A nonlinear shallow water wave equation containing the famous Degasperis−Procesi and Fornberg−Whitham models is investigated. The novel derivation is that we establish the L2 bounds of solutions from the equation if its initial value belongs to space L2(R). The L∞ bound of the solution is derived. The techniques of doubling the space variable are employed to set up the L1 local stability of short time solutions.
Citation: Jun Meng, Shaoyong Lai. L1 local stability to a nonlinear shallow water wave model[J]. Electronic Research Archive, 2024, 32(9): 5409-5423. doi: 10.3934/era.2024251
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A nonlinear shallow water wave equation containing the famous Degasperis−Procesi and Fornberg−Whitham models is investigated. The novel derivation is that we establish the L2 bounds of solutions from the equation if its initial value belongs to space L2(R). The L∞ bound of the solution is derived. The techniques of doubling the space variable are employed to set up the L1 local stability of short time solutions.
Consider the equation
ut−utxx+βux+muux=3αuxuxx+αuuxxx, | (1.1) |
in which constants m>0, α>0, and β∈R. Equation (1.1) characterizes the hydrodynamical dynamics of shallow water waves and is a special model derived in Constantin and Lannes [1]. In fact, the nonlinear shallow water wave model holds great significance for the scientific community due to its application in tsunami modeling and forecasting, a critical scientific problem with global implications for coastal communities. The investigation of shallow water wave equations may aid scientists in comprehending and predicting the behavior of tsunamis.
If m=32, β=−1, and α=32, Eq (1.1) reduces to the Fornberg−Whitham (FW) model [2,3]
ut−utxx+32uux=ux+92uxuxx+32uuxxx. | (1.2) |
Many works have been carried out to discuss various dynamical behaviors of the FW equation. Sufficient and necessary conditions, guaranteeing that the wave breaking of Eq (1.2) happens, are found out in Haziot [4]. The sufficient conditions of wave breaking and discontinuous traveling wave solutions to the FW model are considered in H¨ormann[5,6]. The continuity solutions of Eq (1.2) in Besov space are explored in Holmes and Thompson [7]. The H¨older continuous solutions to the FW model are in detail investigated in Holmes [8]. Ma et al. [9] provide sufficient conditions to ensure the occurrence of wave breaking for a range of nonlocal Whitham type equations. On the basis of L2(R) conservation law, Wu and Zhang [10] investigate the wave breaking of the Fornberg−Whitham equation. Comparing to the previous wave breaking results for the FW model, Wei [11] gives a novel sufficient condition to guarantee that the wave breaking for Eq (1.2) happens.
Suppose that m=4, β=0, and α=1, Eq (1.1) becomes the well-known Degasperis−Procesi (DP) equation [12]
ut−utxx+4uux=3uxuxx+uuxxx. | (1.3) |
Many works have been carried out to study the dynamical characteristics of Eq (1.3). For instances, the integrability of the DP equation is derived in Degasperis and Procesi [12] and Degasperis et al. [13]. Escher et al. [14] investigate the existence of global weak solutions for the DP model. Liu et al. [15] prove the well-posedness of global strong solutions and blow-up phenomena for Eq (1.3) under certain conditions. Yin [16] considers the Cauchy problem for a periodic generalized Degasperis−Procesi model. The large-time asymptotic behavior of the periodic entropy solutions for the DP equation is discussed in Conclite and Karlsen [17]. Various kinds of traveling wave solutions for Eq (1.3) are presented in [18,19,20]. In the Sobolev space Hs(R) with s>32, Lai and Wu [21] discuss the local existence for a partial differential equation involving the DP and Camassa−Holm(CH) models. The investigation of wave speed for the DP model is carried out in Henry [22]. The dynamical properties of CH equations are presented in [23,24,25,26]. For dynamical features of other nonlinear models, which are closely relevant to the DP and FW models, we refer the reader to [27,28,29,30].
As we know, the L2 conservation law derived from the DP or FW equation takes an essential role in investigating the dynamical features of the DP and FW models. We derive that Eq (1.1) possesses the following L2 conservation law:
∫R1+ξ2mα+ξ2|ˆu(ξ)|2dξ=∫R1+ξ2mα+ξ2|^u0(ξ)|2dξ∼∥u0∥2L2(R), | (1.4) |
where u(0,x)=u0∈Hs(R) endowed with the index s>32 is the initial value of u.
A natural question is that as the shallow water wave model (1.1) generalizes the famous Fornberg−Whitham equation (1.2) and Degasperis−Procesi model (1.3), what kinds of dynamical characteristics of DP and FW models still hold for Eq (1.1). For this purpose, the key element of this work is that we derive L2(R) conservation law for (1.1). Using (1.4) and the technique of transport equation, we establish the boundedness of the solutions for Eq (1.1). Employing the approach called doubling the space variable in Kruˇzkov [31], we investigate the L1(R) stability of short-time strong solutions provided that u0(x) belongs to the space Hs(R)∩L1(R) with s>32. To our knowledge, this L1(R) stability of Eq (1.1) has never been established in literatures.
The organization of this job is that Section 2 prepares several Lemmas. The L1(R) stability of short time solution to Eq (1.1) is established in Section 3.
For the nonlinear shallow water wave equation (1.1), we write out its initial problem
{ut−utxx+βux+muux=3αuxuxx+αuuxxx,u(0,x)=u0(x). | (2.1) |
Utilizing inverse operator A−2=(1−∂2∂x2)−1, we obtain the equivalent form of (2.1), which reads as
{ut+αuux=−βA−2ux+α−m2A−2(u2)x,u(0,x)=u0(x). | (2.2) |
In fact, for any function D(x)∈Lr(R) with 1≤r≤∞, we have
A−2D(x)=12∫Re−|x−z|D(z)dz. |
Writing Qu=βA−2u+m−α2A−2(u2) and Ju=βA−2∂xu+m−α2∂xA−2(u2) yields
ut+α2(u2)x+Ju=0. | (2.3) |
We define L∞=L∞(R) with the standard norm ∥h∥L∞=infm(e)=0supx∈R∖e|h(t,x)|. For any real number s, we let Hs=Hs(R) denote the Sobolev space with the norm defined by
∥h∥Hs=(∫∞−∞(1+|ξ|2)s|ˆh(t,ξ)|2dξ)12<∞, |
where ˆh(t,ξ)=∫∞−∞e−ixξh(t,x)dx. For T>0 and nonnegative number s, let C([0,T);Hs(R) denote the Frechet space of all continuous Hs-valued functions on [0,T).
Lemma 2.1. ([21]) Provided that s>32 and initial value u0(x)∈Hs(R), then there has a unique solution u which belongs to the space C([0,T);Hs(R))∩C1([0,T);Hs−1(R)), in which T represents maximal existence time for solution u*.
*In the sense of Lemma 2.1, for s>32, the maximal existence time T means limt→T∥u(t,⋅)∥Hs(R)=∞.
Lemma 2.2. Suppose that m>0, α>0, u0∈Hs(R), and s>32. Let u be the solution of (2.1). Set y=u−∂2u∂x2 and Y=(mα−∂2∂x2)−1u. Then
∫RyYdx=∫R1+ξ2mα+ξ2|ˆu(ξ)|2dξ=∫R1+ξ2mα+ξ2|^u0(ξ)|2dξ∼∥u0∥2L2(R). | (2.4) |
Moreover,
{∥u∥L2≤√αm∥u0∥L2,ifmα≤1,∥u∥L2≤√mα∥u0∥L2,ifmα≥1. | (2.5) |
Proof. We have u=mαY−∂2xxY and ∂2xxY=mαY−u. Utilizing integration by parts and Eq (1.1) yields
ddt∫RyYdx=∫RytYdx+∫RyYtdx=2∫RYytdx=2∫R[(−m2u2)x−βux+α2∂3xxx(u2)]Ydx=2∫R[(−m2u2)xY−βuxY+α2(u2)x∂2xxY]dx=∫R[(−mu2)xY−2βuxY+α(u2)x(mαY−u)]dx=∫R(−2βuxY−α(u2)xu)dx=2β∫RuYxdx=2β∫R(mαY−∂2xxY)Yxdx=0. |
Utilizing the above identity and the Parserval identity gives rise to (2.4). Inequality (2.5) is derived directly from (2.4).
For each time t∈[0,T), we write the transport system
{qt=αu(t,q),q(0,x)=x. | (2.6) |
The next lemma demonstrates that q(t,x) possesses the feature of increasing diffeomorphism.
Lemma 2.3. Provided that T is defined as in Lemma 2.1 and u0∈Hs(R) endowed with s≥3, then system (2.6) possesses a unique q belonging to C1([0,T)×R). In addition, qx(t,x)>0 in the region [0,T)×R.
Proof. Employing Lemma 2.1 derives that ux∈C2(R) and ut∈C1[0,T) if (t,x)∈[0,T)×R. Subsequently, it is concluded that solution u(t,x) and its slope ux(t,x) possess boundness and are Lipschitz continuous in the region [0,T)×R. Using the theorem of existence and uniqueness for ODE guarantees that system (2.6) possesses a unique solution q∈C1([0,T)×R).
Making use of system (2.6) gives rise to ddtqx=αux(t,q)qx and qx(0,x)=1. Thus, we have
qx(t,x)=e∫t0αux(τ,q(τ,x))dτ. |
If T′<T, we acquire
sup(t,x)∈[0,T′)×R|ux(t,x)|<∞, |
implying that it must have a constant C0>0 to ensure qx(t,x)≥e−C0t. The proof is finished.
For writing concisely in the following discussions, we utilize notations L∞=L∞(R), L1=L1(R), and L2=L2(R).
Lemma 2.4. Assume t∈[0,T], s>32, and u0∈Hs(R). Then
∥u(t,x)∥L∞≤∥u0∥L∞+(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t, | (2.7) |
in which c0=max(√αm,√mα).
Proof. Set η(x)=12e−∣x∣. Utilizing the density arguments utilized in [15], we only need to deal with the case s=3 to verify Lemma 2.4. For u0∈H3(R), using Lemma 2.1 ensures the existence of u belonging to H3(R). Applying system (2.2) arises
ut+αuux=(α−m)η⋆(uux)−βη⋆ux, | (2.8) |
where ⋆ stands for the convolution. Using ∫Re2|x−z|dz=1, we acquire
|η(x)⋆ux|=12|−∫x−∞e−x+zu(t,z)dz+∫∞xex−zu(t,z)dz|≤12∫Re−|x−z||u(t,z)|dz≤12(∫Re−2|x−z|dz)12(∫Ru2(t,z)dz)12≤12∥u∥L2≤c02∥u0∥L2. | (2.9) |
We have
|η⋆(uux)|=|12∫∞−∞e−∣x−z∣uuzdz|=12|∫x−∞e−x+zuuzdz+12∫+∞xex−zuuzdz|=|−14∫x−∞e−∣x−z∣u2dz+14∫∞xe−∣x−z∣u2dz|≤14∫∞−∞e−∣x−z∣u2dz≤14c20∥u0∥2L2 | (2.10) |
and
du(t,q(t,x))dt=ut(t,q(t,x))+ux(t,q(t,x))dq(t,x)dt=ut(t,q(t,x))+αuux(t,q(t,x)). | (2.11) |
Combining with (2.8)–(2.11) and Lemma 2.2 gives rise to
∣du(t,q(t,x))dt∣≤|m−α|4∫∞−∞e−∣q(t,x)−z∣u2dz+∣βη⋆ux∣≤|m−α|4∫∞−∞u2dz+|β|2∣∫∞−∞e−∣q(t,x)−z∣uzdz∣≤|m−α|4∥u∥2L2+|β|2∥u∥L2≤|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2. | (2.12) |
From (2.12), we have
{du(t,q(t,x))dt≤|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2,du(t,q(t,x))dt≥−(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2). | (2.13) |
Integrating (2.13) on the interval [0,t] yields
{u(t,q(t,x))−u0≤(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t,u(t,q(t,x))−u0≥−(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t. | (2.14) |
From the first inequality in (2.14), we have
∥u(t,q(t,x))∥L∞≤(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t+∥u0∥L∞. | (2.15) |
Using the second inequality in (2.14) gives rise to
|u(t,q(t,x))|≥|u0−(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t.|≥−(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t−|u0|, |
from which we have
∥u(t,q(t,x))∥L∞≥−(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t−∥u∥L∞. | (2.16) |
Utilizing (2.15) and (2.16), we obtain
∥u(t,q(t,x))∥L∞≤∥u0∥L∞+(|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2)t. | (2.17) |
Utilizing Lemma 2.3 and (2.17) yields (2.7).
Lemma 2.5. If u0∈L2(R), then
{∥Qu(t,⋅)∥L∞(R)≤|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2,∥Ju(t,⋅)∥L∞(R)≤|β|c02∥u0∥L2+|α−m|c204∥u0∥2L2, | (2.18) |
in which c0=max(√αm,√mα).
Proof. From (2.3), we have
Qu=m−α4∫Re−|x−z|u2(t,z)dz+β2∫Re−|x−z|u(t,z)dz, | (2.19) |
Ju=m−α4∫Re−|x−z|sgn(z−x)u2(t,z)dz+β2∫Re−|x−z|sgn(z−x)u(t,z)dz. | (2.20) |
Utilizing (2.9), (2.19), (2.20), Lemma 2.2, and the Schwartz inequality, we obtain (2.18).
Lemma 2.6. Let u0,v0∈Hs(R),s>32. Provided that functions u and v satisfy system (2.2), for any g(t,x)∈C∞0([0,∞)×(−∞,∞)), then
∫∞−∞|Ju(t,x)−Jv(t,x)||g(t,x)|dx≤c(1+t)∫∞−∞|u(t,x)−v(t,x)|dx, | (2.21) |
in which c>0 depends on m,α,β,g,∥u0∥L2 and ∥v0∥L2.
Proof. Applying the Tonelli Theorem and Lemmas 2.2 and 2.4 gives rise to
∫∞−∞|Ju(t,x)−Jv(t,x)||g(t,x)|dx≤|β|2∫∞−∞∫∞−∞e−|x−z||sgn(z−x)||u−v||g(t,x)|dzdx+|m−α|2∫∞−∞|∂xA−2(u2−u2)||g(t,x)|dx≤c∫∞−∞|u−v|dz∫∞−∞e−|x−z||g(t,x)|dx+|m−α|4|∫∞−∞∫∞−∞e−|x−z||sgn(z−x)||u2−v2|dz|g(t,x)|dx|≤c∫∞−∞|u(t,z)−v(t,z)|dz+|m−α|4∫∞−∞|(u−v)(u+v)|dz|∫∞−∞|g(t,x)|dx|≤c(1+t)∫∞−∞|u(t,z)−v(t,z)|dz, |
from which we acquire (2.21).
Suppose that function γ(y) is infinitely differentiable on R such that γ(y)≥0, γ(y)=0 when |y|≥1, and ∫∞−∞γ(y)dy=1. For arbitrary constant h>0, set γh(y)=γ(h−1y)h≥0. Thus, γh(y) belongs to C∞(−∞,∞) and
|γh(y)|≤ch,∫∞−∞γh(y)dy=1;γh(y)=0if|y|≥h. |
Suppose that G(x) is locally integrable in R. Its mean function is written as
Gh(x)=1h∫∞−∞γ(x−yh)G(y)dy,h>0. |
For the Lebesgue point x0 of G(x), it has
limh→01h∫|x−x0|≤h|G(x)−G(x0)|dx=0. | (2.22) |
If x is an arbitrary Lebesgue point of G(x), it has limh→0Gh(x)=G(x). Provided that point x is not Lebesque point of G(x), (2.22) always holds. Thus, Gh(x)→G(x) (h→0) is valid almost everywhere.
We illustrate the notation of a characteristic cone. Suppose that N>maxt∈[0,T]∥W(t,⋅)∥L∞<∞, 0≤t≤T0=min(T,R0N−1) and ℧={(t,x):|x|<R0−Nt}. We write that Sτ represents the cross section of ℧ endowed with t=τ,τ∈[0,T0]. For r>0,ρ>0, set Kr={x:|x|≤r}. Let θT=[0,T]×R and D1={(t,x,τ,y)||t−τ2|≤h, ρ≤t+τ2≤T−ρ, |x−y2|≤h, |x+y2|≤r−ρ}.
Lemma 2.7. [31] If function Q(t,x) is measurable and bounded in ΩT=[0,T]×Kr, for h∈(0,ρ), ρ∈(0,min[r,T]), setting
Hh=1h2⨌D1|Q(t,x)−Q(τ,y)|dxdtdydτ, |
then limh→0Hh=0.
Lemma 2.8. [31] Provided that |∂M(u)∂u| is bounded and
L(u,v)=sgn(u−v)(M(u)−M(v)), |
then for any functions u and v, function L(u,v)) obeys the Lipschitz condition.
Lemma 2.9. Suppose that u0(x)∈Hs(R) endowed with s>32. Provided that u satisfies (2.2), g(t,x)∈C∞0(θT) and g(0,x)=0, for every constant k, then
∬θT{|u−k|gt+sgn(u−k)α2[u2−k2]gx−sgn(u−k)Jug}dxdt=0. |
Proof. Assume that Ψ(u) is a convex downward and twice smooth function for −∞<u<∞. Let g(t,x)∈C∞0(θT). Using Ψ′(u)g(t,x) to multiply Eq (2.3), integrating over the domain θT, we transfer the derivatives to g and acquire
∬θT{Ψ(u)gt+α[∫ukΨ′(y)ydy]gx−Ψ′(u)Ju(t,x)g}dtdx=0, | (2.23) |
in which for any constant k, the identity ∫∞−∞[∫ukΨ′(y)ydy]gxdx=−∫∞−∞[gΨ′(u)uux]dx is utilized. We have the expression
∫∞−∞[∫ukΨ′(y)ydy]gxdx=∫∞−∞[12Ψ′(u)u2−12Ψ′(k)k2−12∫uky2Ψ″(y)dy]gxdx. | (2.24) |
Let Ψh(u) be the mean function of |u−k| and set Ψ(u)=Ψh(u). Letting h→0 and employing the features of sgn(u−k), (2.23), and (2.24) complete the proof.
Actually, the derivation of Lemma 2.9 can also be found in [31].
Utilizing the bounded property of solution u(t,x) for system (2.2), we investigate the L1(R) local stability of u(t,x), which is written in the following theorem.
Theorem 3.1. Suppose that u and v satisfy Eq (1.1) endowed with initial values u0,v0∈Hs(R)∩L1(R) (s>32), respectively. Let t∈[0,T]. Then there is a CT depending on ∥u0∥L2(R),∥v0∥L2(R), T,α,β and m, to satisfy
∥u(t,⋅)−v(t,⋅)∥L1(R)≤CT∥u0−v0∥L1(R). | (3.1) |
Proof. Utilizing Lemmas 2.1 and 2.4 deduces that u and v remain bounded and continuous in [0,T]×R. Set ⊎={(t,x)}=[ρ,T−2ρ]×Kr−2ρ, where 0<2ρ≤min(T,r), and θT=[0,T]×R. Assume b(t,x)∈C∞0([0,∞)×R) associated with b(t,x)=0 outside ⊎.
For h≤ρ, we construct the function
g=b(t+τ2,x+y2)γh(t−τ2)γh(x−y2)=b(...)λh(∗), |
in which (...)=(t+τ2,x+y2) and (∗)=(t−τ2,x−y2). By the definition of function γ(y), we have
gt+gτ=bt(...)λh(∗),gx+gy=bx(...)λh(∗). |
Choosing k=v(τ,y) in Lemma 2.9 and applying the methods called doubling the space variables in [31] yield
⨌θT×θT{|u(t,x)−v(τ,y)|gt+sgn(u(t,x)−v(τ,y))α2(u2(t,x)−v2(τ,y))gx−sgn(u(t,x)−v(τ,y))Ju(t,x)g}dtdxdτdy=0. | (3.2) |
Taking k=u(t,x) in Lemma 2.9 gives rise to
⨌θT×θT{|v(τ,y)−u(t,x)|gτ+sgn(v(τ,y)−u(t,x))α2(u2(t,x)−v2(τ,y))gy−sgn(v(τ,y)−u(t,x))Jv(τ,y)g}dτdydtdx=0. | (3.3) |
Using (3.2) and (3.3) yields
0≤⨌θT×θT{|u(t,x)−v(τ,y)|(gt+gτ)+sgn(u(t,x)−v(τ,y))α2(u2(t,x)−v2(τ,y))(gx+gy)}dxdtdydτ+|⨌θT×θTsgn(u(t,x)−v(t,x))(Ju(t,x)−Jv(τ,y))gdxdtdydτ|.=P1+P2+|⨌θT×θTP3dxdtdydτ|. | (3.4) |
On the basis of the approaches in [31], we aim to verify the inequality
0≤∬θT{|u(t,x)−v(t,x)|bt+sgn(u(t,x)−v(t,x))α2(u2(t,x)−v2(t,x))bx}dxdt+|∬θTsgn(u(t,x)−v(t,x))[Ju(t,x)−Jv(t,x)]bdxdt|. | (3.5) |
We write the integrands of P1 and P2 in (3.4) as
Yh=Y(t,x,τ,y,u(t,x),v(τ,y))λh(∗). |
Using Lemma 2.4, we obtain ∥u∥L∞<CT and ∥v∥L∞<CT. From Lemmas 2.7 and 2.8, for both functions u and v, it is deduced that Yh obeys the Lipschitz condition. Combining function g, we find Yh=0 outside region ⊎ and
⨌θT×θTYhdxdtdydτ=⨌θT×θT[Y(t,x,τ,y,u(t,x),v(τ,y))−Y(t,x,t,x,u(t,x),v(t,x))]λh(∗)dxdtdydτ+⨌θT×θTY(t,x,t,x,u(t,x),v(t,x))λh(∗)dxdtdydτ=G11(h)+G12. | (3.6) |
Utilizing |λ(∗)|≤ch2 yields
|G11(h)|≤c[h+1h2⨌D1|u(t,x)−v(τ,y)|dxdtdydτ], | (3.7) |
in which c does not rely on h. Employing Lemma 2.9 deduces that G11(h)→0 when h→0. Now we consider G12. Substituting t−τ2=δ,x−y2=ω, we have
∫h−h∫∞−∞λh(δ,ω)dδdω=1 | (3.8) |
and
G12=22∬θTY(t,x,t,x,u(t,x),v(t,x)){∫h−h∫∞−∞λh(δ,ω)dδdω}dxdt=4∬θTY(t,x,t,x,u(t,x),v(t,x))dxdt. | (3.9) |
From (3.6)–(3.9), we obtain
limh→0⨌θT×θTYhdxdtdydτ=4∬θTY(t,x,t,x,u(t,x),v(t,x))dxdt. | (3.10) |
Note that
P3=sgn(u(t,x)−v(τ,y))(Ju(t,x)−Jv(τ,y))b(...)λh(∗)=¯P3(t.x,τ,y)λh(∗) |
and
⨌θT×θTP3dxdtdydτ=⨌θT×θT[¯P3(t.x,τ,y)−¯P3(t.x,t,x)]λh(∗)dxdtdydτ+⨌θT×θT¯P3(t.x,t,x)λh(∗)dxdtdydτ=G21(h)+G22. | (3.11) |
We obtain
|G21(h)|≤c(h+1h2×⨌D1|Ju(t,x)−Jv(τ,y)|dxdtdydτ). |
Using Lemmas 2.5 and 2.7 derives G21(h)→0 when h→0. Applying (3.8) gives rise to
G22=22∬θT¯P3(t,x,t,x){∫h−h∫∞−∞λh(δ,ω)dδdω}dxdt=4∬θT¯P3(t,x,t,x)dxdt=4∬θTsgn(u−v)(Ju−Jv)b(t,x)dxdt. | (3.12) |
Employing (3.6), (3.10)–(3.12), we obtain inequality (3.5).
Set
F(t)=∫∞−∞|u−v|dx. |
In order to prove the inequality (3.1), we define
Ah(z)=∫z−∞γh(z)dz(A′h(z)=γh(z)≥0). |
In (3.5), provided that two numbers ρ<τ1, τ1,ρ∈(0,T0), and h<min(ρ,T0−τ1), we set
b(t,x)=[Ah(t−ρ)−Ah(t−τ1)]B(t,x), |
where
B(t,x)=Bε(t,x)=1−Aε(|x|+Nt−R0+ε),ε>0. |
Provided that (t,x) does not belong to ⊎, then b(t,x)=0. If (t,x) does not belong to ℧, we have B(t,x)=0. It arises for (t,x)∈℧ that
0=Bt+N|Bx|≥Bt+NBx. |
Using the above analysis and (3.5) yields
0≤∫T00∫∞−∞{[γh(t−ρ)−γh(t−τ1)]Bε|u−v|}dxdt+∫T00∫∞−∞[Ah(t−ρ)−Ah(t−τ1)]|[Ju−Jv]b(t,x)|dxdt, |
which together with Lemma 2.6 (when ε→∞ and R0→∞) gives rise to
0≤∫T00{[γh(t−ρ)−γh(t−τ1)]∫∞−∞|u−v|dx}dt+c(1+T0)∫T00[Ah(t−ρ)−Ah(t−τ1)]∫∞−∞|u−v|dxdt. | (3.13) |
The property of γh(z) for h≤min(ρ,T0−ρ) derives that
|∫T00γh(t−ρ)F(t)dt−F(ρ)|=|∫T00γh(t−ρ)(F(t)−F(ρ))dt|≤c1h∫ρ+hρ−h|F(t)−F(ρ)|dt→0,whenh→0, |
in which c>0 is independent of h.
Setting
Z(ρ)=∫T00Ah(t−ρ)F(t)dt=∫T00∫t−ρ−∞γh(z)F(t)dzdt, |
we derive that
Z′(ρ)=−∫T00γh(t−ρ)F(t)dt→−F(ρ),whenh→0. |
Thus, we acquire
Z(ρ)→Z(0)−∫ρ0F(z)dz,whenh→0. | (3.14) |
and
Z(τ1)→Z(0)−∫τ10F(z)dz,whenh→0. | (3.15) |
Using (3.14) and (3.15) directly deduces that
Z(ρ)−Z(τ1)→∫τ1ρF(z)dz,whenh→0. | (3.16) |
Sending τ1→t,ρ→0, from (3.13) and (3.16), we have
∫∞−∞|u−v|dx≤∫∞−∞|u0−v0|dx+c(1+T0)∫t0∫∞−∞|u−v|dxdt. | (3.17) |
Utilizing (3.17) and the Gronwall inequality leads to the inequality (3.1).
Remark: We establish the L1 local stability of strong solutions for the nonlinear shallow water wave equation (1.1) provided that its initial value belongs to the space Hs(R)∩L1(R) with s>32. The asymptotic or uniform stability of strong solutions for Eq (1.1) deserves to be investigated. To study the asymptotic stability, we need to find certain restrictions on the initial data, which may be our future works.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Thanks are given to the reviewers for their valuable suggestions and comments, which led to the meaningful improvement of this paper. This work is supported by the Natural Science Foundation of Xinjiang Autonomous Region (Nos. 2024D01A07 and 2020D01B04).
The authors declare no conflicts of interest.
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