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Review Special Issues

Reversing the nutrient drain through urban insect farming—opportunities and challenges

  • Received: 05 September 2018 Accepted: 19 November 2018 Published: 28 November 2018
  • Cities consume the majority of proteins produced globally but have unsustainable, linear food systems from production to consumption to disposal, resulting in significant nutrient losses. The industrial rearing of insects is a promising strategy for converting otherwise lost nutrients back into protein-rich animal feed and fertilizer, particularly to supplement local food production. The black soldier fly (BSF), Hermetia illucens, has been identified as a candidate for industrial rearing. BSF has a superior feed conversion ratio and cycle-time compared to other edible insects and can convert and recover nutrients from a vast variety of organic materials to protein, oil and chitin making it an attractive solution for the management of urban organic solid waste. With an increasing awareness of the environmental urgency and interest in the economic potential of the technology, this review discusses the technological factors confounding the upscaling of insect farming in urban and peri-urban contexts using BSF as a case study. These include the challenges of feed homogenisation and pre-treatment, of integrating insect life-cycle factors (e.g. mating) with bioprocess engineering concepts (which complicates automation), of meeting the nutritional requirements of the larvae at different stages of growth in order to maximize bioconversion and product quality, and of elucidating the impact of microbiome on complex behaviours and bioconversion. A multidisciplinary effort is therefore required to lead urban insect farming to full development to ultimately contribute to future food security.

    Citation: Yingyu Law, Leo Wein. Reversing the nutrient drain through urban insect farming—opportunities and challenges[J]. AIMS Bioengineering, 2018, 5(4): 226-237. doi: 10.3934/bioeng.2018.4.226

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  • Cities consume the majority of proteins produced globally but have unsustainable, linear food systems from production to consumption to disposal, resulting in significant nutrient losses. The industrial rearing of insects is a promising strategy for converting otherwise lost nutrients back into protein-rich animal feed and fertilizer, particularly to supplement local food production. The black soldier fly (BSF), Hermetia illucens, has been identified as a candidate for industrial rearing. BSF has a superior feed conversion ratio and cycle-time compared to other edible insects and can convert and recover nutrients from a vast variety of organic materials to protein, oil and chitin making it an attractive solution for the management of urban organic solid waste. With an increasing awareness of the environmental urgency and interest in the economic potential of the technology, this review discusses the technological factors confounding the upscaling of insect farming in urban and peri-urban contexts using BSF as a case study. These include the challenges of feed homogenisation and pre-treatment, of integrating insect life-cycle factors (e.g. mating) with bioprocess engineering concepts (which complicates automation), of meeting the nutritional requirements of the larvae at different stages of growth in order to maximize bioconversion and product quality, and of elucidating the impact of microbiome on complex behaviours and bioconversion. A multidisciplinary effort is therefore required to lead urban insect farming to full development to ultimately contribute to future food security.


    Consider the equation

    $ ututxx+βux+muux=3αuxuxx+αuuxxx, $ (1.1)

    in which constants $ m > 0 $, $ \alpha > 0 $, and $ \beta\in\mathbb{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 = \frac{3}{2} $, $ \beta = -1 $, and $ \alpha = \frac{3}{2} $, Eq (1.1) reduces to the Fornberg$ - $Whitham (FW) model [2,3]

    $ ututxx+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$ \ddot{o} $rmann[5,6]. The continuity solutions of Eq (1.2) in Besov space are explored in Holmes and Thompson [7]. The H$ \ddot{o}lder $ 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 $ L^2(\mathbb{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 $, $ \beta = 0 $, and $ \alpha = 1 $, Eq (1.1) becomes the well-known Degasperis$ - $Procesi (DP) equation [12]

    $ ututxx+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 $ H^s(\mathbb{R}) $ with $ s > \frac{3}{2} $, Lai and Wu [21] discuss the local existence for a partial differential equation involving the DP and Camassa$ - $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 $ L^2 $ 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 $ L^2 $ conservation law:

    $ R1+ξ2mα+ξ2|ˆu(ξ)|2dξ=R1+ξ2mα+ξ2|^u0(ξ)|2dξ∼∥u02L2(R), $ (1.4)

    where $ u(0, x) = u_0\in H^s(\mathbb{R}) $ endowed with the index $ s > \frac{3}{2} $ 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 $ L^2(\mathbb{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 Kru$ \check{z} $kov [31], we investigate the $ L^1(\mathbb{R}) $ stability of short-time strong solutions provided that $ u_0(x) $ belongs to the space $ H^s(\mathbb{R})\cap L^1(\mathbb{R}) $ with $ s > \frac{3}{2} $. To our knowledge, this $ L^1(\mathbb{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 $ L^1(\mathbb{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

    $ {ututxx+βux+muux=3αuxuxx+αuuxxx,u(0,x)=u0(x). $ (2.1)

    Utilizing inverse operator $ \mathbb{A}^{-2} = (1-\frac{\partial^2}{\partial x^2})^{-1} $, we obtain the equivalent form of (2.1), which reads as

    $ {ut+αuux=βA2ux+αm2A2(u2)x,u(0,x)=u0(x). $ (2.2)

    In fact, for any function $ D(x)\in L^{r}(\mathbb{R}) $ with $ 1\leq r\leq \infty $, we have

    $ A2D(x)=12Re|xz|D(z)dz. $

    Writing $ Q_u = \beta\mathbb{A}^{-2}u+\frac{m-\alpha}{2}\mathbb{A}^{-2}(u^2) $ and $ J_u = \beta\mathbb{A}^{-2}\partial_xu+\frac{m-\alpha}{2}\partial_x\mathbb{A}^{-2}(u^2) $ yields

    $ ut+α2(u2)x+Ju=0. $ (2.3)

    We define $ L^{\infty} = L^{\infty}(\mathbb{R}) $ with the standard norm $ \parallel h\parallel_{L^{\infty}} = \inf\limits_{m(e) = 0}\sup\limits_{x\in \mathbb{R}\backslash e}|h(t, x)| $. For any real number $ s $, we let $ H^s = H^s(\mathbb{R}) $ denote the Sobolev space with the norm defined by

    $ hHs=((1+|ξ|2)s|ˆh(t,ξ)|2dξ)12<, $

    where $ \hat{h}(t, \xi) = \int_{-\infty}^{\infty}e^{-ix\xi}h(t, x)dx $. For $ T > 0 $ and nonnegative number $ s $, let $ C([0, T);H^s(\mathbb{R}) $ denote the Frechet space of all continuous $ H^s $-valued functions on $ [0, T) $.

    Lemma 2.1. ([21]) Provided that $ s > \frac{3}{2} $ and initial value $ u_0(x)\in H^s(\mathbb{R}) $, then there has a unique solution $ u $ which belongs to the space $ C([0, T);H^s(\mathbb{R}))\cap C^1([0, T);H^{s-1}(\mathbb{R})) $, in which $ T $ represents maximal existence time for solution $ u $*.

    *In the sense of Lemma 2.1, for $ s > \frac{3}{2} $, the maximal existence time $ T $ means $ \lim\limits_{t\rightarrow T}\parallel u(t, \cdot) \parallel_{H^s(\mathbb{R})} = \infty $.

    Lemma 2.2. Suppose that $ m > 0 $, $ \alpha > 0 $, $ u_0\in H^s(\mathbb{R}) $, and $ s > \frac{3}{2} $. Let $ u $ be the solution of (2.1). Set $ y = u-\frac{\partial^2u}{\partial x^2} $ and $ Y = (\frac{m}{\alpha}-\frac{\partial^2}{\partial x^2})^{-1}u $. Then

    $ RyYdx=R1+ξ2mα+ξ2|ˆu(ξ)|2dξ=R1+ξ2mα+ξ2|^u0(ξ)|2dξ∼∥u02L2(R). $ (2.4)

    Moreover,

    $ {uL2αmu0L2,ifmα1,uL2mαu0L2,ifmα1. $ (2.5)

    Proof. We have $ u = \frac{m}{\alpha}Y-\partial_{xx}^2Y $ and $ \partial_{xx}^2Y = \frac{m}{\alpha}Y-u $. Utilizing integration by parts and Eq (1.1) yields

    $ ddtRyYdx=RytYdx+RyYtdx=2RYytdx=2R[(m2u2)xβux+α23xxx(u2)]Ydx=2R[(m2u2)xYβuxY+α2(u2)x2xxY]dx=R[(mu2)xY2βuxY+α(u2)x(mαYu)]dx=R(2βuxYα(u2)xu)dx=2βRuYxdx=2βR(mαY2xxY)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\in[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 $ u_0\in H^s(\mathbb{R}) $ endowed with $ s\geq 3 $, then system (2.6) possesses a unique $ q $ belonging to $ C^1([0, T)\times \mathbb{R}) $. In addition, $ q_x(t, x) > 0 $ in the region $ [0, T)\times \mathbb{R} $.

    Proof. Employing Lemma 2.1 derives that $ u_x\in C^2(\mathbb{R}) $ and $ u_t\in C^1[0, T) $ if $ (t, x)\in [0, T)\times \mathbb{R} $. Subsequently, it is concluded that solution $ u(t, x) $ and its slope $ u_x(t, x) $ possess boundness and are Lipschitz continuous in the region $ [0, T)\times \mathbb{R} $. Using the theorem of existence and uniqueness for ODE guarantees that system (2.6) possesses a unique solution $ q\in C^1([0, T)\times \mathbb{R}) $.

    Making use of system (2.6) gives rise to $ \frac{d}{dt}q_x = \alpha u_x(t, q)q_x $ and $ q_x(0, x) = 1 $. Thus, we have

    $ qx(t,x)=et0αux(τ,q(τ,x))dτ. $

    If $ T' < T $, we acquire

    $ \sup\limits_{(t,x)\in [0,T')\times R}|u_x(t, x)| < \infty, $

    implying that it must have a constant $ C_0 > 0 $ to ensure $ q_x(t, x)\geq e^{-C_0t} $. The proof is finished.

    For writing concisely in the following discussions, we utilize notations $ L^\infty = L^\infty(\mathbb{R}) $, $ L^1 = L^1(\mathbb{R}) $, and $ L^2 = L^2(\mathbb{R}) $.

    Lemma 2.4. Assume $ t\in [0, T] $, $ s > \frac{3}{2} $, and $ u_0\in H^s(\mathbb{R}) $. Then

    $ u(t,x)L≤∥u0L+(|β|c02u0L2+|αm|c204u02L2)t, $ (2.7)

    in which $ c_0 = \max\Big(\sqrt{\frac{\alpha}{m}}, \sqrt{\frac{m}{\alpha}}\Big) $.

    Proof. Set $ \eta(x) = \frac{1}{2}e^{-\mid x\mid} $. Utilizing the density arguments utilized in [15], we only need to deal with the case $ s = 3 $ to verify Lemma 2.4. For $ u_0\in H^3(\mathbb{R}) $, using Lemma 2.1 ensures the existence of $ u $ belonging to $ H^3(\mathbb{R}) $. Applying system (2.2) arises

    $ ut+αuux=(αm)η(uux)βηux, $ (2.8)

    where $ \star $ stands for the convolution. Using $ \int_{\mathbb{R}}e^{2|x-z|}dz = 1 $, we acquire

    $ |η(x)ux|=12|xex+zu(t,z)dz+xexzu(t,z)dz|12Re|xz||u(t,z)|dz12(Re2|xz|dz)12(Ru2(t,z)dz)1212uL2c02u0L2. $ (2.9)

    We have

    $ |η(uux)|=|12exzuuzdz|=12|xex+zuuzdz+12+xexzuuzdz|=|14xexzu2dz+14xexzu2dz|14exzu2dz14c20u02L2 $ (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∣≤|mα|4eq(t,x)zu2dz+βηux|mα|4u2dz+|β|2eq(t,x)zuzdz|mα|4u2L2+|β|2uL2|β|c02u0L2+|αm|c204u02L2. $ (2.12)

    From (2.12), we have

    $ {du(t,q(t,x))dt|β|c02u0L2+|αm|c204u02L2,du(t,q(t,x))dt(|β|c02u0L2+|αm|c204u02L2). $ (2.13)

    Integrating (2.13) on the interval $ [0, t] $ yields

    $ {u(t,q(t,x))u0(|β|c02u0L2+|αm|c204u02L2)t,u(t,q(t,x))u0(|β|c02u0L2+|αm|c204u02L2)t. $ (2.14)

    From the first inequality in (2.14), we have

    $ u(t,q(t,x))L(|β|c02u0L2+|αm|c204u02L2)t+u0L. $ (2.15)

    Using the second inequality in (2.14) gives rise to

    $ |u(t,q(t,x))||u0(|β|c02u0L2+|αm|c204u02L2)t.|(|β|c02u0L2+|αm|c204u02L2)t|u0|, $

    from which we have

    $ u(t,q(t,x))L(|β|c02u0L2+|αm|c204u02L2)tuL. $ (2.16)

    Utilizing (2.15) and (2.16), we obtain

    $ u(t,q(t,x))L≤∥u0L+(|β|c02u0L2+|αm|c204u02L2)t. $ (2.17)

    Utilizing Lemma 2.3 and (2.17) yields (2.7).

    Lemma 2.5. If $ u_0\in L^2(\mathbb{R}) $, then

    $ {Qu(t,)L(R)|β|c02u0L2+|αm|c204u02L2,Ju(t,)L(R)|β|c02u0L2+|αm|c204u02L2, $ (2.18)

    in which $ c_0 = \max\Big(\sqrt{\frac{\alpha}{m}}, \sqrt{\frac{m}{\alpha}}\Big) $.

    Proof. From (2.3), we have

    $ Qu=mα4Re|xz|u2(t,z)dz+β2Re|xz|u(t,z)dz, $ (2.19)
    $ Ju=mα4Re|xz|sgn(zx)u2(t,z)dz+β2Re|xz|sgn(zx)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 $ u_0, v_0\in H^s(\mathbb{R}), s > \frac{3}{2} $. Provided that functions $ u $ and $ v $ satisfy system (2.2), for any $ g(t, x)\in C_0^{\infty}([0, \infty)\times (-\infty, \infty)) $, then

    $ |Ju(t,x)Jv(t,x)||g(t,x)|dxc(1+t)|u(t,x)v(t,x)|dx, $ (2.21)

    in which $ c > 0 $ depends on $ m, \alpha, \beta, g, \parallel u_0 \parallel_{L^2} $ and $ \parallel v_0 \parallel_{L^2} $.

    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|2e|xz||sgn(zx)||uv||g(t,x)|dzdx+|mα|2|xA2(u2u2)||g(t,x)|dxc|uv|dze|xz||g(t,x)|dx+|mα|4|e|xz||sgn(zx)||u2v2|dz|g(t,x)|dx|c|u(t,z)v(t,z)|dz+|mα|4|(uv)(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 $ \gamma(y) $ is infinitely differentiable on $ \mathbb{R} $ such that $ \gamma(y)\geq 0 $, $ \gamma(y) = 0 $ when $ |y|\geq 1 $, and $ \int_{-\infty}^\infty \gamma(y)dy = 1 $. For arbitrary constant $ h > 0 $, set $ \gamma_h(y) = \frac{\gamma(h^{-1}y)}{h}\geq 0 $. Thus, $ \gamma_h(y) $ belongs to $ C^\infty(-\infty, \infty) $ and

    $ |γh(y)|ch,γh(y)dy=1;γh(y)=0if|y|h. $

    Suppose that $ G(x) $ is locally integrable in $ \mathbb{R} $. Its mean function is written as

    $ Gh(x)=1hγ(xyh)G(y)dy,h>0. $

    For the Lebesgue point $ x_0 $ of $ G(x) $, it has

    $ limh01h|xx0|h|G(x)G(x0)|dx=0. $ (2.22)

    If $ x $ is an arbitrary Lebesgue point of $ G(x) $, it has $ \lim\limits_{h\rightarrow 0}G^h(x) = G(x) $. Provided that point $ x $ is not Lebesque point of $ G(x) $, (2.22) always holds. Thus, $ G^h(x)\rightarrow G(x) $ ($ h\rightarrow 0 $) is valid almost everywhere.

    We illustrate the notation of a characteristic cone. Suppose that $ N > \max\limits_{t\in [0, T]}\parallel W(t, \cdot)\parallel_{L^\infty} < \infty $, $ 0\leq t\leq T_0 = min(T, R_0N^{-1}) $ and $ \mho = \{(t, x): |x| < R_0-Nt\} $. We write that $ S_\tau $ represents the cross section of $ \mho $ endowed with $ t = \tau, \tau\in [0, T_0] $. For $ r > 0, \rho > 0 $, set $ K_{r} = \Big\{x: |x|\leq r\Big\} $. Let $ \theta_{T} = [0, T]\times\mathbb{R} $ and $ D_1 = \Big\{(t, x, \tau, y)\Big| |\frac{t-\tau}{2}|\leq h$, $\rho\leq\frac{t+\tau}{2}\leq T-\rho$, $|\frac{x-y}{2}|\leq h$, $|\frac{x+y}{2}|\leq r-\rho\Big\} $.

    Lemma 2.7. [31] If function $ Q(t, x) $ is measurable and bounded in $ \Omega_T = [0, T]\times K_r $, for $ h\in(0, \rho) $, $ \rho\in (0, \min[r, T]) $, setting

    $ Hh=1h2D1|Q(t,x)Q(τ,y)|dxdtdydτ, $

    then $ \lim\limits_{h\rightarrow 0}H_h = 0 $.

    Lemma 2.8. [31] Provided that $ |\frac{\partial M(u)}{\partial 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 $ u_0(x)\in H^s(\mathbb{R}) $ endowed with $ s > \frac{3}{2} $. Provided that $ u $ satisfies (2.2), $ g(t, x)\in C_0^\infty(\theta_T) $ and $ g(0, x) = 0 $, for every constant $ k $, then

    $ θT{|uk|gt+sgn(uk)α2[u2k2]gxsgn(uk)Jug}dxdt=0. $

    Proof. Assume that $ \Psi(u) $ is a convex downward and twice smooth function for $ -\infty < u < \infty $. Let $ g(t, x)\in C_0^\infty(\theta_T) $. Using $ \Psi'(u)g(t, x) $ to multiply Eq (2.3), integrating over the domain $ \theta_T $, we transfer the derivatives to $ g $ and acquire

    $ θT{Ψ(u)gt+α[ukΨ(y)ydy]gxΨ(u)Ju(t,x)g}dtdx=0, $ (2.23)

    in which for any constant $ k $, the identity $ \int_{-\infty}^{\infty}\Big[\int_k^u\Psi'(y)ydy\Big]g_xdx = -\int_{-\infty}^{\infty}\Big[g\Psi'(u)uu_x\Big]dx $ is utilized. We have the expression

    $ [ukΨ(y)ydy]gxdx=[12Ψ(u)u212Ψ(k)k212uky2Ψ(y)dy]gxdx. $ (2.24)

    Let $ \Psi^h(u) $ be the mean function of $ |u-k| $ and set $ \Psi(u) = \Psi^h(u) $. Letting $ h\rightarrow 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 $ L^1(\mathbb{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 $ u_0, v_0\in H^s(\mathbb{R})\cap L^1(\mathbb{R}) $ $ (s > \frac{3}{2}) $, respectively. Let $ t\in[0, T] $. Then there is a $ C_T $ depending on $ \parallel u_0\parallel_{L^2(\mathbb{R})}, \parallel v_0\parallel_{L^2(\mathbb{R})} $, $ T, \alpha, \beta $ and $ m $, to satisfy

    $ u(t,)v(t,)L1(R)CTu0v0L1(R). $ (3.1)

    Proof. Utilizing Lemmas 2.1 and 2.4 deduces that $ u $ and $ v $ remain bounded and continuous in $ [0, T]\times\mathbb{R} $. Set $ \uplus = \{(t, x)\} = [\rho, T-2\rho]\times K_{r-2\rho} $, where $ 0 < 2\rho\leq \min(T, r) $, and $ \theta_T = [0, T]\times\mathbb{R} $. Assume $ b(t, x)\in C_0^{\infty}([0, \infty)\times\mathbb{R}) $ associated with $ b(t, x) = 0 $ outside $ \uplus $.

    For $ h\leq \rho $, we construct the function

    $ g=b(t+τ2,x+y2)γh(tτ2)γh(xy2)=b(...)λh(), $

    in which $ (...) = (\frac{t+\tau}{2}, \frac{x+y}{2}) $ and $ (\ast) = (\frac{t-\tau}{2}, \frac{x-y}{2}) $. By the definition of function $ \gamma(y) $, we have

    $ gt+gτ=bt(...)λh(),gx+gy=bx(...)λh(). $

    Choosing $ k = v(\tau, 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))gxsgn(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))gysgn(v(τ,y)u(t,x))Jv(τ,y)g}dτdydtdx=0. $ (3.3)

    Using (3.2) and (3.3) yields

    $ 0θT×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

    $ 0θT{|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 $ P_1 $ and $ P_2 $ in (3.4) as

    $ Yh=Y(t,x,τ,y,u(t,x),v(τ,y))λh(). $

    Using Lemma 2.4, we obtain $ \parallel u\parallel_{L^\infty} < C_T $ and $ \parallel v\parallel_{L^\infty} < C_T $. From Lemmas 2.7 and 2.8, for both functions $ u $ and $ v $, it is deduced that $ Y_h $ obeys the Lipschitz condition. Combining function $ g $, we find $ Y_h = 0 $ outside region $ \uplus $ 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 $ |\lambda(\ast)|\leq \frac{c}{h^2} $ yields

    $ |G11(h)|c[h+1h2D1|u(t,x)v(τ,y)|dxdtdydτ], $ (3.7)

    in which $ c $ does not rely on $ h $. Employing Lemma 2.9 deduces that $ G_{11}(h)\rightarrow 0 $ when $ h\rightarrow 0 $. Now we consider $ G_{12} $. Substituting $ \frac{t-\tau}{2} = \delta, \frac{x-y}{2} = \omega $, we have

    $ hhλh(δ,ω)dδdω=1 $ (3.8)

    and

    $ G12=22θTY(t,x,t,x,u(t,x),v(t,x)){hhλh(δ,ω)dδdω}dxdt=4θTY(t,x,t,x,u(t,x),v(t,x))dxdt. $ (3.9)

    From (3.6)–(3.9), we obtain

    $ limh0θT×θTYhdxdtdydτ=4θTY(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 $ G_{21}(h)\rightarrow 0 $ when $ h\rightarrow 0 $. Applying (3.8) gives rise to

    $ G22=22θT¯P3(t,x,t,x){hhλh(δ,ω)dδdω}dxdt=4θT¯P3(t,x,t,x)dxdt=4θTsgn(uv)(JuJv)b(t,x)dxdt. $ (3.12)

    Employing (3.6), (3.10)–(3.12), we obtain inequality (3.5).

    Set

    $ F(t)=|uv|dx. $

    In order to prove the inequality (3.1), we define

    $ Ah(z)=zγh(z)dz(Ah(z)=γh(z)0). $

    In (3.5), provided that two numbers $ \rho < \tau_1 $, $ \tau_1, \rho\in (0, T_0) $, and $ h < min(\rho, T_0-\tau_1) $, we set

    $ b(t,x)=[Ah(tρ)Ah(tτ1)]B(t,x), $

    where

    $ B(t,x)=Bε(t,x)=1Aε(|x|+NtR0+ε),ε>0. $

    Provided that $ (t, x) $ does not belong to $ \uplus $, then $ b(t, x) = 0 $. If $ (t, x) $ does not belong to $ \mho $, we have $ B(t, x) = 0 $. It arises for $ (t, x)\in \mho $ that

    $ 0=Bt+N|Bx|Bt+NBx. $

    Using the above analysis and (3.5) yields

    $ 0T00{[γh(tρ)γh(tτ1)]Bε|uv|}dxdt+T00[Ah(tρ)Ah(tτ1)]|[JuJv]b(t,x)|dxdt, $

    which together with Lemma 2.6 (when $ \varepsilon\rightarrow \infty $ and $ R_0\rightarrow \infty $) gives rise to

    $ 0T00{[γh(tρ)γh(tτ1)]|uv|dx}dt+c(1+T0)T00[Ah(tρ)Ah(tτ1)]|uv|dxdt. $ (3.13)

    The property of $ \gamma_h(z) $ for $ h\leq \min(\rho, T_0-\rho) $ derives that

    $ |T00γh(tρ)F(t)dtF(ρ)|=|T00γh(tρ)(F(t)F(ρ))dt|c1hρ+hρh|F(t)F(ρ)|dt0,whenh0, $

    in which $ c > 0 $ is independent of $ h $.

    Setting

    $ Z(ρ)=T00Ah(tρ)F(t)dt=T00tργh(z)F(t)dzdt, $

    we derive that

    $ Z(ρ)=T00γh(tρ)F(t)dtF(ρ),whenh0. $

    Thus, we acquire

    $ Z(ρ)Z(0)ρ0F(z)dz,whenh0. $ (3.14)

    and

    $ Z(τ1)Z(0)τ10F(z)dz,whenh0. $ (3.15)

    Using (3.14) and (3.15) directly deduces that

    $ Z(ρ)Z(τ1)τ1ρF(z)dz,whenh0. $ (3.16)

    Sending $ \tau_1\rightarrow t, \rho\rightarrow 0 $, from (3.13) and (3.16), we have

    $ |uv|dx|u0v0|dx+c(1+T0)t0|uv|dxdt. $ (3.17)

    Utilizing (3.17) and the Gronwall inequality leads to the inequality (3.1).

    Remark: We establish the $ L^1 $ local stability of strong solutions for the nonlinear shallow water wave equation (1.1) provided that its initial value belongs to the space $ H^s(\mathbb{R})\cap L^1(\mathbb{R}) $ with $ s > \frac{3}{2} $. 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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