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.
The Caginalp phase-field system
∂u∂t−Δu+f(u)=θ, | (1.1) |
∂θ∂t−Δθ=−∂u∂t, | (1.2) |
has been introduced in [1] in order to describe the phase transition phenomena in certain class of material. In this context,
ψ=∫Ω(12|∇u|2+F(u)−uθ−12θ2)dx, | (1.3) |
where
H=u+θ. | (1.4) |
Then, the evolution equation for the order parameter
∂u∂t=−δuψ, | (1.5) |
where
∂H∂t=−divq, | (1.6) |
where
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=φ−Δφ. | (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
∂u∂t−Δu+f(u)=g(u)(φ−Δφ), | (1.9) |
∂φ∂t−Δ∂φ∂t−Δφ=−g(u)∂u∂t, | (1.10) |
u=φ=0on∂Ω, | (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
In our case, to obtain equations (1.9) and (1.10), the total free energy reads in terms of the conductive temperature
ψ(u,θ)=∫Ω(12|∇u|2+F(u)−G(u)θ−12θ2)dx, | (2.1) |
where
H=G(u)+θ=G(u)+φ−Δφ, | (2.2) |
which yields thanks to (1.6), the energy equation,
∂φ∂t−Δ∂φ∂t+divq=−g(u)∂u∂t. | (2.3) |
Considering the usual Fourier law (
Remark 2.1. We can note that we still have an infinite speed of propagation here.
Before stating the existence result, we make some assumptions on nonlinearities
|G(s)|2≤c1F(s)+c2,c0,c1,c2≥0, | (3.1) |
|g(s)s|≤c3(|G(s)|2+1),c3≥0, | (3.2) |
c4sk+2−c5≤F(s)≤f(s)s+c0≤c6sk+2−c7,c4,c6>0,c5,c7≥0, | (3.3) |
|g(s)|≤c8(|s|+1),|g′(s)|≤c9c8,c9≥0, | (3.4) |
|f′(s)|≤c10(|s|k+1),c10≥0, | (3.5) |
where
Theorem 3.1. We assume that (3.1)-(3.4) hold true. For all initial data
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
12ddt(‖∇u‖2+2∫ΩF(u)dx)+‖∂u∂t‖2=∫Ωg(u)∂u∂t(φ−Δφ)dx. | (3.6) |
Multiplying (1.10) by
12ddt(‖φ‖2+2‖∇φ‖2+‖Δφ‖2)+‖∇φ‖2+‖Δφ‖2=−∫Ωg(u)∂u∂t(φ−Δφ)dx. | (3.7) |
Now, summing (3.6) and (3.7), we are led to,
ddt(‖∇u‖2+2∫ΩF(u)dx+‖φ‖2+2‖∇φ‖2+‖Δφ‖2)+2(‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2)=0. | (3.8) |
Multiplying (1.9) by
12ddt‖u‖2+‖∇u‖2+∫Ωf(u)udx=∫Ωg(u)u(φ−Δφ)dx. | (3.9) |
Using (3.2)-(3.3), (3.9) becomes
12ddt‖u‖2+‖∇u‖2+c∫ΩF(u)dx≤c′∫Ω|G(u)|2dx+12(‖φ‖2+‖Δφ‖2)+c″. | (3.10) |
Adding (3.8) and (3.10), one has
dE1dt+2(‖∇u‖2+c∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2)+‖Δφ‖2≤c′∫Ω|G(u)|2dx+‖φ‖2+c″, | (3.11) |
where
E1=‖u‖2+‖∇u‖2+2∫ΩF(u)dx+‖φ‖2+2‖∇φ‖2+‖Δφ‖2 | (3.12) |
enjoys
E1≤c(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c′ | (3.13) |
and
E1≤c″(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c‴. | (3.14) |
Multiplying now (1.10) by
12ddt‖∇φ‖2+‖∂φ∂t‖2+‖∇∂φ∂t‖2=−∫Ωg(u)∂u∂t∂φ∂tdx. | (3.15) |
Taking into account (3.4) and using Hölder's inequality, we get
12ddt‖∇φ‖2+12‖∂φ∂t‖2+‖∇∂φ∂t‖2≤c(‖∇u‖2+1)‖∂u∂t‖2 | (3.16) |
and then, summing (3.11) and (3.16), we have
dE2dt+2(‖∇u‖2+c∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2+12‖Δφ‖2+12‖∂φ∂t‖2+‖∇∂φ∂t‖2)≤c∫Ω|G(u)|2dx+‖φ‖2+c″(‖∇u‖2+1)‖∂u∂t‖2+c‴, | (3.17) |
where
E2=E1+‖∇φ‖2 | (3.18) |
satisfies similar estimates as
We deduce from (3.1) and (3.17)
dE2dt+c(‖∂φ∂t‖2+‖∇∂φ∂t‖2)≤c′E2+c″, | (3.19) |
which achieve the proof.
For more regularity on solutions, we make following additional assumptions:
f(0)=0andf′(s)≥−c,c≥0. | (3.20) |
We have:
Theorem 3.2. Under assumptions of Theorem 3.1 and assuming that (3.20) is satisfied. For every initial data
Proof. As above proof, we focus on a priori estimates.
We multiply (1.10) by
12ddt‖∇φ‖2+‖∇∂φ∂t‖2+‖Δ∂φ∂t‖2=∫Ωg(u)∂u∂tΔ∂φ∂tdx. | (3.21) |
Thanks to (3.4) and Hölder's inequality:
∫Ωg(u)∂u∂tΔ∂φ∂tdx≤c∫Ω(|u|+1)|∂u∂t||Δ∂φ∂t|dx≤c(‖∇u‖2+1)‖∂u∂t‖2+12‖Δ∂φ∂t‖2 | (3.22) |
and then,
12ddt‖∇φ‖2+‖∇∂φ∂t‖2+12‖Δ∂φ∂t‖2≤c(‖∇u‖2+1)‖∂u∂t‖2. | (3.23) |
Differentiating (1.9) with respect to time, we get
∂2u∂t2−Δ∂u∂t+f′(u)∂u∂t=g′(u)∂u∂t(φ−Δφ)+g(u)(∂φ∂t−Δ∂φ∂t). | (3.24) |
Multiplying (3.24) by
12ddt‖∂u∂t‖2+‖∇∂u∂t‖2+∫Ωf′(u)|∂u∂t|2dx=∫Ωg′(u)|∂u∂t|2(φ−Δφ)dx+∫Ωg(u)∂u∂t(∂φ∂t−Δ∂φ∂t)dx. | (3.25) |
Using (1.10), we write,
∫Ωg(u)∂u∂t(∂φ∂t−Δ∂φ∂t)dx=∫Ωg(u)∂u∂t(−g(u)∂u∂t+Δφ)dx=−∫Ω|g(u)∂u∂t|2dx+∫Ωg(u)∂u∂tΔφdx. | (3.26) |
Owing to (3.26), (3.25) reads
12ddt‖∂u∂t‖2+‖∇∂u∂t‖2+∫Ωf′(u)|∂u∂t|2dx=∫Ωg′(u)|∂u∂t|2(φ−Δφ)dx+∫Ωg(u)∂u∂tΔφdx−∫Ω|g(u)∂u∂t|2dx, | (3.27) |
since
∫Ωg′(u)|∂u∂t|2(φ−Δφ)dx≤c∫Ω|∂u∂t|2(|φ|+|Δφ|)dx≤12‖∇∂u∂t‖2+c(‖φ‖2+‖Δφ‖2), | (3.28) |
∫Ωg(u)∂u∂tΔφdx=−∫Ωg′(u)∇u∂u∂t∇φdx−∫Ωg(u)∇∂u∂t∇φdx | (3.29) |
and then,
|∫Ωg′(u)∇u∂u∂t∇φdx|≤c∫Ω|∇u||∂u∂t||∇φ|dx≤16‖∇∂u∂t‖2+c‖∇u‖2‖Δφ‖2 | (3.30) |
and
|∫Ωg(u)∇∂u∂t∇φdx|≤c∫Ω(|u|+1)|∇∂u∂t||∇φ|dx≤16‖∇∂u∂t‖2+c(‖∇u‖2+1)‖∇φ‖2. | (3.31) |
Furthemore,
∫Ω|g(u)∂u∂t|2dx≤c∫Ω(|u|+1)2|∂u∂t|2dx≤c(‖∇u‖2+‖u‖2+1)‖∂u∂t‖2. | (3.32) |
Now, collecting (3.27)–(3.32) and owing to (3.20), we are led to
ddt‖∂u∂t‖2+c‖∇∂u∂t‖2≤c′(‖u‖2H1(Ω)+1)(‖∂u∂t‖2+‖φ‖2H2(Ω)). | (3.33) |
Adding (3.19),
dE3dt+c(‖∂u∂t‖2H1(Ω)+‖∂φ∂t‖2H2(Ω))≤c′E3+c″, | (3.34) |
where
E3=E2+ε1‖∇φ‖2+ε2‖∂u∂t‖2 | (3.35) |
enjoys
E3≥c(‖u‖2H(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c′ | (3.36) |
and
E3≤c″(‖u‖2H(Ω)+‖u‖k+2k+2+‖φ‖2H2(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
Proof. We suppose the existence of two solutions
∂u∂t−Δu+f(u1)−f(u2)=g(u1)(φ−Δφ)+(g(u1)−g(u2))(φ2−Δφ2), | (3.38) |
∂φ∂t−Δ∂φ∂t−Δφ=−g(u1)∂u∂t−(g(u1)−g(u2))∂u2∂t, | (3.39) |
u|∂Ω=φ|∂Ω=0, | (3.40) |
u|t=0=u01−u02,φ|t=0=φ01−φ02, | (3.41) |
with
Multiplying (3.38) by
12ddt‖∇u‖2+‖∂u∂t‖2+∫Ω(f(u1−f(u2)))∂u∂tdx=∫Ωg(u1)(φ−Δφ)∂u∂tdx+∫Ω(g(u1)−g(u2))(φ2−Δφ2)∂u∂tdx. | (3.42) |
Multiplying (3.39) by
12ddt(‖φ‖2+‖∇φ‖2)+‖∇φ‖2=−∫Ωg(u1)∂u∂tφdx−∫Ω(g(u1)−g(u2))∂u2∂tφdx. | (3.43) |
Multiplying (3.39) by
12ddt(‖∇φ‖2+‖Δφ‖2)+‖Δφ‖2=∫Ωg(u1)∂u∂tΔφdx+∫Ω(g(u1)−g(u2))∂u2∂tΔφdx. | (3.44) |
Finally, adding (3.42), (3.43) and (3.44), we get
dE4dt+‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2+∫Ω(f(u1)−f(u2))∂u∂tdx=∫Ω(g(u1)−g(u2))(φ2−Δφ2)∂u∂tdx−∫Ω(g(u1)−g(u2))(φ−Δφ)∂u2∂tdx, | (3.45) |
where
E4=‖∇u‖2+‖φ‖2+2‖∇φ‖2+‖Δφ‖2. | (3.46) |
Now, owing to (3.5), and applying Hölder's inequality for
∫Ω(f(u1)−f(u2))∂u∂tdx≤c∫Ω(|u2|k+1)|u||∂u∂t|dx≤c(‖∇u2‖2k+1)‖∇u‖2+‖∂u∂t‖2, | (3.47) |
we also get, thanks to (3.4), and applying Hölder's inequality,
∫Ω(g(u1)−g(u2))(φ2−Δφ2)∂u∂tdx≤c∫Ω|u||φ2−Δφ2||∂u∂t|dx≤c‖∇u‖2(‖φ2‖2+‖Δφ2‖2)+‖∂u∂t‖2 | (3.48) |
and
∫Ω(g(u1)−g(u2))(φ−Δφ)∂u2∂tdx≤c∫Ω|u||∂u∂t||φ−Δφ|dx≤c‖∂u2∂t‖2(‖φ‖2+‖Δφ‖2)+‖∇u‖2. | (3.49) |
From (3.45)-(3.49), we deduce a differential inequality of the type
dE4dt+c‖∂u∂t‖2≤c(‖∇u2‖2k+‖∂u2∂t‖2+‖φ2‖2+‖Δφ2‖2+1)E4. | (3.50) |
In particular,
dE4dt≤cE4 | (3.51) |
and then applying the Gronwall's lemma to (3.51), we end the proof.
This section is devoted to the existence of bounded absorbing sets for the semigroup
∀ϵ>0,|G(u)|2≤ϵF(s)+cϵ,s∈R. | (4.1) |
We then have
Theorem 4.1. Under the assumptions of the Theorem 3.3 and assuming that (4.1) holds true. Then,
Proof. Going from (3.8) and (3.10), we get, summing (3.8) and
dE5dt+2(c‖∇u‖2+δ∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2)≤2c′δ∫Ω|G(u)|2dx+δ(‖φ‖2+‖Δφ‖2)+c″≤2c′δ∫Ω|G(u)|2dx+δ(c‖∇φ‖2+‖Δφ‖2)+c″, | (4.2) |
where
E5=δ‖u‖2+‖∇u‖2+2∫ΩF(u)dx+‖φ‖2+2‖∇φ‖2+‖Δφ‖2 | (4.3) |
satisfies
E5≥c(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c′ | (4.4) |
and
E5≤c″(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c‴. | (4.5) |
From (4.2) and owing to (4.1), we obtain
dE5dt+2(c‖∇u‖2+δ∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2)≤Cϵ∫ΩF(u)dx+δ(c‖∇φ‖2+‖Δφ‖2)+C′ϵ, | (4.6) |
where
2δ≥Cϵand2>cδ, | (4.7) |
we then deduce from (4.6),
dE5dt+c(E5+‖∂u∂t‖2)≤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):Φ⟶Φ,(u0,φ0)↦(u(t),φ(t)),∀t≥0, | (4.9) |
where
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
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
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)=∫t0∫D(z)e−ws(usu,1+φ(φ,1+φ,1s)+φsφ,1)dads, | (5.4) |
where
Fw(z+h,t)−Fw(z,t)=e−wt2∫R(z,z+h)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+∫t0∫R(z,z+h)e−ws(|us|2+|∇φ|2+|Δφ|2)dxds+w2∫t0∫R(z,z+h)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds, | (5.5) |
where
Hence,
∂Fw∂t(z,t)=e−wt2∫D(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)da+∫t0∫D(z)e−ws(|us|2+|∇φ|2+|Δφ|2)dads+w2∫t0∫D(z)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dads. | (5.6) |
We consider a second function, namely,
Gw(z,t)=∫t0∫D(z)e−ws(usu,1+φ(θ,1+φ,1s))dads, | (5.7) |
where
Similarly, we have
Gw(z+h,t)−Gw(z,t)=e−wt2∫R(z,z+h)(|u|2+|∇θ|2)dx+∫t0∫R(z,z+h)e−ws(|∇u|2+f(u)u+uΔφ+|φ|2+|∇φ|2)dxds+w2∫t0∫R(z,z+h)e−ws(|u|2+|∇θ|2)dxds+∫t0∫R(z,z+h)e−ws(G(u)−g(u)u)φdxds | (5.8) |
and then
∂Gw∂t(z,t)=e−wt2∫D(z)(|u|2+|∇θ|2)da+∫t0∫D(z)e−ws(|∇u|2+f(u)u+uΔφ+|φ|2+|∇φ|2)dads+w2∫t0∫D(z)e−ws(|u|2+|∇θ|2)dads+∫t0∫D(z)e−ws(G(u)−g(u)u)φdads. | (5.9) |
We choose
2F(u)+τu2≥C1u2,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
For
w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)−g(u)u)φ+w2|φ|2≥C3(|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
∂Hw∂t(z,t)≥C4∫t0∫D(z)e−ws(|u|2+|∇u|2+|us|2+|φ|2+|∇φ|2+|Δφ|2+|∇θ|2)dads. | (5.14) |
We give now an estimate of
|Fw|≤(∫t0∫D(z)e−wsu2sdads)1/2(e−wsu2,1)1/2+(∫t0∫D(z)e−wsφ2dads)1/2(e−wsφ2,1)1/2+(∫t0∫D(z)e−wsφ2dads)1/2(e−wsφ2,1s)1/2+(∫t0∫D(z)e−wsφ2sdads)1/2(e−wsφ2,1)1/2≤C5∫t0∫D(z)e−ws(|∇u|2+|us|2+|φ|2+|∇φ|2+|φs|2+|∇φs|2)dads,C5>0. | (5.15) |
Similarly,
|Gw|≤(∫t0∫D(z)e−wsu2dads)1/2(∫t0∫D(z)e−wsu2,1dads)1/2+(∫t0∫D(z)e−wsφ2dads)1/2(∫t0∫D(z)e−wsθ2,1dads)1/2+(∫t0∫D(z)e−wsφ2sdads)1/2(∫t0∫D(z)e−wsφ2,1dads)1/2≤C6∫t0∫D(z)e−ws(|u|2+|∇u|2+|φ|2+|∇φ|2+|∇θ|2)dads,C6>0. | (5.16) |
We then deduce the existence of a positive constant
|Hw|≤C7∂Hw∂z. | (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
Hw(z,t)≥Hw(z0,t)eC−17(z−z0),z≥z0. | (5.18) |
The estimate (5.18) gives information in terms of measure defined in the cylinder. Actually, from (5.18), we deduce that
e−wt2∫R(0,z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+τe−wt2∫R(0,z)(|u|2+|∇θ|2)dx+∫t0∫R(0,z)e−ws(|us|2+|∇φ|2+|Δφ|2)dxds+τ∫t0∫R(0,z)e−ws(|∇u|2+f(u)u+g(u)uΔφ+|φ|2+2|∇φ|2)dxds+w2∫t0∫R(0,z)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds+τw2∫t0∫R(0,z)e−ws(|u|2+|∇θ|2)dx+τ∫t0∫R(0,z)e−ws(G(u)−g(u)u)φdxds | (5.19) |
tends to infinity exponentially fast. On the other hand, if
−Hw(z,t)≤−Hw(0,t)eC−17z,z≥0, | (5.20) |
where
Ew(z,t)=e−wt2∫R(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+τe−wt2∫R(z)(|u|2+|∇θ|2)dx+∫t0∫R(z)e−ws(|us|2+|∇φ|2+|Δφ|2)dxds+τ∫t0∫R(z)e−ws(|∇u|2+f(u)u+g(u)uΔφ+|φ|2+2|∇φ|2)dxds+w2∫t0∫R(z)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds+τw2∫t0∫R(z)e−ws(|u|2+|∇θ|2)dx+τ∫t0∫R(z)e−ws(G(u)−g(u)u)φdxds | (5.21) |
and
Finally, setting
Ew(z,t)=12∫R(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+τ12∫R(z)(|u|2+|∇θ|2)dx+∫t0∫R(z)(|us|2+|∇φ|2+|Δφ|2)dxds+τ∫t0∫R(z)(|∇u|2+f(u)u+g(u)uΔφ+|φ|2+2|∇φ|2)dxds+w2∫t0∫R(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds+τw2∫t0∫R(z)(|u|2+|∇θ|2)dx+τ∫t0∫R(z)(G(u)−g(u)u)φdxds. | (5.22) |
We have the following result
Theorem 5.2. Let
Ew(z,t)≤Ew(0,t)ewt−C−17z,z≥0, | (5.23) |
where the energy
The author would like to thank Alain Miranville for his advices and for his careful reading of this paper.
The author declares no conflicts of interest in this paper.
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