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A hybridized weak Galerkin finite element scheme for general second-order elliptic problems

  • Received: 01 February 2020 Revised: 01 March 2020
  • Primary: 65N15, 65N30; Secondary: 35J50

  • In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

    Citation: Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems[J]. Electronic Research Archive, 2020, 28(2): 821-836. doi: 10.3934/era.2020042

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  • In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.



    1. Introduction

    The Caginalp phase-field system

    utΔu+f(u)=θ, (1.1)
    θtΔθ=ut, (1.2)

    has been introduced in [1] in order to describe the phase transition phenomena in certain class of material. In this context, θ denotes the relative temperature (relative to the equilibrium melting temperature), and u is the phase-field or order parameter, f is a given function (precisely, the derivaritve of a double-well potential F). This system has received much attention (see for example, [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [22], [29], [33] and [41]). These equations can be derived by introducing the (total Ginzburg-Landau) free energy:

    ψ=Ω(12|u|2+F(u)uθ12θ2)dx, (1.3)

    where Ω is the domain occupied by the system (here, we assume that it is a bounded and smooth domain of Rn, n=1,2 or 3, with boundary Ω), and the enthalpy

    H=u+θ. (1.4)

    Then, the evolution equation for the order parameter u is given by:

    ut=δuψ, (1.5)

    where δu stands for the variational derivative with respect to u, which yields (1.1). Then, we have the energy equation

    Ht=divq, (1.6)

    where q is the heat flux. Assuming finally the classical Fourier law for heat conduction, which prescribes the heat flux as

    q=θ, (1.7)

    we obtain (1.2). Now, a well-known side effect of the Fourier heat law is the infinite speed of propagation of thermal disturbances, deemed physically unreasonable and thus called paradox of heat conduction (see, for example, [9]). In order to account for more realistic features, several variations of (1.7), based, for example, on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed in the context of the Caginalp phase-field system (see, for example, [19], [20], [21], [23], [24], [25], [26], [27], [28], [30], [31], [35], [36], [37], [38], [44], [45] and [46]).

    A different approach to heat conduction was proposed in the Sixties (see, [47], [48] and [49]), where it was observed that two temperatures are involved in the definition of the entropy: the conductive temperature θ, influencing the heat conduction contribution, and the thermodynamic temperature, appearing in the heat supply part. For time-independent models, it appears that these two temperatures coincide in absence of heat supply. Actually, they are different generally in the time depedent case see, for example, [19] and references therein for more discussion on the subject. In particular, this happens for non-simple materials. In that case, the two temperatures are related as follows (see [42], [43]):

    θ=φΔφ. (1.8)

    Our aim in this paper is to study a generalization of the Caginalp phase-field system based on this two temperatures theory and the usual Fourier law with a nonlinear coupling.

    The purpose of our study is the following initial and boundary value problem

    utΔu+f(u)=g(u)(φΔφ), (1.9)
    φtΔφtΔφ=g(u)ut, (1.10)
    u=φ=0onΩ, (1.11)
    u|t=0=u0, φ|t=0=φ0. (1.12)

    The paper is organized as follows. In Section 2, we give the derivation of the model. The Section 3 states existence, regularity and uniqueness results. In Section 4, we address the question of dissipativity properties of the system. The last section, analyzes the spatial behavior of solutions in a semi-infinite cylinder, assuming their existence.

    Thoughout this paper, the same letters c,c,c, and sometimes c denote constants which may change from line to line and also .p will denote the usual Lp norm and (.,.) the usual L2 scalar product. More generally, we will denote by .X the norm in the Banach space X. When there is no possible confusion, . will be noted instead of .2.


    2. Derivation of the model

    In our case, to obtain equations (1.9) and (1.10), the total free energy reads in terms of the conductive temperature θ,

    ψ(u,θ)=Ω(12|u|2+F(u)G(u)θ12θ2)dx, (2.1)

    where f=F and g=G, and (1.5) yields, in view of (1.8), the evolution equation for the order parameter (1.9). Furthermore, the enthalpy now reads

    H=G(u)+θ=G(u)+φΔφ, (2.2)

    which yields thanks to (1.6), the energy equation,

    φtΔφt+divq=g(u)ut. (2.3)

    Considering the usual Fourier law (q=φ), one has (1.10).

    Remark 2.1. We can note that we still have an infinite speed of propagation here.


    3. Existence and uniqueness of solutions

    Before stating the existence result, we make some assumptions on nonlinearities f and g:

    |G(s)|2c1F(s)+c2,c0,c1,c20, (3.1)
    |g(s)s|c3(|G(s)|2+1),c30, (3.2)
    c4sk+2c5F(s)f(s)s+c0c6sk+2c7,c4,c6>0,c5,c70, (3.3)
    |g(s)|c8(|s|+1),|g(s)|c9c8,c90, (3.4)
    |f(s)|c10(|s|k+1),c100, (3.5)

    where k is an integer.

    Theorem 3.1. We assume that (3.1)-(3.4) hold true. For all initial data (u0,φ0)H10(Ω)Lk+2(Ω)×H10(Ω)H2(Ω), the problem (1.9)-(1.12) possesses at least one solution (u,φ) with the following regularity uL(0,T;H10(Ω))Lk+2(Ω),utL2(0,T;L2(Ω)),φL(0,T;H10(Ω)H2(Ω)) and φtL2(0,T;H10(Ω)).

    Proof. The proof is based on the Galerkin scheme. Here, we just make formally computations to get a priori estimates, having in mind that these estimates can be rigourously justified using the Galerkin scheme see, for example, [10], [11] and [40] for details.

    Multiplying (1.9) by ut and integrating over Ω, we get

    12ddt(u2+2ΩF(u)dx)+ut2=Ωg(u)ut(φΔφ)dx. (3.6)

    Multiplying (1.10) by φΔφ and integrating over Ω, we have

    12ddt(φ2+2φ2+Δφ2)+φ2+Δφ2=Ωg(u)ut(φΔφ)dx. (3.7)

    Now, summing (3.6) and (3.7), we are led to,

    ddt(u2+2ΩF(u)dx+φ2+2φ2+Δφ2)+2(ut2+φ2+Δφ2)=0. (3.8)

    Multiplying (1.9) by u and integrating over Ω, we obtain

    12ddtu2+u2+Ωf(u)udx=Ωg(u)u(φΔφ)dx. (3.9)

    Using (3.2)-(3.3), (3.9) becomes

    12ddtu2+u2+cΩF(u)dxcΩ|G(u)|2dx+12(φ2+Δφ2)+c. (3.10)

    Adding (3.8) and (3.10), one has

    dE1dt+2(u2+cΩF(u)dx+ut2+φ2)+Δφ2cΩ|G(u)|2dx+φ2+c, (3.11)

    where

    E1=u2+u2+2ΩF(u)dx+φ2+2φ2+Δφ2 (3.12)

    enjoys

    E1c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c (3.13)

    and

    E1c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c. (3.14)

    Multiplying now (1.10) by φt and integrating over Ω, we have

    12ddtφ2+φt2+φt2=Ωg(u)utφtdx. (3.15)

    Taking into account (3.4) and using Hölder's inequality, we get

    12ddtφ2+12φt2+φt2c(u2+1)ut2 (3.16)

    and then, summing (3.11) and (3.16), we have

    dE2dt+2(u2+cΩF(u)dx+ut2+φ2+12Δφ2+12φt2+φt2)cΩ|G(u)|2dx+φ2+c(u2+1)ut2+c, (3.17)

    where

    E2=E1+φ2 (3.18)

    satisfies similar estimates as E1.

    We deduce from (3.1) and (3.17)

    dE2dt+c(φt2+φt2)cE2+c, (3.19)

    which achieve the proof.

    For more regularity on solutions, we make following additional assumptions:

    f(0)=0andf(s)c,c0. (3.20)

    We have:

    Theorem 3.2. Under assumptions of Theorem 3.1 and assuming that (3.20) is satisfied. For every initial data (u0,φ0)H10(Ω)Lk+2(Ω)×H10(Ω)H2(Ω), the problem (1.9)-(1.12) admits at least one solution (u,φ) such that uL(0,T;H10(Ω))Lk+2(Ω),utL(0,T;L2(Ω))L2(0,T;H10(Ω)), φL(0,T;H10(Ω)H2(Ω)) and φtL2(0,T;H10(Ω)H2(Ω)).

    Proof. As above proof, we focus on a priori estimates.

    We multiply (1.10) by Δφt and have, integrating over Ω,

    12ddtφ2+φt2+Δφt2=Ωg(u)utΔφtdx. (3.21)

    Thanks to (3.4) and Hölder's inequality:

    Ωg(u)utΔφtdxcΩ(|u|+1)|ut||Δφt|dxc(u2+1)ut2+12Δφt2 (3.22)

    and then,

    12ddtφ2+φt2+12Δφt2c(u2+1)ut2. (3.23)

    Differentiating (1.9) with respect to time, we get

    2ut2Δut+f(u)ut=g(u)ut(φΔφ)+g(u)(φtΔφt). (3.24)

    Multiplying (3.24) by ut and integrating over Ω, we obtain

    12ddtut2+ut2+Ωf(u)|ut|2dx=Ωg(u)|ut|2(φΔφ)dx+Ωg(u)ut(φtΔφt)dx. (3.25)

    Using (1.10), we write,

    Ωg(u)ut(φtΔφt)dx=Ωg(u)ut(g(u)ut+Δφ)dx=Ω|g(u)ut|2dx+Ωg(u)utΔφdx. (3.26)

    Owing to (3.26), (3.25) reads

    12ddtut2+ut2+Ωf(u)|ut|2dx=Ωg(u)|ut|2(φΔφ)dx+Ωg(u)utΔφdxΩ|g(u)ut|2dx, (3.27)

    since

    Ωg(u)|ut|2(φΔφ)dxcΩ|ut|2(|φ|+|Δφ|)dx12ut2+c(φ2+Δφ2), (3.28)
    Ωg(u)utΔφdx=Ωg(u)uutφdxΩg(u)utφdx (3.29)

    and then,

    |Ωg(u)uutφdx|cΩ|u||ut||φ|dx16ut2+cu2Δφ2 (3.30)

    and

    |Ωg(u)utφdx|cΩ(|u|+1)|ut||φ|dx16ut2+c(u2+1)φ2. (3.31)

    Furthemore,

    Ω|g(u)ut|2dxcΩ(|u|+1)2|ut|2dxc(u2+u2+1)ut2. (3.32)

    Now, collecting (3.27)–(3.32) and owing to (3.20), we are led to

    ddtut2+cut2c(u2H1(Ω)+1)(ut2+φ2H2(Ω)). (3.33)

    Adding (3.19), ε1(3.23) and ε2(3.33), with εi>0,i=1,2, small enough, we obtain

    dE3dt+c(ut2H1(Ω)+φt2H2(Ω))cE3+c, (3.34)

    where

    E3=E2+ε1φ2+ε2ut2 (3.35)

    enjoys

    E3c(u2H(Ω)+uk+2k+2+φ2H2(Ω))c (3.36)

    and

    E3c(u2H(Ω)+uk+2k+2+φ2H2(Ω))c. (3.37)

    We complete the proof applying Gronwall's lemma.

    We now give a uniqueness result

    Theorem 3.3. Under assumptions of Theorem 3.2 and assuming that (3.5) holds true. The problem (1.9)-(1.12) has a unique solution (u,φ), with the above regularity.

    Proof. We suppose the existence of two solutions (u1,φ1) and (u2,φ2) to problem (1.9)-(1.11) associated to initial conditions (u01,φ01) and (u02,φ02), respectively. We then have

    utΔu+f(u1)f(u2)=g(u1)(φΔφ)+(g(u1)g(u2))(φ2Δφ2), (3.38)
    φtΔφtΔφ=g(u1)ut(g(u1)g(u2))u2t, (3.39)
    u|Ω=φ|Ω=0, (3.40)
    u|t=0=u01u02,φ|t=0=φ01φ02, (3.41)

    with u=u1u2, φ=φ1φ2, u0=u01u02 and φ0=φ01φ02.

    Multiplying (3.38) by ut and integrating over Ω, we have

    12ddtu2+ut2+Ω(f(u1f(u2)))utdx=Ωg(u1)(φΔφ)utdx+Ω(g(u1)g(u2))(φ2Δφ2)utdx. (3.42)

    Multiplying (3.39) by φ and integrating over Ω, one has

    12ddt(φ2+φ2)+φ2=Ωg(u1)utφdxΩ(g(u1)g(u2))u2tφdx. (3.43)

    Multiplying (3.39) by Δφ and integrating over Ω, we obtain

    12ddt(φ2+Δφ2)+Δφ2=Ωg(u1)utΔφdx+Ω(g(u1)g(u2))u2tΔφdx. (3.44)

    Finally, adding (3.42), (3.43) and (3.44), we get

    dE4dt+ut2+φ2+Δφ2+Ω(f(u1)f(u2))utdx=Ω(g(u1)g(u2))(φ2Δφ2)utdxΩ(g(u1)g(u2))(φΔφ)u2tdx, (3.45)

    where

    E4=u2+φ2+2φ2+Δφ2. (3.46)

    Now, owing to (3.5), and applying Hölder's inequality for k=2, when n=3, we can write

    Ω(f(u1)f(u2))utdxcΩ(|u2|k+1)|u||ut|dxc(u22k+1)u2+ut2, (3.47)

    we also get, thanks to (3.4), and applying Hölder's inequality,

    Ω(g(u1)g(u2))(φ2Δφ2)utdxcΩ|u||φ2Δφ2||ut|dxcu2(φ22+Δφ22)+ut2 (3.48)

    and

    Ω(g(u1)g(u2))(φΔφ)u2tdxcΩ|u||ut||φΔφ|dxcu2t2(φ2+Δφ2)+u2. (3.49)

    From (3.45)-(3.49), we deduce a differential inequality of the type

    dE4dt+cut2c(u22k+u2t2+φ22+Δφ22+1)E4. (3.50)

    In particular,

    dE4dtcE4 (3.51)

    and then applying the Gronwall's lemma to (3.51), we end the proof.


    4. Dissipativity properties of the system

    This section is devoted to the existence of bounded absorbing sets for the semigroup S(t),t0. To this end, we consider a more restrictive assumption on G, namely,

    ϵ>0,|G(u)|2ϵF(s)+cϵ,sR. (4.1)

    We then have

    Theorem 4.1. Under the assumptions of the Theorem 3.3 and assuming that (4.1) holds true. Then, uL(R+;H10(Ω))Lk+2(Ω), φL(R+;H10(Ω)H2(Ω)).

    Proof. Going from (3.8) and (3.10), we get, summing (3.8) and δ(3.10), with δ>0, as small as we need,

    dE5dt+2(cu2+δΩF(u)dx+ut2+φ2+Δφ2)2cδΩ|G(u)|2dx+δ(φ2+Δφ2)+c2cδΩ|G(u)|2dx+δ(cφ2+Δφ2)+c, (4.2)

    where

    E5=δu2+u2+2ΩF(u)dx+φ2+2φ2+Δφ2 (4.3)

    satisfies

    E5c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c (4.4)

    and

    E5c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c. (4.5)

    From (4.2) and owing to (4.1), we obtain

    dE5dt+2(cu2+δΩF(u)dx+ut2+φ2+Δφ2)CϵΩF(u)dx+δ(cφ2+Δφ2)+Cϵ, (4.6)

    where Cϵ and Cϵ are positive constants which depend on ϵ. Now, choosing ϵ and δ such that:

    2δCϵand2>cδ, (4.7)

    we then deduce from (4.6),

    dE5dt+c(E5+ut2)c, (4.8)

    we complete the proof applying the Gronwall's lemma.

    Remark 4.2. It follows from theorems 3.1, 3.2 and 4.1 that we can define the family solving operators:

    S(t):ΦΦ,(u0,φ0)(u(t),φ(t)),t0, (4.9)

    where Φ=H10(Ω)×H10(Ω)H2(Ω), and (u,φ) is the unique solution to the problem (1.9)-(1.12). Moreover, this family of solving operators forms a continuous semigroup i.e., S(0)=Id and S(t+τ)=S(t)S(τ),t,τ0. And then, it follows from (4.8) that S(t) is dissipative in Φ, it means that it possesses a bounded absorbing set B0Φ i.e., BΦ(bounded),t0=t0(B) suchthat tt0 implies S(t)BB0. (see, e.g., [32], [34] for details).


    5. Spatial behavior of solutions

    The aim of this section is to study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist. This study is motivated by the possibility of extending results obtained above to the case of unbounded domains like semi-infinite cylinders. To do so, we will study the behavior of solutions in a semi-infinite cylinder denoted R=(0,+)×D, where D is a smooth bounded domain of Rn1, n being the space dimension. We then consider the problem defined by the system (1.9)-(1.10) in the semi-infinite R, with n=3. Furthermore, we endow to this system following boundary conditions:

    u=φ=0on(0,+)×D×(0,T) (5.1)

    and

    u(0,x2,x3;t)=h(x2,x3;t),φ(0,x2,x3;t)=l(x2,x3;t)on{0}×D×(0,T), (5.2)

    where T>0 is a given final time.

    We also consider following initial data

    u|t=0=φ|t=0=0onR. (5.3)

    Let us suppose that such solutions exist. We consider the function

    Fw(z,t)=t0D(z)ews(usu,1+φ(φ,1+φ,1s)+φsφ,1)dads, (5.4)

    where D(z)={xR:x1=z}, u,1=ux1, us=us and w is a positive constant. Using the divergence theorem and owing to (5.1), we have

    Fw(z+h,t)Fw(z,t)=ewt2R(z,z+h)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+t0R(z,z+h)ews(|us|2+|φ|2+|Δφ|2)dxds+w2t0R(z,z+h)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds, (5.5)

    where R(z,z+h)={xR:z<x1<z+h}.

    Hence,

    Fwt(z,t)=ewt2D(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)da+t0D(z)ews(|us|2+|φ|2+|Δφ|2)dads+w2t0D(z)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dads. (5.6)

    We consider a second function, namely,

    Gw(z,t)=t0D(z)ews(usu,1+φ(θ,1+φ,1s))dads, (5.7)

    where θ=t0φ(s)ds.

    Similarly, we have

    Gw(z+h,t)Gw(z,t)=ewt2R(z,z+h)(|u|2+|θ|2)dx+t0R(z,z+h)ews(|u|2+f(u)u+uΔφ+|φ|2+|φ|2)dxds+w2t0R(z,z+h)ews(|u|2+|θ|2)dxds+t0R(z,z+h)ews(G(u)g(u)u)φdxds (5.8)

    and then

    Gwt(z,t)=ewt2D(z)(|u|2+|θ|2)da+t0D(z)ews(|u|2+f(u)u+uΔφ+|φ|2+|φ|2)dads+w2t0D(z)ews(|u|2+|θ|2)dads+t0D(z)ews(G(u)g(u)u)φdads. (5.9)

    We choose τ large enough such as

    2F(u)+τu2C1u2,C1>0. (5.10)

    Now, we focus on the nonliear part i.e.,

    w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)g(u)u)φ+w2|φ|2. (5.11)

    We assume that G(s)g(s)sc(|s|k+2+s2).

    For τ large enough, we have F(u)+τ2|u|2C2(|u|k+2+|u|2),C2>0. Thus, for wτ, we deduce that

    w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)g(u)u)φ+w2|φ|2C3(|u|2+|φ|2+|Δφ|2). (5.12)

    Taking into account previous choices, it clearly appears that the following function

    Hw=Fw+τGw (5.13)

    satisfies

    Hwt(z,t)C4t0D(z)ews(|u|2+|u|2+|us|2+|φ|2+|φ|2+|Δφ|2+|θ|2)dads. (5.14)

    We give now an estimate of |Hw| in terms of Hwt. Applying Cauchy-Schwarz's inequality, one has

    |Fw|(t0D(z)ewsu2sdads)1/2(ewsu2,1)1/2+(t0D(z)ewsφ2dads)1/2(ewsφ2,1)1/2+(t0D(z)ewsφ2dads)1/2(ewsφ2,1s)1/2+(t0D(z)ewsφ2sdads)1/2(ewsφ2,1)1/2C5t0D(z)ews(|u|2+|us|2+|φ|2+|φ|2+|φs|2+|φs|2)dads,C5>0. (5.15)

    Similarly,

    |Gw|(t0D(z)ewsu2dads)1/2(t0D(z)ewsu2,1dads)1/2+(t0D(z)ewsφ2dads)1/2(t0D(z)ewsθ2,1dads)1/2+(t0D(z)ewsφ2sdads)1/2(t0D(z)ewsφ2,1dads)1/2C6t0D(z)ews(|u|2+|u|2+|φ|2+|φ|2+|θ|2)dads,C6>0. (5.16)

    We then deduce the existence of a positive constant C7=C5+τC6C4 such that

    |Hw|C7Hwz. (5.17)

    Remark 5.1. The inequality (5.17) is well known in the study of spatial estimates and leads to the Phragmén-Lindelöf alternative (see, e.g., [9], [39]).

    In particular, if there exist z00 such that Fw(z0,t)>0, then the solution satisfies

    Hw(z,t)Hw(z0,t)eC17(zz0),zz0. (5.18)

    The estimate (5.18) gives information in terms of measure defined in the cylinder. Actually, from (5.18), we deduce that

    ewt2R(0,z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+τewt2R(0,z)(|u|2+|θ|2)dx+t0R(0,z)ews(|us|2+|φ|2+|Δφ|2)dxds+τt0R(0,z)ews(|u|2+f(u)u+g(u)uΔφ+|φ|2+2|φ|2)dxds+w2t0R(0,z)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds+τw2t0R(0,z)ews(|u|2+|θ|2)dx+τt0R(0,z)ews(G(u)g(u)u)φdxds (5.19)

    tends to infinity exponentially fast. On the other hand, if Hw(z,t)0, for every z0, we deduce that the solution decreases and we get an inequality of the type

    Hw(z,t)Hw(0,t)eC17z,z0, (5.20)

    where

    Ew(z,t)=ewt2R(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+τewt2R(z)(|u|2+|θ|2)dx+t0R(z)ews(|us|2+|φ|2+|Δφ|2)dxds+τt0R(z)ews(|u|2+f(u)u+g(u)uΔφ+|φ|2+2|φ|2)dxds+w2t0R(z)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds+τw2t0R(z)ews(|u|2+|θ|2)dx+τt0R(z)ews(G(u)g(u)u)φdxds (5.21)

    and R(z)={xR:x1>z}.

    Finally, setting

    Ew(z,t)=12R(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+τ12R(z)(|u|2+|θ|2)dx+t0R(z)(|us|2+|φ|2+|Δφ|2)dxds+τt0R(z)(|u|2+f(u)u+g(u)uΔφ+|φ|2+2|φ|2)dxds+w2t0R(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds+τw2t0R(z)(|u|2+|θ|2)dx+τt0R(z)(G(u)g(u)u)φdxds. (5.22)

    We have the following result

    Theorem 5.2. Let (u,φ) be a solution to the problem given by (1.9)-(1.10), boundary conditions (5.1)-(5.2) and initial data (5.3). Then, either this solution satisfies (5.18), or it satisfies

    Ew(z,t)Ew(0,t)ewtC17z,z0, (5.23)

    where the energy Ew is given by (5.22).


    Acknowledgments

    The author would like to thank Alain Miranville for his advices and for his careful reading of this paper.


    Conflict of interest

    The author declares no conflicts of interest in this paper.




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