Loading [MathJax]/jax/output/SVG/jax.js
Research article

Global existence and finite time blow-up for a class of fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity

  • Received: 15 October 2020 Accepted: 21 December 2020 Published: 23 December 2020
  • MSC : 39A13, 34B18, 34A08

  • In this paper, we study the initial-boundary value problem for a class of fractional p-Laplacian Kirchhoff diffusion equation with logarithmic nonlinearity. For both subcritical and critical states, by means of the Galerkin approximations, the potential well theory and the Nehari manifold, we prove the global existence and finite time blow-up of the weak solutions. Further, we give the growth rate of the weak solutions and study ground-state solution of the corresponding steady-state problem.

    Citation: Fugeng Zeng, Peng Shi, Min Jiang. Global existence and finite time blow-up for a class of fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity[J]. AIMS Mathematics, 2021, 6(3): 2559-2578. doi: 10.3934/math.2021155

    Related Papers:

    [1] Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for ψ-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244
    [2] Ugyen Samdrup Tshering, Ekkarath Thailert, Sotiris K. Ntouyas . Existence and stability results for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point fractional integral boundary conditions. AIMS Mathematics, 2024, 9(9): 25849-25878. doi: 10.3934/math.20241263
    [3] Najla Alghamdi, Bashir Ahmad, Esraa Abed Alharbi, Wafa Shammakh . Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions. AIMS Mathematics, 2024, 9(5): 12964-12981. doi: 10.3934/math.2024632
    [4] Murugesan Manigandan, R. Meganathan, R. Sathiya Shanthi, Mohamed Rhaima . Existence and analysis of Hilfer-Hadamard fractional differential equations in RLC circuit models. AIMS Mathematics, 2024, 9(10): 28741-28764. doi: 10.3934/math.20241394
    [5] Arjumand Seemab, Mujeeb ur Rehman, Jehad Alzabut, Yassine Adjabi, Mohammed S. Abdo . Langevin equation with nonlocal boundary conditions involving a ψ-Caputo fractional operators of different orders. AIMS Mathematics, 2021, 6(7): 6749-6780. doi: 10.3934/math.2021397
    [6] Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut . On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function. AIMS Mathematics, 2022, 7(5): 7817-7846. doi: 10.3934/math.2022438
    [7] J. Vanterler da C. Sousa, E. Capelas de Oliveira, F. G. Rodrigues . Ulam-Hyers stabilities of fractional functional differential equations. AIMS Mathematics, 2020, 5(2): 1346-1358. doi: 10.3934/math.2020092
    [8] Nichaphat Patanarapeelert, Jiraporn Reunsumrit, Thanin Sitthiwirattham . On nonlinear fractional Hahn integrodifference equations via nonlocal fractional Hahn integral boundary conditions. AIMS Mathematics, 2024, 9(12): 35016-35037. doi: 10.3934/math.20241667
    [9] Leping Xie, Jueliang Zhou, Haiyun Deng, Yubo He . Existence and stability of solution for multi-order nonlinear fractional differential equations. AIMS Mathematics, 2022, 7(9): 16440-16448. doi: 10.3934/math.2022899
    [10] Xiaoming Wang, Rizwan Rizwan, Jung Rey Lee, Akbar Zada, Syed Omar Shah . Existence, uniqueness and Ulam's stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives. AIMS Mathematics, 2021, 6(5): 4915-4929. doi: 10.3934/math.2021288
  • In this paper, we study the initial-boundary value problem for a class of fractional p-Laplacian Kirchhoff diffusion equation with logarithmic nonlinearity. For both subcritical and critical states, by means of the Galerkin approximations, the potential well theory and the Nehari manifold, we prove the global existence and finite time blow-up of the weak solutions. Further, we give the growth rate of the weak solutions and study ground-state solution of the corresponding steady-state problem.



    Modeling, simulation and optimization of critical infrastructure systems such as traffic, electricity, water or natural gas networks play an increasingly important role in our society. Many of these problems involve aspects of dynamic energy transportation and distribution in networks. The optimization with respect to efficiency, robustness, or environmental performance requires the use of high-resolution dynamical models in form of time-dependent differential equations. As a particular case, we consider transportation and distribution of natural gas in a network of pipelines, where the high-resolution models refer to the one-dimensional Euler gas equations coupled with further dynamics representing active or passive network elements such as compressors, resistors and control valves.

    Detailed models for gas flow in pipeline networks are well established, see e.g., [4,9,17]. Similar maturity has been achieved for time-simulation methods, see e.g., [1,16,29], the analysis of stationary states [22], the time-continuous optimal control of compressors using adjoint techniques [26,37], and feedback stabilization based on classical solutions [15]. If the discrete nature of control valves being open or closed is to be taken into account, then the optimization needs to deal with mixed integer and continuous type variables simultaneously. Similar decisions are often to be taken into account also in other critical infrastructure systems.

    In industrial practice the treatment of switched systems is often carried out by first applying a full space-time discretization of the systems and then using a mixed-integer nonlinear programming tool that incorporates the switches as extra variables to be optimized, see e.g., [7,5,47]. Another approach is to restrict the optimization to the stationary case where special purpose techniques can be successfully applied [44]. A temporal expansion of these techniques on a full network currently seems to be out of reach, see, e.g., [23,36]. We proceed in yet another way via a model switching approach, see [24]. Since networks already provide a natural spatial partition by the edges representing pipes, it seems natural to minimize a global model error by selecting either one of several models in a dynamic model hierarchy [16,18] or a stationary model hierarchy [37] for each pipe as a function of time. To do this, it is important to identify regions in the time expanded network problem where stationary models still provide a reasonable approximation in the sense that the global error remains small. A particular difficulty that the model switching problem for gas networks has in common with most of the other mentioned applications is the high-resolution model being a partial differential equation (typically a system of balance laws). While the computation of optimal switching for ordinary differential equations and differential-algebraic equations is theoretically and numerically well studied, see [11,19,27,28,33,38,51,52,53,54], the corresponding theory involving certain types of partial differential equations is still under development [45,46].

    Our main contribution in this paper is to provide a theoretical and numerical framework for solving the model switching problem using the example of gas networks. We show that the problem can be casted in the sense of switching among a family of abstract evolution equations on an appropriate Banach space. This allows us to use adjoint based gradient representations for switching time and mode sequence variations recently developed in [45] to characterize locally optimal solutions for the model switching problem by introducing the notion of first order stationarity. This, in turn, motivates a two stage gradient descent approach conceptually introduced and analyzed in [2] for the optimization of switching sequences in the context of ordinary differential equations for numerical solutions of the model switching problem. We provide results for a proof-of-concept implementation for a gas network comprising 340 km of pipes on a 30 min time horizon.

    Our approach relies on a semigroup property for networked transport systems. For related results without nodal control, see [6,30,41], and boundary conditions of a delay-differential-type are considered in [48,49]. In [20,21], classical solutions for a system separating a semigroup equation from a linear nodal control condition are derived. The recent work [31] considers mild solutions if the nodal control is semilinear. In contrast, we derive a semigroup formulation for the entire system, allowing for semilinear dynamics both on the edges and the boundary control, based on a characteristic decomposition of the system.

    The paper is organized a follows. In Section 2 we provide a more detailed description of the common gas network models. In particular, we briefly discuss a semilinear simplification of the Euler gas equations as the most detailed model on a pipe and consider the corresponding stationary solutions. Moreover, we introduce the model switching problem. In Section 3, we show that the equations on a network coupling these models for the pipes, along with appropriate coupling conditions on the nodes allowing further network elements such as valves and compressors, is well-posed in the sense of being equivalently represented by a nonlinear perturbation of a strongly continuous semigroup in a suitable Sobolev space. In Section 4 we apply the well-posedness result to define an appropriate system of adjoint equations and to derive a stationarity concept based on switching-time gradients and mode-insertion gradients as a first order optimality condition for model optimality. Further, we present details of a conceptual algorithm alternating between switching-time optimization and mode-insertion in order to compute a solution of the model switching problem in the sense of our stationarity concept. In Section 5 we present a numerical study. In Section 6 we discuss applications and directions of future work.

    The motion of a compressible non-viscous gas in long high-pressure pipelines is described by the one-dimensional isothermal Euler equations. They are given by a system of nonlinear hyperbolic partial differential equations (PDEs) and consist of the continuity equation and the balance of moments (see, e.g., [9,34,35,50])

    tϱ+x(ϱv)=0,t(ϱv)+x(P+ϱv2)=θϱv|v|gϱh, (1)

    where ϱ denotes the density in kgm3, v the velocity in ms, P the pressure of the gas in kgms2, g the gravitational constant and h the slope of the pipe. Furthermore θ=λ2D, where λ is the friction coefficient of the pipe, and D is the diameter of the pipe. The conserved, respectively balanced, quantities of the system are the density ϱ and the flux q=ϱv. Pressure and density are related by the constitutive law for a real gas

    P=RsϱT0Z(P,T0),

    where Z(P,T0) is the gas compressibility factor at constant temperature T0 and Rs is the specific gas constant. For an ideal gas one has Z(P)1. This system of equations yields a hierarchy of simplified models for pipe dynamics [9,17,25,42].

    For instationary situations, we consider the semilinear PDE model

    tϱ+xq=0,tq+c2xϱ=θq|q|ϱghϱ, (2)

    with a constant speed of sound c=P/ϱ obtained under the assumption of a constant gas compressibility factor Z(P,T0)ˉZ. This model neglects inertia effects and assumes small velocities |v|c. For natural gas, indeed one typically has |v|10ms and c340ms, see e.g., [17]. This PDE exhibits two simple characteristics with speeds λ1=c and λ2=c.

    For stationary situations, we consider the model

    xq=0,c2xϱ=θq|q|ϱghϱ, (3)

    obtained from (2) with tϱ=tq=0. Here, the flux q is constant in space and time q(t,x)=ˉq and ϱ(t,x)=ˉϱ(x) with ˉϱ being a solution of the momentum equation in (3), which is a Bernoulli-equation. A solution of the momentum equation can therefore be obtained algebraically

    ˉϱ(x)={ˉϱ(0)22θˉq|ˉq|c2x,if h=0ˉϱ(0)2exp(2ghx)2θˉq|ˉq|c2exp(2ghx)12gh,if h0. (4)

    Analogous considerations apply also for the case of non-isothermal flow, see e.g. [9,17]. More generally, similar model hierarchies also exist for other infrastructure systems such as water distribution networks [24].

    For m,nN we consider a network of pipes that we model by a metric graph G=(V,E) with nodes V=(v1,,vm) and edges E=(e1,,en)V×V. For each edge eE, call e(1) the left node and e(2) the right node of e. We demand the incident nodes of every edge to be different, so e(1)e(2) for any eE and thus self-loops are not allowed. On the other hand, if vV is any node, then we define

    the set of ingoing edges byδ+v={eE|e(2)=v},the set of outgoing edges byδv={eE|e(1)=v},the set of incident edges byδv=δvδ+v.

    The number |δv| then is called the degree of node vV.

    With each edge ejE of such a network, we associate a pipe model from the hierarchy in Section 2.1 and a given pipe length Lj>0. Furthermore, depending on the role of each node in the network, we impose appropriate coupling conditions for the gas density and flow at the boundary of pipes corresponding to edges being incident to that, see [13]. To this end, we define for vV and ejδv

    x(v,ej)={0, if ejδv,1, if ejδ+v.

    For each node vV, we then impose a transmission condition for the density and a balance equation for the fluxes at the node. The transmission condition states that the density variables ϱj weighted by given factors α(0,)m×2 coincide for all incident edges eδv and can be expressed as

    αkx(v,ek)ϱk(t,Lkx(v,ek))=αlx(v,el)ϱl(t,Llx(v,el)),    ek,elδv,t[0,T].

    The nodal balance equation for a given outflow function qv:[0,T]R is similar to a classical Kirchhoff condition for the fluxes qj and can be written as

    ejδ+vqj(t,Lj)ejδvqj(t,0)=qv(t),     t[0,T].

    The choice of α corresponds to the nodal types in the network. Prototypically, we will consider the following cases:

    Junctions: nodes v such that qv0 and αkx(v,ek)=1 for all ekδv.

    Boundary nodes: nodes v such that αkx(v,ek)=1 for all ekδv, but qv0.

    We also refer to v as an entry node, if qv<0, or an exit node, if qv>0.

    Compressors: nodes v with qv0 and |δ+v|=|δv|=1. A description established via the characteristic diagram based on measured specific changes in adiabatic enthalpy Had of the compression process yields the model

    Had=ˉZT0Rsκκ1((ϱl(0,t)ϱk(Lk,t))κ1κ1),     ekδ+v,elδv,t[0,T],

    where κ is a compressor specific constant, ˉZ is the gas compressibility factor that is assumed to be constant and Had is within flow dependent and compressor specific bounds obtained from the characteristic diagram. In consistency with the pipe models, we assume that Had is given by a known reference ˉHad. Then we get

    ϱl(0,t)=ˉαϱk(Lk,t),    ekδ+v,elδv,t[0,T]

    with a compressor specific factor

    ˉα=(1+(κ1)ˉHadκˉZT0Rs)κκ1. (5)

    This yields αk1=1 and αl0=ˉα. The compression ratio of centrifugal and turbo compressors mostly lies within the range ˉα[1,2], with ˉα1.3 being a typical value, see [13], [39,Chapter 4] and [12,Chapter 5.1]. For realistic long distance gas networks, compressors are set along the pipes in a distance of around 100-300 km, see [12,Chapter 5.3].

    We note that there are other conceivable ways to model gas pipe junctions, including geometry conditions, leading to possibly nonlinear coupling conditions, see e.g., [3,9,40]. Furthermore, gas networks typically involve additional network components such as valves, resistors, gas coolers, etc. For appropriate coupling conditions we refer to [44].

    We now discuss switching between the two models (2) and (3) in order to efficiently resolve the dynamics of the gas flow in a network. The idea is that, with the exception of locally high fluctuation, in realistic scenarios, the solution to (2) is on big parts of the network close to the stationary model (3). In these regions we thus can freeze the solution with an acceptable loss in accuracy to save computational effort. By comparison with the solution fully simulated with (2) we then can set up a cost functional measuring the deviation of the partially frozen solution in some appropriate norm. Adding a performance function measuring the time steps where the costly model (2) is calculated, we can set up the optimization problem of weighting the accuracy against the computational effort. Solving this problem enables us to identify a time-dependent model selection for the simulation of gas dynamics on networks that can be used in further examinations to get a cheaply solvable system.

    In order to apply adjoint-based gradient methods to solve this problem efficiently, we consider (3) as a limit of appropriately perturbed instationary semilinear dynamics. To this end, we note that, after a linear transformation, system (2) can be written for t[0,T] and x[0,L], L,T>0, as partially diagonalized system

    wt(t,x)+Dwx(t,x)=g(w(t,x)),   D=[c00c],   (t,x)(0,T)×(0,L),

    with the characteristic variable w, see Section 3 for details. Let us assume for the moment that g:R2R2 is globally Lipschitz-continuous. The same transformation applied to (3), together with given boundary values w1C([0,T],R2) yields the ODE-system

    Dwx(t,x)=g(w(t,x)),(t,x)(0,T)×(0,L),w1(t,1)=w11(t),t[0,T],w2(t,0)=w12(t),t[0,T]. (6)

    To define an appropriate hyperbolic system to approximate (6), for ε>0 and any initial values w0L1([0,L],R2) we set up the auxiliary system

    εwt(t,x)+Dwx(t,x)=εD1g(w(t,x)),(t,x)(0,T)×(0,L),w(0,x)=w0(x),x(0,L),w1(t,1)=w11(t),t[0,T],w2(t,0)=w12(t),t[0,T]. (7)

    Referring to [8,Chapter 3.4] for the definition of a broad solution to system (7), we get the following result:

    Lemma 2.1. For any ε>0 there is a unique solution ˉwC([0,T]×[0,L],R2) to (6) and a unique broad solution w:[0,T]×[0,L]R2 to (7). The restriction of w to the set

    Ω={(t,x)[0,T]×[0,L]|Lcεt<x<cεt}

    is a continuous function. Furthermore, for any t0(0,T) the function w converges uniformly on [t0,T]×[0,L] for ε0 to ˉw, i.e.

    limε0sup(t,x)[t0,T]×[0,L]|w(t,x)ˉw(t,x)|=0.

    Proof. The global existence and uniqueness of ˉw follows from standard ODE-theory. For (t,x)Ω, the broad solution w can be written as

    w1(t,x)=w11(tεc(Lx))+ttεc(Lx)1cg1(w(s,x+cε(ts)))ds,w2(t,x)=w12(tεcx)+ttεcx1cg2(w(s,cε(ts)))dy.

    Its existence and the continuity on Ω is shown in [8,Theorem 3.1,Theorem 3.4]. Given a fixed t0(0,T), we obviously can choose ε small enough such that [t0,T]×[0,L]Ω. We then have

    w2(t,x)=w12(tεcx)+ttεcx1cg2(w(s,cε(ts)))dyw12(t)+x01cg2(w(s,y))dy=ˉw2(t,x)

    for ε0 and all (t,x)[t0,T]×[0,L]. The convergence is uniform, since both w12 and g2(w) are continuous and defined on a compact set, thus uniformly continuous. The same argumentation holds for w1.

    Retransforming (7) back to the original variables ϱ and q, we get

    ϱt+1εqx=θc2q|q|ϱghc2ϱ,qt+c2εϱx=0. (8)

    By Lemma 2.1, this system yields an approximation to model (3).

    Now let G=(V,E) be a network, where the type of each node vV is given by the parameters α and qv as in Section 2.2. On each edge ej of length Lj we have an initial gas density ϱj0 and gas flow qj0. For an increasing sequence of switching times τ=(τk)k=0,,N+1[0,) and a finite sequence μ=(μk)k=1,,N, where μk{0,1}n denotes for each edge ejE, whether model (2) (μk(j)=1) or (8) (μk(j)=0) is used on the time interval (τk1,τk), consider the PDE-system

    [tϱj(t,x)tqj(t,x)]+1εμk(j)[01c20][xϱj(t,x)xqj(t,x)]=fμk(j)([ϱj(t,x)qj(t,x)]) (9)

    where x[0,Lj] for j=1,,n and t(τk1,τk) for k=1,,N+1. Here,

    εμk(j)={ˉε, if μk(j)=0,1, if μk(j)=1

    for a fixed ˉε>0 with ˉε1 and

    f0(ϱ,q)=[θc2q|q|ϱghc2ϱ0]   and   f1(ϱ,q)=[0θq|q|ϱghϱ] (10)

    denote the right-hand sides in (8) and (2), respectively. Let ϱj(0,x)=ϱj0(x), qj(0,x)=qj0(x) for x[0,Lj] be given initial states. Moreover, at any time point t[τ0,τN+1], density and flux additionally satisfy the node coupling conditions

    αix(v,ei)ϱi(t,Lix(v,ei))=αjx(v,ej)ϱj(t,Ljx(v,ej)),    ei,ejδv,ejδ+vqj(t,Lj)ejδvqj(t,0)=qv(t) (11)

    for each vV, explained in Section 2.2.

    Denote by zd=(ϱ1d,q1d,,ϱnd,qnd) the reference solution to (9), (11) for the choice N=1, μ=1 and τ=(0,T), which corresponds to the fine model (2) being fully solved on the complete network and the existence of which will be proven in Section 3. For any other z=(ϱ1,q1,,ϱn,qn) then define the cost functional

    J(μ,τ,z)=nj=1T0Lj0γ1(ϱj(t,x)ϱjd(t,x))2+γ2(qj(t,x)qjd(t,x))2dxdt=+γ3Nk=1nj=11Ljτk+1τk(εμk(j)ˉε)2dt,=+γ4N (12)

    with γ1,,γ40, where the first term measures the deviation of z from zd, the second term penalizes using the fine model (2) and the third term penalizes the number of switching time points. Note that, since longer pipes mean more computational effort when using the fine model, the lengths Lj of the pipes enter into the cost as well. For later reference, we set

    l(t,z)=nj=1Lj0γ1(ϱj(t,x)ϱjd(t,x))2+γ2(qj(t,x)qjd(t,x))2dx,J1=12T0l(t,z)dt,J2=γ3Nk=1nj=11Ljτk+1τk(εμk(j)ˉε)2dt+γ4N, (13)

    then J=J1+J2.

    The challenge now is to choose the sequences μ and τ such that, with z being the corresponding solution to (9), (11), the cost functional J is minimized. Hence our objective is to solve the minimization problem

    min(μ,τ)   J(μ,τ,z)s.t.   z solves (9),(11) for the switching sequence (μ,τ). (14)

    In this section, we will set up an abstract formulation of the PDE-system in (9) for τ=(0,T) and any fixed choice of modes per edge and prove that the linear part generates a C0-semigroup. By nonlinear perturbation theory and induction, this yields a well-posedness result of (9) for any finite switching sequence.

    For each j{1,,n}, we now consider the initial boundary value problem for zj=(ϱj,qj) on pipe ejE given by

    zjt(t,x)+Ajzjx(t,x)=fj(zj(t,x)),t[0,T],x[0,Lj],zj(0,x)=zj0(x),x[0,Lj], (15)

    and the coupling conditions

    αkx(v,ek)zk1(t,Lkx(v,ek))=αlx(v,el)zl1(t,Llx(v,el))    ek,elδv, (16)
    ejδ+vzj2(t,Lj)ejδvzj2(t,0)=qv0(t) (17)

    for each node vV. Here, Lj>0, zj0:[0,Lj]R2, fjC1(R2,R2) is globally Lipschitz-continuous and

    Aj=1εj[01c2j0]      for some εj,cj>0

    for each j{1,,n}. Moreover, α(0,)m×2 and qv0:[0,)R is a given outflow for each vV. Introduce the space

    Z=[nj=1L2([0,Lj],R2)]L2([0,),Rm),

    the vectors

    z=((z1),,(zn),qv1,,qvm),z0=((z10),,(zn0),qv10,,qvm0)

    the operators

    A=[A1An]x,     B=1mx     andf(z)=(f1(z1),,fn(zn),0,,0). (18)

    Moreover, define the operator

    diag(A,B)=[A00B]

    on the domain

    D([A00B])={z=((z1),,(zn),qv1,,qvm)Z|z is absolutely continuous,αkx(v,ek)zk1(Lkx(v,ek))=αlx(v,el)zl1(Llx(v,el))    vV,ek,elδv,ejδ+vzj2(Lj)ejδvzj2(0)=qv(0)    vV} (19)

    of all vectors of absolutely continuous functions that satisfy the boundary conditions (16) and (17). Note that, though the inflow functions qv is only evaluated at the origin, it also is shifted in time due to the transport-type evolution represented by operator B - this together enforces the originally time-dependent coupling condition (17). We then can reformulate system (15)-(17) as the abstract initial-value problem

    ˙z(t)+[A00B]z(t)=f(z),   t[0,T],z(0)=z0. (20)

    Now we can state the following result:

    Theorem 3.1. The operator

    (D([A00B]),[A00B])

    is the infinitesimal generator of a strongly continuous semigroup on Z.

    Proof. Note that diag(A,B) is a densely defined, closed operator on Z with a nonempty resolvent set (for instance 0ρ(diag(A,B))). Following [43,Chapter 4,Theorem 1.3], to prove the claimed semigroup property, it thus suffices to show that the homogeneous system

    ˙z(t)+[A00B]z(t)=0,    t[0,T],z(0)=z0. (21)

    has a unique solution zC1([0,T],Z) for any T>0 and every choice z0D(diag(A,B)). So let

    z0=(ϱ10,q10,,ϱn0,qn0,qv10,,qvm0)D(diag(A,B))

    be given and first assume TˉT:=min{εjLj2cj|j=1,,n}>0. We recognise in the operator B a transport equation with velocity 1 for the variables qv1,,qvm with the well-known unique solution

    qvk(t,s)=qvk0(s+t).

    Next, for each note vV we can set

    αv=(ejδ+vαj1cj+ejδvαj0cj)>0

    and

    ϱv(t)=α1v[ejδ+v(cjϱj0(Ljcjεjt)+qj0(Ljcjεjt))+ejδv(cjϱj0(cjεjt)qj0(cjεjt))+qv(t,0)].

    Since ϱ1,q1,,ϱn,qn,qv1,,qvm are absolutely continuous, so is ϱv for each vV. For each edge ejE, j=1,,n, construct the functions ϱj,qj:[0,T]×[0,1]R as follows: for t(0,T] set

    ϱj(t,0)=αj0ϱej(1)(t),qj(t,0)=αj0cjϱej(1)(t)+cjϱj0(cjεjt)+qj0(cjεjt),ϱj(t,Lj)=αj1ϱej(2)(t),qj(t,Lj)=αj1cjϱej(2)(t)cjϱj0(Ljcjεjt)+qj0(Ljcjεjt).

    Substituting the coupling conditions stated in (19), one can indeed show that, with these definitions,

    limt0ϱj(t,x)=ϱj0(x)      and      limt0qj(t,x)=qj0(x)       for x{0,Lj}.

    We skip these calculations here for brevity. Next, set

    [ϱjqj](t,x)=12[1εjcjcjεj1][ϱjqj](cjεjt,x)+12[1εjcjcjεj1][ϱjqj](cjεjt,x)

    for (t,x)(0,T]×(0,Lj), where

    [ϱjqj](t,x)={[ϱj0qj0](xt), if xt[0,Lj],[ϱjqj](εj(tx)cj,0), if xt<0,[ϱjqj](εj(t(Ljx))cj,Lj), if xt>Lj.

    The above considerations show that (ϱj,qj) is continuous and piecewise absolutely continuous, thus absolutely continuous everywhere. To see that these functions in fact solve system (21), first note that the initial condition and the coupling condition (16) are satisfied by construction. Moreover, for each vV we have

    ejδ+vqj(t,Lj)ejδvqj(t,0)=ejδ+v[αj1cjϱej(2)(t)cjϱj0(Ljcjεjt)+qj0(Ljcjεjt)]=ejδv[αj0cjϱej(1)(t)+cjϱj0(cjεjt)+qj0(cjεjt)]=(ejδ+vαj1cj+ejδvαj0cj)ϱv(t)=ejδ+v[cjϱj0(Ljcjεjt)qj0(Ljcjεjt)]ejδv[cjϱj0(cjεjt)+qj0(cjεjt)]=αvϱv(t)(αvϱv(t)qv(t))=qv(t)

    Observe that ϱj and qj are continuous at the boundaries x=0 and x=Lj and that

    Ajx[ϱjqj](t,x)=12Aj[1εjcjcjεj1]x[ϱjqj](cjεjt,x)+12Aj[1εjcjcjεj1]x[ϱjqj](cjεjt,x)=εj2cj[cjεj1cj2ε2jcjεj]t[ϱjqj](cjεjt,x)+εj2cj[cjεj1cj2ε2jcjεj]t[ϱjqj](cjεjt,x)=t[ϱjqj](t,x)

    for a.e. (t,x)(0,T]×(0,Lj), thus the above construction indeed yields a solution to (21) for TˉT with z(t)D for t[0,T]. For the case T>ˉT note that the same steps can be repeated successively to expand the solution for the times [ˉT,2ˉT],[2ˉT,3ˉT] and so on. To prove uniqueness of the solution, consider the case z0=0 of initial data and introduce the energy

    E(t)=12ejELj0(ϱj(t,x))2+1c2j(qj(t,x))2dx.

    Obviously, we have E(t)=0 if and only if ϱj(t,.)=qj(t,.)=0 for all j=1,,n, E(t)0 for all t0 and E(0)=0. Furthermore,

    ddtE(t)=ejELj0ϱj(t,x)tϱj(t,x)+1c2jqj(t,x)tqj(t,x)dx=ejE1εjLj0ϱj(t,x)xqj(t,x)+qj(t,x)xϱj(t,x)dxP.I.=ejE1εj[qj(t,Lj)ϱj(t,Lj)qj(t,0)ϱj(t,0)]=vV[ejδ+v1εjqj(t,Lj)ϱj(t,Lj)ejδv1εjqj(t,0)ϱj(t,0)]=vVϱv(t)[ejδ+vαj1εjqj(t,Lj)ejδvαj0εjqj(t,0)]=0,

    since ϱv0 for all vV. But then E0, hence z0. This concludes the proof for the semigroup property.

    We now can apply standard arguments from semigroup theory for further discussion of (20): by [43,Chapter 6,Theorem 1.2 and 1.5], if the inhomogeneity f is continuously differentiable and globally Lipschitz-continuous, (20) has a unique classical solution z. Though these assumptions do not apply to the friction term f stated in (2), we can use the following technical consideration to still get a unique solution: choose any ˉϱ,ˉq>0 and define the modified right-hand side

    ˜f(z)=[0θmin{z2|z2|,ˉq}max{z1,ˉϱ}ghz1].

    The function ˜f is continuously differentiable and globally Lipschitz-continuous as a function mapping L2([0,L],R) onto itself for any L>0. If we replace f by the ˜f in (9), we then have a system fitting the assumptions made above for each fixed k=1,,N and thus we have a unique solution. If we choose ˉϱ sufficiently small and ˉq sufficiently large, then the numerical simulation with realistic data shows that both the boundaries ˉϱ and ˉq will in fact never be reached by ϱ and q, respectively. In this case, however, the solution coincides with the unique solution to the original problem (9). The same argumentation holds for the right-hand side in (8). In summary, this proves the necessary results on well-posedness of the system (20) that we need to examine the optimization problem (14).

    By the results developed in Section 3, we find that the optimization problem (14) is of the form

    minμ,τJ(μ,τ,z)s.t.˙z(t)=Aμkz(t)+fμk(t,z(t)),k{1,,N},t(τk1,τk),z(τk)=gμk,μk+1(z(τk)),k{1,,N},z(τ0)=z0,τT(0,T). (22)

    where Aμk is a strongly continuous semigroup, fμk is a semilinear perturbation and gμk,μk+1=id is a transition map that can be chosen trivially here for each k=1,,N. The ordering cone T(0,T) is for fixed T>0 defined by

    T(0,T)={τ=(τ1,,τN)RN0=τ0τ1τNτN+1=T}.

    In general, problem (22) does not have a unique global minimum. Instead, we will define a concept of stationarity fitted to the hybrid nature of the problem and set up an algorithm capable of finding such stationary points.

    First, however, we apply on system (22) some results from the preliminary work in [45]: by Theorem 3.1 and the subsequent remarks, [45,Lemma 2] can be applied, yielding a control-to-state-map (μ,τ)z(μ,τ) that can be substituted into J to define the reduced cost functional Φ(μ,τ)=J(μ,τ,z(μ,τ)). We can now apply [45,Theorem 8] to show that Φ is continuously differentiable with respect to the switching time τk. Recalling the shortened notation introduced in (13), we find that, while J2 can be differentiated directly with respect to (μ,τ), the term J1 depends on the solution z=z(μ,τ). Again by [45,Theorem 8], we can state gradient formulae for the derivatives of J1 based on the adjoint equation

    [pt(t)γt(t)]+[Aμk00B][p(t)γ(t)]=[[fμkz(z(t))]p(t)+lz(z(t))0],t(τk1,τk),[p(T)γ(T)]=0,k=1,,N+1. (23)

    Here, fμz:ZL(Z) and lz:ZZ denote the derivatives of the friction term fμ and l as in (13) with respect to z. Substituting the definitions made in (18) yields that in fact γ0 and p=(pj)j=1,,n again can be partitioned into edgewise defined functions pj=(pj1,pj2) for j=1,,n satisfying

    [pj1pj2]t+1εμk(j)[0c2j10][pj1pj2]x=(fμk(j)z(zj))[pj1pj2]+γ1[ϱjϱjdqjqjd][pj1pj2](T,x)=0, (24)

    where

    f0z(ϱ,q)=[θc2qj|qj|ϱj|ϱj|gh2θc2|qj|ϱj00]     and     f1z(ϱ,q)=[00θqj|qj|ϱj|ϱj|gh2θ|qj|ϱj]

    are the derivatives of f0 and f1 in (10), as well as the coupling conditions

    αjx(v,ek)pj1(t,Lkx(v,ek))=αkx(v,el)pk1(t,Llx(v,el)),      ek,elδv, (25)
    ejδ+vpj2(t,Lj)ejδvpj2(t,0)=0. (26)

    By [45,Lemma 6], system (23) (and thus (24), (25), (26) as well) have a unique mild solution and, applying [45,Theorem 8], the switching time gradient is given by

    Φτk=ejE[Lj0pj(τk,x)[(Aμk)j(Aμk1)j]zj(τk,x)dx+γ3[(εμk(j)ˉε)2(εμk1(j)ˉε)2]]. (27)

    If the mode sequence μ is fixed, then we can conclude that a switching sequence τ=(τk)k=0,,N+1 is a KKT-point of the minimization problem (14), if the following holds: For n{1,,N} set a(τ,n)=min{m{0,,n}|τm=τn} and b(τ,n)=max{m{n,,N+1}|τm=τn}, then

    nj=a(τ,n)Φτj(τ)0     and     b(τ,n)j=nΦτj(τ)0. (28)

    Similarly, by [45,Theorem 10], the sensitivity of the cost function with respect to introducing a new mode μ on an infinitesimal time interval at the point τ can be represented by the mode insertion gradient given by

    Φμ(τ)=ejE[Lj0pj(τ,x)[(Aμk)j(Aμ)j]zj(τ,x)dx+γ3[(εμk(j)ˉε)2(εμˉε)2]], (29)

    where μk is the original mode at time τ. In summary, a switching sequence (μ,τ) is called stationary, if τ is a KKT-point for μ fixed and if Φμ(τ)0 for all modes μ and all times τ[0,T]. Any global minimum of the problem (14) then is stationary.

    In order to compute such stationary switching signals, we consider a conceptual algorithm originally proposed for optimal mode scheduling in hybrid ODE-dynamical systems [2]. The main idea is a two phase approach as follows: the algorithm is initialized with a switching sequence (μ0,τ0), for instance μ0=1 and τ0=(0,T) where no switching occurs and the system is solved by keeping the mode constant at 1.

    In a first phase, the positions of available switching time points are optimized, while conserving their order, by using a projected-gradient method with Armijo step size. To this end a projection P onto the ordering cone T(0,T) is used. In a second phase, a new mode μ is inserted at a time point τ where Φμ(τ)<0. If no such point exists, the switching sequence is stationary in the above sense, otherwise the algorithm continues with the first phase again.

    A more precise description of the procedure is given in Algorithm 1. Note that Algorithm 1 does not necessarily terminate with the global solution to the minimization problem (22), but with a stationary point in the above sense. A convergence analysis for this algorithm can be found in [2]. A possible implementation of the projection P can be found in [19].

    Algorithm 1 Two-phase gradient descent for stationary switching sequences
    Require: Initial switching sequence (μ0,τ0) with N modes, Armijo-parameters β(0,1) and γ(0,1)
    1: Set k=0, solve (20) for z and (24) for p.
    2: Calculate the switching time gradient Φτk=(Φτkn)n=1,,N1 in (27).
    3: while τk does not satisfy (28) do
    4:     Find a step size sk=max{βl|l=0,1,2,} such that
         Φ(P(τkskΦτk))Φ(τk)γΦτk[τkP(τkskΦτk)]
    5:     Set τkP(τkskΦτk).
    6:     Solve (9) for z and (24) for p.
    7:     Calculate the switching time gradient Φτk=(Φτkn)n=1,,N1 in (27).
    8: end while
    9: if the mode insertion gradient Φμ(τ)0 in (29)   μ{0,1},τ[0,T] then
    10:    return (μ,τ)=(μk,τk)
    11: end if
    12: Find mode μ and τ[0,T] with Φμ(τ)<0 in (29).
    13: Find n{0,,N} such that τ[˜τkn,˜τkn+1).
    14: Set μk+1(μk1,,μkn,μ,μkn+1,,μkN)
    15: Set τk+1(τk1,,τkn,τ,τ,τkn+1,,τkN)
    16: Set kk+1, NN+1 and go to 2.

     | Show Table
    DownLoad: CSV

    As a proof of concept we consider a gas network outlined in Figure 1b. At the boundary node N1 gas is supplied to the network whereas nodes N2 and N3 are customer nodes where gas can possibly be taken out of the network. This can be seen as a simplified example of a big regional gas pipeline network with a local distribution subnetwork. In the particular scenario we are looking at, there is gas transported from the supply N1 to node N2 to satisfy a given demand while N3 is inactive. All pipes are assumed to be horizontal (h=0), the compressor is assumed to be running at a constant compression factor ˉα, compare (5), and at initial time the network is assumed to be stationary with constant densities ϱ0 on the outer circle (thus αϱ0 on the inner circle) and flux q0 everywhere, moreover the outflows at the nodes N1 and N2 are outlined in Figure 1c. See the table in Figure 1a for specific values for those and other constants.

    Figure 1.  A gas network with a supply node N1 and two costumer nodes N2 and N3.

    Due to the almost decoupled inner and outer circle, it can be expected that the numerical solution highly varies on the outer circle connecting N1 and N2 but is near to constant on the inner circle. This is confirmed by the simulation of the full model, see Figure 2 for a snapshot of the simulation at the time t=900 s. We therefore can suspect that the simulation on the inner circle can be widely frozen with only some small losses in the accuracy of the solution. Indeed, letting ˉz be the distinguished solution to (9), (11) resulting from freezing the solution completely on the inner edges 6 to 10, our simulations show that the L2-errors of the density and the flux relative to the respective maximum values in zd is less than 1% for ϱ and less than 6.5% for q; see Figure 3d. Moreover, compared to a full simulation zd about half of the computation time in the sense of J2 defined in (13) is saved.

    Figure 2.  Snapshot of the fully simulated solution showing density (solid, blue) and flux (dashed, red, scaled by 0.05). On the outer pipes 1 to 5 we see a lot of fluctuation due to the oscillatory boundary flows. The pipes 6 to 9 of the inner circle, however, remain nearly constant.
    Figure 3.  (A): resulting optimized switching sequence showing, for each time step from t0=0 s to T=1800 s and each edge e1,,e10, if the solution is calculated with the fine model (white) or frozen (black). (B), (C): filtered results with two different filters. (D): L2-error relative to maximum values of the solution ˉz corresponding to freezing edges 6 to 10 completely.

    In our implementation the system (9), (11) is solved via splitting of the hyperbolic part and the friction term. For the simulation of the hyperbolic part we use the 2-step-Richtmyer-method with artificial viscosity, see [32,Chapter 18.1]. Given a system matrix A and a discretization zk=(zk1,,zkn) with spatial step size Δx of the solution at time point tk, it computes the discretized values zk+1 at time tk+1=tk+Δt by the explicit finite-volume-scheme

    zk+12j+12=12(zkj+1+zkj)Δt2ΔxA(zkj+1zkj),zk+1j=zkjΔtΔxA(zk+12j+12zk+12j12)+ε(zkj+12zkj+zkj1)

    for j=1,,n, where ε0.05 introduces an additional, minor but stabilizing smoothing to the solution. The discretization is chosen in a way such that the cells zk1 and zkn are centered at the boundary points x=0 and x=1, respectively. In order to handle the boundary values and the inner cells simultaneously with the same scheme, we add appropriate ghost cells on each end of the spatial domain. The appropriate ghost cell values (ze)k0,(ze)kn+1 for each eE are derived from the coupling conditions. We mention without further proof that these can be solved explicitly for each node vV by first setting the weighted mean value

    zv=2|δv|(eδ+v(ze)kn1+[1001]eδv(ze)k2)

    and then using the ghost cell values

    (ze)kn+1=[1001](zv(ze)kn1)         for all eδ+v,(ze)k0=zv[1001](ze)k2         for all eδv.

    Here, in each time step, we incorporate only those edges where the solution is actively calculated. In the special case if vV is a compressor, we only have one ingoing edge e+δ+v and one outgoing edge eδv and instead have to set

    zv=21+ˉα((ze+)kn1+[1001](ze)k2),(ze+)kn+1=[1001](zv(ze+)kn1),(ze)k0=[ˉα001]zv[1001](ze)k2.

    For the friction term, we add an explicit Runge-Kutta-step using the classical Runge-Kutta-scheme of order 2 or midpoint rule, see [10,Chapter 23]. The same methods are used on the same discretization grid backwards in time to solve the adjoint system stated in (24), (25), (26).

    To evaluate the gradient formulae (27) and (29), we use the trapezoidal rule over the spatial grid, see [14,Chapter 2.1]. The expression [(Aμk)j(Aμk1)j]zj(τk,x) occurring in both (27) and (29) represents the difference of the time derivatives of the solution depending on which mode μk is switched to. In the numerical scheme, this is realized by calculating a time step of the solution for each choice of μk and then substracting the forward difference quotients.

    The fixed temporal discretization grid for the actual solution is supplemented by a grid of switching time points that may vary due to the superordinated optimization where the data needed for the gradient formulae is calculated. For our study we start the optimization with the fully frozen solution, where in no time step the model (2) is actually calculated, and iteratively insert regions of active numerical solving wherever the gradients indicate a major loss of accuracy. Applying the projected-gradient method with sequential mode insertion described in Algorithm 1 yields the results shown in Figure 3a. We observe that it is in fact almost unnecessary to calculate the fine model on the edges 6 to 10 of the inner circle and that the algorithm indeed approximates the distinguished solution ˉz as expected. Note that Algorithm 1 does not remove switching points during its process, because doing so might lead to the iteration being caught in an infinite loop, where the same switching points are added and removed repeatedly. This way, however, there remain scattered short intervals where the mode is switched twice almost instantly, which can be interpreted as the algorithm eliminating the respective intermediate modes. We suggest a post-processing step to remove this scattering, for instance by applying the following filtering rule with dN a fixed parameter: if within d time steps after a switching point in the numerical solution the mode is switched again, then both switching time points are removed. Relative to the initial cost, we get for the optimized solution a cost reduction of 96.4% with 65 switching points, for d=5 shown in Figure 3b a cost reduction of 96.3% with 19 switching points and for d=10 shown in Figure 3c a cost reduction of 95.1% with 7 switching points.

    We have presented an application of the theory of switching systems to a model hierarchy for the dynamics of gas in a pipeline network. A semigroup formulation was given for the model on gas networks including time-dependent outflow at each node as well as a linearized model for compressors that allowed us to prove the existence of unique solutions. Using adjoint-based gradient representations for switching abstract evolution systems, we implemented a two stage projected-gradient descent method for the optimization of switching between different levels of accuracy in the hierarchy. As an example, we optimized the simulation of a small supply network by freezing the calculation on edges whenever the numerical error made is small compared to the computational costs.

    Our prototypical approach can be applied in a similar fashion to realistic industrial networks. The technique can also be extended to identify a model switching strategy for a reduced model using a range of different parameter configurations such as representing different boundary flow scenarios, compressor settings and valve positions. The resulting reduced simulation model can then be used for a time expanded mixed-integer optimization technique based on full discretization with a minimum of variables or within a bilevel optimisation method which optimises a cost functional on the outer level with optimal efficiency on the lower level as proposed, for example, in [24]. Further directions include a combination of this method in a network of submodel hierarchies such as those used in [37] for the optimisation of stationary models. Moreover, it remains an open question, whether the chosen numerical method lead to a consistent discretization of the optimality system. Well-balanced finite-volume schemes also seem to be an promising alternative approach in this context.

    The authors thank the Deutsche Forschungsgemeinschaft for their support within the projects A03 and B03 in the Sonderforschungsbereich/-Transregio 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks. Furthermore, the authors are indebted to the anonymous reviewers who considerably improved the exposition of this paper.



    [1] M. Xiang, D. Yang, B. Zhang, Degenerate Kirchhoff-type fractional diffusion problem with logarithmic nonlinearity, Asymptotic Anal., 118 (2020), 313-329. doi: 10.3233/ASY-191564
    [2] M. Xiang, D. Hu, D. Yang, Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity, Nonlinear Anal., 198 (2020), 1-20.
    [3] H. Ding, J. Zhou, Global existence and blow-Up for a parabolic problem of Kirchhoff type with Logarithmic nonlinearity, Appl. Math. Optim., 8 (2019), 1-57.
    [4] O. C. Alves, T. Boudjeriou, Existence of solution for a class of nonlocal problem via dynamical methods, Nonlinear Anal., 3 (2020), 1-17.
    [5] M. Xiang, B. Zhang, H. Qiu, Existence of solutions for a critical fractional Kirchhoff type problem in N, Sci. China Math., 60 (2017), 1647-1660. doi: 10.1007/s11425-015-0792-2
    [6] D. Lu, S. Peng, Existence and concentration of solutions for singularly perturbed doubly nonlocal elliptic equations, Commun. Contemp. Math., 22 (2020), 1-37.
    [7] W. Qing, Z. Zhang, The blow-up solutions for a Kirchhoff-type wave equation with different-sign nonlinear source terms in high energy Level, Per. Oc. Univ. China., 48 (2018), 232-236.
    [8] H. Guo, Y. Zhang, H. Zhou, Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential, Commun. Pure Appl. Anal., 17 (2018), 1875-1897. doi: 10.3934/cpaa.2018089
    [9] Q. Lin, X. Tian, R. Xu, M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst., 13 (2020), 2095-2107.
    [10] M. Xiang, V. D. Dulescu, B. Zhang, Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity, 31 (2018), 3228-3250. doi: 10.1088/1361-6544/aaba35
    [11] J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265(2015), 807-818.
    [12] E. Piskin, F. Ekinci, General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms, Math. Methods Appl. Sci., 11 (2019), 5468-5488.
    [13] Y. Yang, J. Li, T. Yu, Qualitative analysis of solutions for a class of Kirchhoff equation with linear strong damping term nonlinear weak damping term and power-type logarithmic source term, Appl. Numer. Math., 141 (2019), 263-285. doi: 10.1016/j.apnum.2019.01.002
    [14] N. Boumaza, G. Billel, General decay and blowup of solutions for a degenerate viscoelastic equation of Kirchhoff type with source term, J. Math. Anal. Appl., 489 (2020), 1-18.
    [15] X. Shao, Global existence and blowup for a Kirchhoff-type hyperbolic problem with logarithmic nonlinearity, Appl. Math. Optim., 7 (2020), 1-38.
    [16] Y. Yang, X. Tian, M. Zhang, J. Shen, Blowup of solutions to degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian, Electron. J. Differ. Equ., 155 (2018), 1-22.
    [17] R. Jiang, J. Zhou, Blowup and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian, Commun. Pure Appl. Anal., 18 (2019), 1205-1226. doi: 10.3934/cpaa.2019058
    [18] M. Xiang, P. Pucci, Multiple solutions for nonhomogeneous Schrodinger-Kirchhoff type equations involving the fractional p-Laplacian in R-N, Calc. Var. Partial Differ. Equ., 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5
    [19] D. Idczak, Sensitivity of a nonlinear ordinary BVP with fractional Dirichlet-Laplace operator, Available from: arXivpreprintarXiv, 2018, 1812.11515.
    [20] M. Xiang, D. Yang, Nonlocal Kirchhoff problems: Extinction and non-extinction of solutions, J. Math. Anal. Appl., 477 (2019), 133-152. doi: 10.1016/j.jmaa.2019.04.020
    [21] N. Pan, B. Zhang, J. Cao, Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian, Nonlinear Anal. Real World Appl., 37 (2017), 56-70. doi: 10.1016/j.nonrwa.2017.02.004
    [22] P. Dimitri, On the Evolutionary Fractional p-Laplacian, Appl. Math. Res. Exp., 2 (2015), 235-273.
    [23] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105
    [24] A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. Theory Methods Appl., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011
    [25] D. Lu, S. Peng, On nonlinear fractional Schrodinger equations with Hartree-type nonlinearity, Appl. Anal., 97 (2018), 255-278. doi: 10.1080/00036811.2016.1260708
    [26] L. Chen, W. Liu, Existence of solutions to boundary value problem with p-Laplace operator for a coupled system of nonlinear fractional differential equations, J. Hubei Univ. Natural Sci., 35 (2013), 48-51.
    [27] G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol. Nat., 16 (2017), 288-297.
    [28] T. Boudjeriou, Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 17 (2020), 162-186. doi: 10.1007/s00009-020-01584-6
    [29] J. Liu, J. Liao, H. Pan, Multiple positive solutions for a Kirchhoff type equation involving two potentials, Math. Methods Appl., 43 (2020), 10346-10356. doi: 10.1002/mma.6702
    [30] L. X. Truong, The Nehari manifold for fractional p Laplacian equation with logarithmic nonlinearity on whole space, Comput. Math. Appl., 78 (2019), 3931-3940. doi: 10.1016/j.camwa.2019.06.024
    [31] A. Ardila, H. Alex, Existence and stability of standing waves for nonlinear fractional Schrodinger equation with logarithmic nonlinearity. Nonlinear Anal., 155 (2017), 52-64. doi: 10.1016/j.na.2017.01.006
    [32] X. Chang, Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494. doi: 10.1088/0951-7715/26/2/479
    [33] E. D. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004
  • This article has been cited by:

    1. Kota Kumazaki, Adrian Muntean, A two-scale model describing swelling in porous materials with elongated internal structures, 2025, 0033-569X, 10.1090/qam/1705
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2340) PDF downloads(95) Cited by(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog