∫R(ρn*∂xG*w(u+v))sgn⁡(ρn*w)dx-3b-a2∫R(ρn*∂xG*wx(ux+vx))sgn⁡(ρn*w)dx-λ∫R(ρn*w)sgn⁡(ρn*w)dx.
Using (5.36)–(5.38) and Young’s inequality to the first term on the right-hand side of (5.44) yields,
(5.45)|∫R(ρn*(wux))sgn⁡(ρn*w)dx|≤∫R|(ρn*(wux))|dx≤∫R|ρn*w||ρn*ux|dx+∫R|ρn*(wux)-(ρn*w)(ρn*ux)|dx≤‖ρn*ux‖L∞∫R|ρn*w|dx+∫R|ρn*(wux)-(ρn*w)(ρn*ux)|dx≤‖ρn‖L1‖ux‖L∞∫R|ρn*w|dx+∫R|ρn*(wux)-(ρn*w)(ρn*ux)|dx≤N2∫R|ρn*w|dx+∫R|ρn*(wux)-(ρn*w)(ρn*ux)|dx.
Similarly, for the second term and the third term on the right-hand side of (5.44), we have
(5.46)|∫R(ρn*(wxv))sgn⁡(ρn*w)dx|≤∫R|(ρn*(wxv))|dx≤∫R|ρn*wx||ρn*v|dx+∫R|ρn*(wxv)-(ρn*wx)(ρn*v)|dx≤N2∫R|ρn*wx|dx+∫R|ρn*(wxv)-(ρn*wx)(ρn*v)|dx,|∫R(ρn*∂xG*[w(u+v)])sgn⁡(ρn*w)dx|≤∫R|ρn*G*[wx(u+v)]|dx+∫R|ρn*G*[w(u+v)x]|dx≤12∫R|ρn*[wx(u+v)]|dx+12∫R|ρn*[w(ux+vx)]|dx≤N2∫R|ρn*wx|dx+N2∫R|ρn*w|dx+∫R|ρn*(wx(u+v))-(ρn*wx)[ρn*(u+v)]|dx+∫R|ρn*[w(u+v)x]-(ρn*w)[ρn*(u+v)x]|dx.
For the last term on the right-hand side of (5.44), we have
(5.47)|∫R(ρn*∂xG*[wx(u+v)x])sgn⁡(ρn*w)dx|≤∫Rρn*∂xG*[|wx|(|ux|+|vx|)]dx≤N∫R|ρn*∂xG*|wx||dx≤N‖∂xG‖L1(R)∫Rρn*|wx|dx≤N∫R|ρn*wx|dx+N[∫R(ρn*|wx|-|ρn*wx|)dx].
From (5.45)–(5.47), for a.e. t∈R+ and all n≥1, we find
(5.48)ddt∫R|ρn*w|dx≤(a+2b4N+λ)∫R|ρn*w|dx+(a+2b)N∫R|ρn*wx|dx+Rn(t),
where
(5.49)Rn(t)⟶0ast⟶∞,|Rn(t)|≤K,n≥1,t∈R+,
where K is a positive constant depending on N and the H1(R)-norms of u(0) and v(0).
In the same way, convoluting (2.1) for u and v with ρn,x and using Lemma 3.4, we get that, for a.e. t∈R+ and all n≥1,
(5.50)ddt∫R|ρn*wx|dx=∫R(ρn*wxt)sgn⁡(ρn,x*w)dx=-b∫R(ρn*wx(ux+vx))sgn⁡(ρn,x*w)dx-b∫R(ρn*vxxw)sgn⁡(ρn,x*w)dx-b∫R(ρn*uwxx)sgn⁡(ρn,x*w)dx-∫R(ρn*∂xxG*[a2(u2-v2)+3b-a2(ux2-vx2)]sgn⁡(ρn,x*w)dx-λ∫R(ρn*wx)sgn⁡(ρn,x*w)dx.
Using the identity∂x2(G*g)=G*g-g for g∈L2(R) and Young’s inequality, we estimate the forth term of the right-hand side of (5.50)
(5.51)|∫R(ρn*∂xxG*[a2(u2-v2)+3b-a2(ux2-vx2)]dx|≤∫R|(ρn*G*[a2(u2-v2)+3b-a2(ux2-vx2)]|dx+∫R|(ρn*[a2(u2-v2)+3b-a2(ux2-vx2)]|dx≤(‖G‖L1(R)+1)∫R|(ρn*[a2w(u+v)+3b-a2wx(ux+vx)]|dx≤a∫R|(ρn*[w(u+v)]|dx+(3b+a)∫R|(ρn*[wx(ux+vx)]|dx≤aN∫R|ρn*w|dx+(3b+a)N∫R|ρn*wx|dx+Rn.
Using (5.36)–(5.38) and Young’s inequality to the first term on the right-hand side of (5.50) gives rise to
(5.52)-b∫R(ρn*wx(ux+vx))sgn⁡(ρn,x*w)dx≤b∫R|ρn*wx(ux+vx)|dx≤b∫R|ρn*wx||ρn*(ux+vx)|dx+b∫R|ρn*wx(ux+vx)-(ρn*wx)(ρn*(ux+vx))|dx≤bN∫R|ρn*wx|dx+Rn.
To treat the second term of the right-hand side of (5.50), we note that
(5.53)|b∫R(ρn*vxxw)sgn⁡(ρn,x*w)dx|≤b∫R|(ρn*w)(ρn*vxx)|dx+b∫R|(ρn*vxxw)-(ρn*w)(ρn*vxx)|dx.
Applying Lemma 3.1, the second expression of the right-hand side of (5.53) can be estimated by a function Rn(t) belonging to (5.49). Making use of the Hölder inequality and (5.1), for a.e. t∈R+ and all n≥1, we have
(5.54)∫R|(ρn*w)(ρn*vxx)|dx≤‖ρn*w‖L∞(R)‖ρn*vxx‖L1(R)≤‖ρn*w‖W1,1(R)‖vxx‖M(R).
It follows from (5.53) and (5.54) that
(5.55)|b∫R(ρn*vxxw)sgn⁡(ρn,x*w)dx|≤bN∫R|ρn*w|dx+bN∫R|ρn*wx|+Rn(t).
Now, we deal with the third term on the right-hand side of (5.50)
(5.56)-b∫R(ρn*uwxx)sgn⁡(ρn,x*w)dx=-b∫R(ρn*u)(ρn*wxx)sgn⁡(ρn,x*w)dx-b∫R[(ρn*uwxx)-(ρn*u)(ρn*wxx)]sgn⁡(ρn,x*w)dx≤-b∫R(ρn*u)∂∂x|ρn*wx|dx+b∫R|(ρn*uwxx)-(ρn*u)(ρn*wxx)|dx=b∫R(ρn*ux)|ρn*wx|dx+Rn.
Therefore, (5.56) implies that, for a.e. t∈R+ and all n≥1,
(5.57)|-b∫R(ρn*uwxx)sgn⁡(ρn,x*w)dx|≤bN∫R|ρn*wx|dx+Rn.
From (5.51), (5.52), (5.55), and (5.57), for a.e. t∈R+ and all n≥1, we deduce that
(5.58)ddt∫R|ρn*wx|dx≤(a+b)N∫R|ρn*w|dx+[(6b+a)N+λ]∫R|ρn*wx|dx+Rn.
Combining with (5.48) and (5.58), we find
(5.59)ddt∫R(|ρn*w|+|ρn*wx|)dx≤[5a+6b4N+λ]∫R|ρn*w|dx+[(2a+8b)N+λ]∫R|ρn*wx|dx+Rn≤[(2a+8b)N+λ]∫R(|ρn*w|+|ρn*wx|)dx+Rn.
It follows from Gronwall’ inequality that, for a.e. t∈R+ and all n≥1,
(5.60)∫R(|ρn*w|+|ρn*wx|)dx≤[∫0tRn(s)ds+∫R(|ρn*w|+|ρn*wx|)(0,x)dx]e[(2a+8b)N+λ]t.
Fix t>0, and let n→∞ in (5.60). Since w=u-v∈W1,1(R) and relation (5.49) holds, making use of Lebesgue’s dominated convergence theorem yields
(5.61)∫R(|w|+|wx|)dx≤[∫R(|w|+|wx|)(0,x)dx]e[(2a+8b)N+λ]t.
Note that w(0)=wx(0)=0; therefore, we obtain u(t,x)=v(t,x) for a.e. (t,x)∈R+×R. This completes the proof of the theorem.



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*Source: 840919-2012-05-22.xml*