Semidiscretization for a nonlocal parabolic problem

PARABOLIC PROBLEM

ABDERRAHMANE EL HACHIMI AND MOULAY RCHID SIDI AMMI

Received 22 February 2004 and in revised form 16 February 2005

A time discretization technique by Euler forward scheme is proposed to deal with a nonlocal parabolic problem. Existence and uniqueness of the approximate solution are proved.

1. Introduction

In this work, we study the time discretization by Euler forward scheme of the nonlocal initial boundary value problem

∂u

∂t − u=λ

f(u)

Ωf(u)dx2

inΩ×]0;T[,

u=0 on∂Ω×]0;T[,

u(0)=u0 inΩ,

(1.1)

withΩ⊂Rd_{(d≥1) a bounded regular domain andλa positive parameter. The} hypothe-ses we will assume on f are the same as in [6]. We recall first that (1.1) arises by reducing the following system of two equations modeling the thermistor problem:

ut= ∇·

k(u)∇u+σ(u)∇ϕ2,

∇σ(u)∇ϕ=0, (1.2)

whereurepresents the temperature generated by the electric current flowing through a conductor,ϕthe electric potential,σ(u) andk(u) are, respectively, the electric and ther-mal conductivities. For more description, we refer to [5,6,7,8,11] among others.

We recall also that the Euler forward method was used by several authors to treat semidiscretization of nonlinear parabolic problems, see [3,4]. Concerning problem (1.1), results of existence and uniqueness of solutions are known under particular forms of

f, we refer to [2] and the references therein. On the other hand, little is known about

Copyright©2005 Hindawi Publishing Corporation

the solutions to the discrete problem

Un−τUn=Un−1+λτ f

Un

ΩfUndx2 inΩ,

Un_{=0 on∂Ω,}

U0=u0 inΩ.

(1.3)

Whereas, semidiscretization has been involved for the equations of the thermistor problem in [1,9]. Our aim here is to continue the study of problem (1.1) initiated in [6], where an a prioriL∞-estimate is derived. In addition to habitual existence and uniqueness questions concerning the solutions of (1.3), we will prove some results of stability and proceed to error estimates analysis. In [1], the authors derived anL2_andH1_{-norm error} by requiring more regularity on the solutionu, for instanceu,utinH2(Ω)∩W1,∞(Ω). Unfortunately, such smoothness is not always possible since the function f is nonlinear.

2. The semidiscrete problem

2.1. Existence and uniqueness. We consider the Euler scheme (1.3), withNτ=T,T >0 fixed, and 1≤n≤N, under the following hypotheses.

(H1) f :R→Ris a locally Lipschitzian function.

(H2) There exist positive constantsσ,c1,c2, andαsuch thatα <4/(d−2) and for all ξ∈R,

σ≤f(ξ)≤c1ξα +1

+c2. (2.1)

In the sequel, we will denote the norms in the spacesL∞(Ω),Lk_{(Ω) by| · |L∞}

(Ω)and| · |k, respectively, (·,·) will denote the associated inner product inL2_{(Ω) or the duality product} betweenH1

0(Ω) and its dualH−1(Ω).

Theorem2.1. Let(H1)-(H2)be satisfied. Then, for eachn, there exists a unique solution

Un_{of (1.3) inH}1

0(Ω)∩L∞(Ω)provided thatτis small enough.

Proof. For simplicity, we writeU=Un_,h(x)=Un−1_{. Then (1.3) becomes}

U−τU=h(x) +λ f(U)

Ωf(U)dx2 inΩ,

U=0 on∂Ω.

(2.2)

Existence. Define the mapS(µ,·) byU=S(µ,v),µ∈[0, 1] if and only if

U−τU=µg(x,v) inΩ,

U=0 on∂Ω,

U0=µu0,

whereg(x,v)=h(x) +λ(f(v)/(_Ωf(v)dx)2_{). For a fixedv∈H}1

0(Ω), (2.3) has a unique solutionU∈H1

0(Ω). Then, for eachµ∈[0, 1], the operatorS(µ,·) is well defined. More-over,S(µ,·) is compact fromH01(Ω) into it self. Indeed, using (H2), we have the estimate

_U2

2+τ∇U 2

2≤c3. (2.4)

We can easily see thatµ→S(µ,v) is continuous and thatS(0,v)=U, for anyv, if and only ifU=0. From Leray-Schauder fixed point theorem, there exists therefore a fixed pointU

ofS(µ,·).

Now, we derive an a priori estimate.

Lemma2.2. Ifu0∈L∞(Ω), then for alln∈ {1,. . .,N},Un∈L∞(Ω).

Proof. The proof is similar to the one used by De Th´elin in [10] concerning a very

differ-ent problem and we will give here only a sketch. Suppose thatd≥2 and define

δ=

    

2d

d−2 if 2< d, 2(α+ 2) ifd=2.

(2.5)

For eachk∈N∗, we consider the number

qk= δ

2

k−1

(δ−γ)−(2−γ)

δ

δ−2, k≥2,

q1=δ,

(2.6)

we have

qk+1=

qk+ 2−γ

δ

2 withγ=α+ 2,∀k∈N∗. (2.7)

Lemma2.3. For allk∈N∗_,Un_∈Lqk₍Ω_{), and moreover}

_Un

∞=limUnqk<+∞. (2.8)

Proof. We prove by recurrence thatU∈Lqk_{. The property is true fork}=_{1, sinceH}1 0(Ω)⊂ Lδ_{(Ω). We show now thatU∈L}qk+1_{. Letm}∈N_{, 1}≤_m≤_{k. Multiplying (2.2) by}|_U|qm−γ_U, using (H2), and Young’s inequality, we get

qm−γ+ 1

Ω|∇U|

2_|U|qm−γ_dx≤c

4|U|qqmm+c5. (2.9)

On the other hand, we have

|U|qm+2−γ qm+1 ≤c6

1 +qm−γ 2

2

Ω|∇U|

Therefore, we obtain

|U|qm+2−γ qm+1 ≤

c7+c8|U|qqmm

qm+ 2−γ

. (2.11)

Thus,

|U|qk+1 qk+1

2/δ

≤c7+c8|U|qqkk

qk+ 2−γ. (2.12)

The rest of the proof follows the same lines as in [10, pages 383-384].

Uniqueness. ConsiderU andV two different solutions of (2.2) and definew=U−V. Then, we have

w−τw= λτ

Ωf(U)dx2

f(U)−f(V)

+λτ

Ωf(U)−f(V)dxΩf(V) +f(U)dx Ωf(U)dx2Ωf(V)dx2

f(V).

(2.13)

Multiplying (2.13) byw, integrating onΩ, and using theL∞-estimate obtained inLemma 2.2, we get

|w|2

2+τ|∇w|22≤c9τ|w|22. (2.14)

Therefore,w=0 ifτ≤1/c9.

We address now the question of stability.

3. Stability

Theorem3.1. Assume(H1)-(H2)hold. Then, there existsc(T,u0)>0depending on data but not onNsuch that for anyn∈ {1,. . .,N},

(a)|Un_|L∞

(Ω)≤c(T,u0); (b)|Un_|2

2+τ n

k=1|∇Uk|22≤c(T,u0); (c)nk=1|Uk−Uk−1|22≤c(T,u0).

Proof. (i) Multiplying (1.3) by|Uk_|m_Uk_{for some integerm≥1, usingLemma 2.2, and} H¨older’s inequality, we obtain after simplification

_Uk

m+2≤Uk−1m+2+c10τ. (3.1)

By induction and taking the limit in the resulting inequality asm→+∞, we get

_Uk

L∞_(Ω)≤c

T,u0

. (3.2)

(ii) Multiplying the first equation of (1.3) byUk _{and using the hypotheses on f, one} easily has

Uk_−Uk−1_,Uk_+τ∇Uk2

Using the elementary identity 2a(a−b)=a2_−b2_{+ (a−b)}2_{and summing fromk=1 to} n, we obtain

_Un2 2+

n

k=1

_Uk_−Uk−12 2+τ

n

k=1

_∇Uk2

2≤u022+τc12

n

k=1 _Uk

1. (3.4)

Then, the inequalities (b)-(c) hold by using the uniform bound ofUn _inL∞ _{which is}

established in part (a).

4. Error estimates for solutions

We will adopt the following notations concerning the time discretization for problem (1.1). We denote the time stepτ=T/N,tn_{=nτ, andI}

n=(tn,tn−1) forn=1,. . .,N. Ifz is a continuous function (resp., summable), defined in (0,T) with values inH−1_{(Ω) or} L2_{(Ω) orH}1

0(Ω), we definezn=z(tn,·),zn=(1/τ)

Inz(t,·)dt,z

0_=z0_{=z(0,·); the error} en=u(t)−Unfor all t∈In and the local errorseunandendefined byeun=un(t)−Un,

en_=un_−Un_.

We have the following theorem.

Theorem4.1. Let(H1)-(H2)hold. Then, the following error bounds are satisfied: (1)en2

L∞_(0,T,H−1_(Ω))+

T

0 |en|2dt≤c13τ, (2)em_H−1

(Ω)≤c14τ1/2, (3)|∇0Ten(t)dt|2≤c15τ1/4.

Proof. For the proof, we consider the following variational formulation of discrete prob-lem (1.3):

Un_−Un−1_,ϕ+τ∇Un_,∇ϕ= λτ ΩfUndx2

fUn_{,ϕ, ∀ϕ∈H}1

0(Ω). (4.1)

Integrating the continuous problem (1.1) overIn, we get

un_−un−1_,ϕ+τ∇un_,∇ϕ=λτ

fun_,ϕ

Ωfundx2

, ∀ϕ∈H1

0(Ω). (4.2)

Substracting (4.2) from (4.1) and adding fromn=1 tomwithm≤N, we obtain

m

n=1

en_−en−1_,ϕ+τ

m

n=1

∇en u,∇ϕ

≤c16τ

m

n=1

f(u)n−fUn_,ϕ

+c17τ

m

n=1

f(Un_,ϕ

.

Let (−)−1_{be the green operator satisfying}

∇(−)−1ψ,∇ϕ=(ψ,ϕ)H−1_(Ω),H1

0(Ω) (4.4)

for allψ∈H−1_(Ω),ϕ∈H1

0(Ω). Choosingϕ=(−)−1(en) as test function, we then ob-tain

I1+I2≤I3+I4, (4.5)

where

I1=

m

n=1

en_−en−1_{, (−)}−1_en_,

I2=τ

m

n=1

en u,en

,

I3≤c16τ

m

n=1

f(u)n−fUn_{, (−)}−1_en

,

I4=c17τ

m

n=1

fUn, (−)−1en .

(4.6)

With the aid of the elementary identity 2a(a−b)=a2_−b2_{+ (a−b)}2_{and the property of} (−)−1_,I

1reduces after straightforward calculations to

I1=1 2e

m2 H−1_(Ω)+

1 2

m

n=1

_en_−en−12

H−1_(Ω). (4.7)

On the other hand,

I2=τ

m

n=1

en u,en

= m

n=1

In

u(t)−Un_,u(t)−Un_dt+ m

n=1

In

u(t)−Un_,un_−u(t)dt

=I21+I22,

(4.8)

where

I21=

m

n=1

In

u(t)−Un,u(t)−Undt= m

n=1

In

_en2 2dt,

I22=

m

n=1

In

u(t),un_−u(t)dt− m

n=1

In

Un_,un_−u(t)dt

=I221 +I222.

We now estimateI1

22.Using the boundedness of∂u/∂s(see [6]), we have _I1

22=

m

n=1

In

u(t),

tn t ∂u ∂sds dt ≤ m

n=1 In tn t ∂u_∂s

H−1_(Ω)ds

_u(t)

H1 0(Ω)dt

≤τ∂u ∂s

L2_(0,tm_,H−1_(Ω))uL 2_(0,tm_,H1

0(Ω))

≤c18τ.

(4.10)

In the same manner, we have

_I2 22≤τ

∂u_∂s

L2_(0,tm_,H−1_(Ω))

τ

m

n=1 _Un2

H1 0(Ω)

1/2

≤c18τ.

(4.11)

Next, we estimate the first term on the right-hand side of (4.5) by using H¨older’s and Young’s inequalities and (H1),

_I3≤

m

n=1

In

f(u)−fUndt, (−)−1en

≤c20τ1/2

m

n=1

In

_f(u)−f

Un22dt 1/2

_en H−1_(Ω)

≤η

m

n=1

In

_f(u)−f

Un2₂dt

+c21

η τ

m

n=1 _en2

H−1_(Ω)

≤c22η

m

n=1

In

_en2 2dt

+c21

η τ

m

n=1 _en2

H−1_(Ω).

(4.12)

Moreover, we have

_{I4≤c23τ+c24τ}m n=1

_en2

H−1_(Ω). (4.13)

Choosing suitableη, we conclude that

_em2 H−1_(Ω)+

m

n=1

_en_−en−12

H−1_(Ω)+ m

n=1

In

_en2 2dt

≤c25τ+c26τ

m

n=1 _en2

H−1_(Ω).

(4.14)

On the other hand, settingym₌m

n=1enH2−1_(Ω), then from (4.14), we get

ym_−ym−1_≤c

By applying the discrete Gronwall inequality, we deduce that ym_{≤c(T). Therefore, we} have

_em

H−1_(Ω)≤c27τ1/2. (4.16)

On the other hand, we have

sup t∈(0,tm)

_en(t)

H−1_(Ω)−c27τ1/2≤ max 1≤n≤m

_en

tn_H−1_(Ω)= max 1≤n≤m

_en

H−1_(Ω). (4.17)

Thus, we get

_en

L∞_(0,T,H−1_(Ω))−c27τ1/2≤ max 1≤n≤m

_en

H−1_(Ω). (4.18)

From the last inequality, we obtain

_en2

L∞_(0,T,H−1_(Ω))+

T

0 _en2

2dt≤c29τ,

m

n=1

_en_−en−12

H−1_(Ω)≤c29τ.

(4.19)

Choosingϕ=τm_n=1(un−Un) in (4.3), we get

τ

Ω

um_−Um

_m

n=1

un_−Un_dx

+τ2

m

n=1

∇un_−Un

2

2

≤c30τ2 Ω m

n=1

f(u)n−fUn _m

n=1

un−Un

dx

+c31τ2

m

n=1

fUn_, m

n=1

un_−Un

. (4.20) Thus, τ2 m

n=1

∇un_−Un

2

2

=∇

tm

0 endt 2 2≤ τ Ω

um_−Um

_m

n=1

un_−Un_dx

+c30τ2 Ω m

n=1

f(u)n−fUn

_m

n=1

un_−Un

dx

+c31τ2

m

n=1

fUn, m

n=1

un−Un

≤I+II+III.

Clearly,

I≤em H−1_(Ω)

_m

n=1

In

_u(t)

H1

0(Ω)dt+τ m

n=1 _Un

H1 0(Ω)

≤c32em_H−1_(Ω)

≤c33τ1/2.

(4.22)

We also get

II≤   Ω _m

n=1

In

f(u)−fUn_dt

2 dx

 

1/2

×   Ω _m

n=1

In

u(t)−Un_dt

2 dx

 

1/2

≤T2 _m

n=1

In

_f(u)−f Un2

2dt 1/2

×

_m

n=1

In

_u(t)−Un2

2dt 1/2

≤T2 _m

n=1

In

_f(u)−f(Un2

2dt 1/2

×

2u2

L2_(0,T,H1 0(Ω))+ 2τ

m

n=1 _Un2

2 1/2

≤c34τ1/2.

(4.23)

The last inequality follows by using simultaneously theL∞-estimate ofu(t) (see [6]),Un_, and the error bound given in (a). Arguing exactly as in the previous estimate, we get

III≤T2 _m

n=1

In

_f Un2

2dt 1/2

×

2u2

L2_(0,T,H1 0(Ω))+ 2τ

m

n=1 _Un2

2 1/2

. (4.24)

Using again the hypothesis (H1) and the estimates above, we obtain

III≤c35τ1/2. (4.25)

Finally collecting these results, it follows that

∇

T

0 endt 2

2≤

c36τ1/2. (4.26)

This completes the proof.

Corollary4.2. Under hypotheses(H1)-(H2), problem (1.3) generates a continuous semi-groupSτdefined bySτUn−1=Un.

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Abderrahmane El Hachimi: UFR Math´ematiques Appliqu´ees et Industrielles, D´epartement de Math´ematiques et Informatique, Facult´e des Sciences, Universit´e Chouaib Doukkali, B.P. 20, El Jadida, Morocco

E-mail address:elhachimi@ucd.ac.ma

Moulay Rchid Sidi Ammi: UFR Math´ematiques Appliqu´ees et Industrielles, D´epartement de Math´ematiques et Informatique, Facult´e des Sciences, Universit´e Chouaib Doukkali, B.P. 20, El Ja-dida, Morocco