Well-Posedness and Numerical Study for Solutions of a ...downloads.hindawi.com/journals/ijde/2018/9754567.pdf · 0.0038859 0.0036268 0.0033678 0.0031087 0.0028496 0.0025906 0.0023315

Research ArticleWell-Posedness and Numerical Study for Solutions ofa Parabolic Equation with Variable-Exponent Nonlinearities

Jamal H Al-Smail Salim A Messaoudi and Ala A Talahmeh

Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals PO Box 546Dhahran 31261 Saudi Arabia

Correspondence should be addressed to Salim A Messaoudi messaoudkfupmedusa

Received 11 October 2017 Accepted 25 December 2017 Published 1 March 2018

Academic Editor Dongfang Li

Copyright copy 2018 Jamal H Al-Smail et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider the following nonlinear parabolic equation 119906119905 minus div(|nabla119906|119901(119909)minus2nabla119906) = 119891(119909 119905) where 119891 Ω times (0 119879) rarr R and theexponent of nonlinearity 119901(sdot) are given functions By using a nonlinear operator theory we prove the existence and uniqueness ofweak solutions under suitable assumptions We also give a two-dimensional numerical example to illustrate the decay of solutions

1 Introduction

Let Ω be a bounded domain in R119899 with a smooth boundary120597Ω We consider the following initial and boundary valueproblem

119906119905 minus div (|nabla119906|119901(119909)minus2 nabla119906) = 119891 (119909 119905) in Ω times (0 119879)119906 (119909 119905) = 0 on 120597Ω times (0 119879)119906 (119909 0) = 1199060 (119909) in Ω

(P)

where119891 Ωtimes(0 119879) rarr R and1199060 Ω rarr R are given functionsThe exponent 119901(sdot) is a given measurable function on Ω suchthat

2 lt 1199011 le 119901 (119909) le 1199012 lt +infin (1)

with

1199011 fl essinf119909isinΩ

119901 (119909) 1199012 fl esssup

119909isinΩ

119901 (119909) (2)

We also assume that 119901(sdot) satisfies the log-Holder continuitycondition

1003816100381610038161003816119901 (119909) minus 119901 (119910)1003816100381610038161003816 le minus 119860log 1003816100381610038161003816119909 minus 1199101003816100381610038161003816

forall119909 119910 isin Ω with 1003816100381610038161003816119909 minus 1199101003816100381610038161003816 lt 120575(3)

where 119860 gt 0 and 0 lt 120575 lt 1 are constants The termdiv(|nabla119906|119901(sdot)minus2nabla119906) is called the 119901(sdot)-Laplacian and denoted byΔ 119901(sdot)119906

The study of partial differential equations involvingvariable-exponent nonlinearities has attracted the attentionof researchers in recent years The interest in studying suchproblems is stimulated and motivated by their applicationsin elastic mechanics fluid dynamics nonlinear elasticityelectrorheological fluids and so forth In particular parabolicequations involving the 119901(sdot)-Laplacian are related to thefield of image restoration and electrorheological fluids whichare characterized by their ability to change the mechanicalproperties under the influence of the exterior electromagneticfield The rigorous study of these physical problems has beenfacilitated by the development of the Lebesgue and Sobolevspaces with variable exponents

Regarding parabolic problems with nonlinearities ofvariable-exponent type many works have appeared We note

HindawiInternational Journal of Differential EquationsVolume 2018 Article ID 9754567 9 pageshttpsdoiorg10115520189754567

2 International Journal of Differential Equations

here that most of the results deal with blow-up and globalnonexistence Let us mention some of these works Forinstance Alaoui et al [1] considered the following nonlinearheat equation

119906119905 (119909 119905) minus div (|nabla119906|119898(119909)minus2 nabla119906) = 119906 |119906|119901(119909)minus2 + 119891 (4)

in a bounded domain in Ω sub R119899 (119899 ge 1) with a smoothboundary 120597Ω Under appropriate conditions on the exponentfunctions 119898 119901 and for 119891 = 0 they showed that any solutionwith nontrivial initial datumblows up in finite timeThey alsogave a two-dimensional numerical example to illustrate theirresult Pinasco [2] studied the following problem

119906119905 minus Δ119906 = 119891 (119906) in Ω times [0 119879)119906 (119909 119905) = 0 on 120597Ω times [0 119879)119906 (119909 0) = 1199060 (119909) in Ω

(5)

whereΩ sub R119899 is a bounded domain with a smooth boundary120597Ω and the source term is of the following form

119891 (119906) = 119886 (119909) 119906119901(119909)or 119891 (119906) = 119886 (119909) int

Ω119906119902(119910) (119910 119905) 119889119910 (6)

with 119901 119902 Ω rarr (1infin) and the continuous function 119886Ω rarr R being given functions satisfying specific conditionsThey established the local existence of positive solutions andproved that solutions with initial data sufficiently large blowup in finite time Parabolic problems with sources like theones in (5) appear in several branches of appliedmathematicsand they have been used to model chemical reactions heattransfer or population dynamics

Recently Shangerganesh et al [3] studied the followingfourth-order degenerate parabolic equation

119906119905 + div (|nablaΔ119906|119901(119909)minus2 nablaΔ119906) = 119891 minus div119892 (7)

in a bounded domainΩ sub R119899 (119899 ge 1) with a smooth bound-ary 120597Ω and proved the existence and uniqueness of weaksolutions of (7) by using the difference and variationmethodsunder suitable assumptions on 119891 119892 and the exponents 119901

Equation (P) is a nonlinear diffusion equation which hasbeen used to study image restoration and electrorheologicalfluids (see [4ndash11]) In particular Bendahmane et al [12]proved the well-posedness of a solution for 1198711-data Akagiand Matsuura [13] gave the well-posedness for 1198712 initialdatum and discussed the long-time behaviour of the solutionusing the subdifferential calculus approach In our paper wegive an alternative proof of the well-posedness of (P)which issimpler than that in [13] using a theory of nonlinear evolutionequations In addition we give a numerical example in 2D toillustrate the decay result obtained in [13]

This paper consists of three sections in addition to theintroduction In Section 2 we recall the definitions of thevariable-exponent Lebesgue spaces 119871119901(sdot)(Ω) the Sobolevspaces1198821119901(sdot)(Ω) as well as some of their propertiesWe also

state without proof a proposition to be used in the proof ofour main result In Section 3 we state and prove the well-posedness of solution to our problem In Section 4 we givea numerical verification of the decay result

2 Preliminaries

We present some preliminary facts about the Lebesgue andSobolev spaceswith variable exponents (see [1 14ndash16]) Let119901 Ω rarr [1infin] be a measurable function where Ω is a domainofR119899We define the Lebesgue space with a variable-exponent119901(sdot) by

119871119901(sdot) (Ω) fl 119906 Ω997888rarr R measurable in Ω 984858119871119901(sdot)(Ω) (120582119906)lt infin for some 120582 gt 0

(8)

where

984858119871119901(sdot)(Ω) (119906) = intΩ|119906 (119909)|119901(119909) 119889119909 (9)

is called a modular Equipped with the Luxembourg-typenorm

119906119871119901(sdot)(Ω) fl inf 120582 gt 0 984858119871119901(sdot)(Ω) (119906120582) le 1 (10)

119871119901(sdot)(Ω) is a Banach space (see [10])

Lemma 1 (Holderrsquos inequality [10]) Let 119901 119902 119904 ge 1 be meas-urable functions defined on Ω such that

1119904 (119910) =

1119901 (119910) +

1119902 (119910) (11)

for ae 119910 isin Ω If 119891 isin 119871119901(sdot)(Ω) and 119892 isin 119871119902(sdot)(Ω) then 119891119892 isin119871119904(sdot)(Ω) and

10038171003817100381710038171198911198921003817100381710038171003817119904(sdot) le 2 10038171003817100381710038171198911003817100381710038171003817119901(sdot) 10038171003817100381710038171198921003817100381710038171003817119902(sdot) (12)

Lemma 2 (see [10]) Let 119901 be a measurable function on ΩThen

(a) 119891119901(sdot) le 1 if and only if 984858119901(sdot)(119891) le 1(b) for 119891 isin 119871119901(sdot)(Ω) if 119891119901(sdot) le 1 then 984858119901(sdot)(119891) le 119891119901(sdot)

and if 119891119901(sdot) ge 1 then 119891119901(sdot) le 984858119901(sdot)(119891)(c) 119891119901(sdot) le 1 + 984858119901(sdot)(119891)

Lemma 3 (see [10]) If (1) holds then

min 1199061199011119901(sdot) 1199061199012119901(sdot) le 984858119901(sdot) (119906)le max 1199061199011119901(sdot) 1199061199012119901(sdot)

(13)

for any 119906 isin 119871119901(sdot)(Ω)

International Journal of Differential Equations 3

We next define the variable-exponent Sobolev space1198821119901(sdot)(Ω) as follows1198821119901(sdot) (Ω) = 119906

isin 119871119901(sdot) (Ω) such that nabla119906 exists |nabla119906| isin 119871119901(sdot) (Ω) (14)

This space is a Banach space with respect to the norm1199061198821119901(sdot)(Ω) = 119906119901(sdot) + nabla119906119901(sdot) Furthermore 1198821119901(sdot)0 (Ω) isthe closure of 119862infin0 (Ω) in1198821119901(sdot)(Ω) The dual of1198821119901(sdot)0 (Ω) isdefined as119882minus11199011015840(sdot)(Ω) by the same way as the usual Sobolevspaces where 1119901(sdot) + 11199011015840(sdot) = 1Lemma 4 (see [10]) Let Ω be a bounded domain of R119899 and119901(sdot) satisfies (1) and (3) and then

119906119901(sdot) le 119862 nabla119906119901(sdot) forall119906 isin 1198821119901(sdot)0 (Ω) (15)

where the positive constant 119862 depends on 119901(sdot) and Ω Inparticular the space 1198821119901(sdot)0 (Ω) has an equivalent norm givenby 1199061198821119901(sdot)(Ω) = nabla119906119901(sdot)Lemma 5 (see [10]) If 119901(sdot) isin 119862(Ω) 119902 Ω rarr [1infin) is acontinuous function and

essinf119909isinΩ

(119901lowast (119909) minus 119902 (119909)) gt 0

with 119901lowast (119909) =

119899119901 (119909)esssup119909isinΩ

(119899 minus 119901 (119909)) if 1199012 lt 119899infin if 1199012 ge 119899

(16)

Then the embedding1198821119901(sdot)0 (Ω) 997893rarr 119871119902(sdot)(Ω) is continuous andcompact

Definition 6 (see [17]) Let119881 be a separable Banach space and119867 be a Hilbert space such that 119881 sub 119867 sub 1198811015840 with continuousembedding and 119881 is dense in 119867 Let 119860 119881 rarr 1198811015840 be anonlinear operator

(1) 119860 is said to be monotone if ⟨119860(119906)minus119860(V) 119906minus V⟩1198811015840times119881 ge0 If in addition we have

⟨119860 (119906) minus 119860 (V) 119906 minus V⟩1198811015840times119881 = 0 forall119906 = V (17)

then 119860 is said to be strictly monotone(2) 119860 is said to be bounded if 119860(119878) is bounded in 1198811015840

whenever 119878 is bounded in 119881(3) 119860 is said to be hemicontinuous if the real function

120582 997891997888rarr ⟨119860 (119906 + 120582V) 119908⟩ (18)

is continuous from R to R for any fixed 119906 V 119908 isin 119881We end this section with a proposition which is exactly

like Theorem 71 [17]Proposition 7 Let 1199060 isin 119867 and 119891 isin 1198711199011015840((0 119879) 1198811015840) Supposethat 119860 119881 rarr 1198811015840 is a bounded monotone and hemicontinuous

(nonlinear) operator satisfying for some 120572 120573 gt 0 and for some119901 gt 1⟨119860 (V) V⟩ ge 120572 V119901119881 minus 120573 forallV isin 119881 (19)

Then the following problem

119906119905 + 119860 (119906) = 119891119906 (sdot 0) = 1199060 (20)

has a unique weak solution

119906 isin 119871119901 ((0 119879) 119881) 119908119894119905ℎ 119906119905 isin 1198711199011015840 ((0 119879) 1198811015840) (21)

where 1119901 + 11199011015840 = 13 Well-Posedness

In this section we state and prove the well-posedness of ourproblem

Theorem 8 Let 1199060 isin 1198712(Ω) 119891 isin 1198711199011015840(sdot)((0 119879)119882minus11199011015840(sdot)(Ω))Assume that (1) and (3) hold Then (P) has a unique weaksolution

119906 isin 1198711199011 ((0 119879) 1198821119901(sdot)0 (Ω)) cap 119871infin ((0 119879) 1198712 (Ω)) 119906119905 isin 11987111990110158402 ((0 119879) 119882minus11199011015840(sdot) (Ω))

(22)

where 1119901(sdot) + 11199011015840(sdot) = 1Proof We verify the conditions of Proposition 7 Let 119881 =1198821119901(sdot)0 (Ω) and equip it with the norm

1199061198821119901(sdot)(Ω) = nabla119906119901(sdot) (23)

So 1198811015840 = 119882minus11199011015840(sdot)(Ω) Define 119860 119881 rarr 1198811015840 by119860 (119906) = minusdiv (|nabla119906|119901(119909)minus2 nabla119906) (24)

Boundedness of 119860 For all 119906 V isin 119881

|⟨119860 (119906) V⟩| = 1003816100381610038161003816100381610038161003816intΩ |nabla119906|119901(119909)minus2 nabla119906 sdot nablaV1003816100381610038161003816100381610038161003816

le intΩ|nabla119906|119901(119909)minus1 |nablaV|

(25)

Holder inequality gives

intΩ|nabla119906|119901(119909)minus1 |nablaV| le 2 10038171003817100381710038171003817|nabla119906|119901(sdot)minus1100381710038171003817100381710038171199011015840(sdot) V119881 (26)

Combining (25) and (26) we obtain

119860 (119906)1198811015840 le 2 10038171003817100381710038171003817|nabla119906|119901(sdot)minus1100381710038171003817100381710038171199011015840(sdot) (27)

Then Lemma 2 implies10038171003817100381710038171003817|nabla119906|119901(sdot)minus1100381710038171003817100381710038171199011015840(sdot) le (1 + 984858119901(sdot) (nabla119906)) (28)


Combining (27) and (28) we arrive at

119860 (119906)1198811015840 le 2 (1 + 984858119901(sdot) (nabla119906)) (29)

Let 119878 sub 119881 such that 119906119881 le 119872 for all 119906 isin 119878That is nabla119906119901(sdot) le119872If 119872 le 1 then Lemma 2 implies 984858119901(sdot)(nabla119906) le 1 and (29)

gives 119860(119906)1198811015840 le 4 lt +infinIf 119872 gt 1 then 984858119901(sdot)(nabla119906) le 1198721199012 Thus (29) implies

119860(119906)1198811015840 le 2(1 +1198721199012) lt +infinHence 119860 is bounded

Monotonicity of 119860 Let 119906 V isin 119881Then

⟨119860 (119906) minus 119860 (V) 119906 minus V⟩1198811015840times119881= minusint div (|nabla119906|119901(119909)minus2 nabla119906 minus |nablaV|119901(119909)minus2 nablaV) (119906 minus V) 119889119909= int (|nabla119906|119901(119909)minus2 nabla119906 minus |nablaV|119901(119909)minus2 nablaV) (nabla119906 minus nablaV) 119889119909

(30)

By using the inequality

(|119886|119901(119909)minus2 119886 minus |119887|119901(119909)minus2 119887) sdot (119886 minus 119887) ge 0 (31)

for all 119886 119887 isin R119899 and ae 119909 isin Ω Thus we obtain ⟨119860(119906) minus119860(V) 119906 minus V⟩ ge 0Hence 119860 is monotoneTo verify (19) we note that for all 119906 isin 119881 we have

⟨119860 (119906) 119906⟩ = intΩ|nabla119906|119901(119909) 119889119909 = 984858119901(sdot) (nabla119906) (32)

If 119906119881 = nabla119906119901(sdot) gt 1 then by Lemma 3 we get

984858119901(sdot) (nabla119906) ge nabla1199061199011119901(sdot) (33)

Combining (32) and (33) we easily see that

⟨119860 (119906) 119906⟩ ge 1199061199011119881 (34)

If 119906119881 = nabla119906119901(sdot) le 1 then by Lemma 2 we obtain

⟨119860 (119906) 119906⟩ = 984858119901(sdot) (nabla119906) ge 119906119881 minus 1 ge 1199061199011119881 minus 1 (35)

Therefore we have

⟨119860 (119906) 119906⟩ ge 1199061199011119881 minus 1 forall119906 isin 119881 (36)

Hemicontinuity of 119860 Let 119906 V 119908 isin 119881 be fixed Let

119892 (120582) = ⟨119860 (119906 + 120582V) 119908⟩= intΩ|nabla119906 + 120582nablaV|119901(119909)minus2 (nabla119906 + 120582nablaV) sdot nabla119908119889119909 (37)

Let 120582119896 rarr 120582 (real) and consider

119892 (120582119896) = intΩ

1003816100381610038161003816nabla119906 + 120582119896nablaV1003816100381610038161003816119901(119909)minus2 (nabla119906 + 120582119896nablaV) sdot nabla119908119889119909 (38)

Since

1003816100381610038161003816nabla119906 + 120582119896nablaV1003816100381610038161003816119901(119909)minus2 (nabla119906 + 120582119896nablaV) sdot nabla119908997888rarr |nabla119906 + 120582nablaV|119901(119909)minus2 (nabla119906 + 120582nablaV) sdot nabla119908 (39)

for ae 119909 isin Ω and

1003816100381610038161003816nabla119906 + 120582119896nablaV1003816100381610038161003816119901(119909)minus1 |nabla119908|le 119862 (|nabla119906|119901(119909)minus1 |nabla119908| + |nablaV|119901(119909)minus1 |nabla119908|) isin 1198711 (Ω) (40)

where 119862 = max21199012minus2 21199012minus2(1 + |119872|1199012minus1) gt 0 then by theclassical dominated convergence theorem

119892 (120582119896) 997888rarr 119892 (120582) as 119896 997888rarr infin (41)

Hence 119860 is hemicontinuousTherefore conditions of Proposition 7 are satisfied and

problem (P) has a unique solution4 Numerical Study

In this section we present some numerical results andapplications of the problem

119906119905 minus div (|nabla119906|119901(119909119910)minus2 nabla119906) = 0 in 119876 = Ω times (0 119879)119906 = 0

on 120597119876 = 120597Ω times (0 119879)119906 (119909 119910 0) = 1199060 (119909 119910) in Ω

(119875lowast)

which is a well-posed problem due toTheorem 8 Our objec-tive is to provide a numerical verification of the followingdecay result

Proposition 9 (see [13]) Assume that (1) and (3) hold Thenthe solution of (119875lowast) satisfies the following

(i) If 1199012 = 2 then there exists a constant 1198882 gt 0 such that119906 (119905)2 le 1003817100381710038171003817119906010038171003817100381710038172 119890minus1198882119905 forall119905 ge 0 (42)

(ii) If 1199012 gt 2 there exists a constant 1198881 gt 0 and 1199051 ge 0 suchthat

119906 (119905)2 le 1003817100381710038171003817119906010038171003817100381710038172 (1 + 1198881119905)minus1(1199012minus2) forall119905 ge 1199051 (43)

We consider two applications to illustrate numerically anexponential decay for the case 119901(119909 119910) = 2 and a polynomialdecay for an exponent function 119901(sdot) satisfying conditions(1)ndash(3)

For this purpose we introduce a numerical scheme for(119875lowast) prove its convergence in Section 41 and show the decayresults in Section 42


41 Numerical Method In this part we present a linearizednumerical scheme to obtain the numerical results of thesystem (119875lowast) and confirm the decay resultsThe system is fullydiscretized through a finite difference method for the timevariable and a finite element Galerkin method for the spacevariable Useful background about the numerical and erroranalysis of these methods is found in [18] More interestinglyin [19] Li and Wang introduced a numerical scheme to solvestrongly nonlinear parabolic systems and proved uncondi-tional error estimates of the scheme Our problem (119875lowast) ishighly nonlinear due to the presence of the gradient andnonlinear exponent in the diffusivity coefficient which canbe zero inside the spatial domain Below we introduce ournumerical scheme for the purpose of confirming the decayresults

The parabolic equation

119906119905 minus div (|nabla119906|119901(119909119910)minus2 nabla119906) = 0 in 119876 = Ω times (0 119879) (44)

is discretized using finite differences for the time derivativeand a finite element method for the 119901(sdot)minusLaplacian termFor this we divide the time interval [0 119879] into 119873 equalsubintervals by

119905119899 = 119899120591 120591 = 119879119873 (45)

and denote by

119906119899 (119909 119910) fl 119906 (119909 119910 119905119899) 119899 = 0 1 119873 (46)

The term 119906119905 is approximated using the first-order forwardfinite difference formula

119906119899+1119905 fl119906119899+1 minus 119906119899

120591 (47)

Semidiscrete Problem A linear semidiscrete formulation of(119875lowast) takes the following form given 119901 and 1199060 find 1199061 1199062 119906119899+1 such that

119906119899+1119905 minus div (1003816100381610038161003816nabla1199061198991003816100381610038161003816119901(119909119910)minus2 nabla119906119899+1) = 0 in Ω119906119899+1 = 0 on 120597Ω1199060 = 1199060 (119909 119910) in Ω

(48)

This problem is elliptic and admits a unique solution [20] forevery 119899 = 0 1 119873 Also the Rothe approximation 119906(119899) tothe exact solution 119906 given by

119906(119899) (119909 119910 119905) fl 119906119894minus1 (119909 119910) + (119905 minus 119905119894minus1) 119906119894 minus 119906119894minus1120591

119905 isin [119905119894minus1 119905119894] 119894 = 1 2 119899(49)

is well defined and 119906(119899) rarr 119906 in 1198712(Ω) as 120591 rarr 0 see [1]Full-Discrete ProblemThe variable119906119899+1 is discretized in spaceby a finite element method For this letΩℎ be a triangulationof Ω with a maximal element size ℎ Let also Vℎ be a test

minus500

minus400

minus300

minus200

minus100

000

100

200

300

400

500

minus500

minus400

minus300

minus200

minus100

000

000

100

200

300400

500

Y

Z

X

Figure 1 Mesh of the domain [minus5 5] times [minus5 5]

function in the linear Lagrangian space 1198751(Ωℎ) such that Vℎ =0 on 120597ΩℎThe semidiscrete problem is then written in a weak form

to define the full-discrete problem given 119901ℎ 119906119899 isin 1198751(Ωℎ)find 119906119899+1ℎ isin 1198751(Ωℎ) such that

intΩℎ

(119906119899+1ℎ minus 119906119899ℎ120591 Vℎ + 1003816100381610038161003816nabla119906119899ℎ1003816100381610038161003816119901ℎ(119909119910)minus2 nabla119906119899+1ℎ sdot nablaVℎ)119889Ωℎ= 0 forallVℎ isin 1198751 (Ωℎ)

(50)

For 119901ℎ ge 2 the above problem has a unique solution 119906119899+1ℎ isin11986710 (Ωℎ) for every nontrivial 119906119899ℎ isin 1198671(Ωℎ) This follows fromthe Lax-Milgram Lemma and the Galerkin approximation119906119899+1ℎ converges to 119906119899+1 in11986710 (Ωℎ) as ℎ rarr 0 see [18]42 Numerical Results In this subsection we present thefollowing numerical applications of (119875lowast)

(1) Exponential decay for 119901ℎ(119909 119910) = 2 we show forsome 119888 gt 0 that

1198921 (119905) fl1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus119888119905 forall119905 ge 0 (51)

(2) Polynomial decay for 119901ℎ(119909 119910) = (15)lceil119909rceil2 + 25 weshow for some 119888 gt 0 and 1199050 gt 0 that1198922 (119905) fl

1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119888119905)minus1(1199012minus2) forall119905 ge 1199050 (52)

Here lceilsdotrceil denote the greatest integer functionIn both applications we set the following parameters

119879 = 100120591 = 01ℎ = 01

Ωℎ = [minus5 5] times [minus5 5] (53)

Figure 1 shows the mesh used for Ωℎ which involves23702 triangles and 12052 vertices


1

095

09

085

08

075

07

065

06

055

05

045

04

035

03

025

02

015

01

005

13888e minus 011

Figure 2 Initial condition 1199060ℎ(119909 119910) = 119890minus05(1199092+1199102)

0082556

0078429

0074301

0070173

0066045

0061917

0057789

0053662

0049534

0045406

0041278

003715

0033023

0028895

0024767

0020639

0016511

0012383

00082556

00041278

28278e minus 038

(a) 1199065ℎ

0031551

0029974

0028396

0026819

0025241

0023664

0022086

0020508

0018931

0017353

0015776

0014198

0012621

0011043

00094654

00078879

00063103

00047327

00031551

00015776

13558e minus 038

(b) 11990610ℎ

22716e minus 039

00051812

00049221

0004663

0004404

00041449

00038859

00036268

00033678

00031087

00028496

00025906

00023315

00020725

00018134

00015543

00012953

00010362

000077717

000051812

000025906

(c) 11990620ℎ

10126e minus 041

23298e minus 00522133e minus 00520968e minus 00519803e minus 00518638e minus 00517474e minus 00516309e minus 00515144e minus 00513979e minus 00512814e minus 00511649e minus 00510484e minus 00593192e minus 00681543e minus 00669894e minus 00658245e minus 00646596e minus 00634947e minus 00623289e minus 00611649e minus 006

(d) 11990650ℎ

Figure 3 Numerical solutions for Application 1


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0

(a) Solid line 119910 = 119890minus01119905 dashed 119910 = 1198921(119905)0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905

Figure 4 Exponential decay

75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25

Figure 5 119901ℎ(119909 119910) = (15)lceil119909rceil2 + 25

The initial condition is taken to be 1199060ℎ(119909 119910) = 119890minus05(1199092+1199102)and projected into 1198751(Ωℎ) see Figure 2

The numerical results are obtained using the noncom-mercial software FreeFem++ [21]

Application 1 119901ℎ(119909 119910) = 2 satisfies the required conditions(1)ndash(3) Figure 3 shows the numerical solutions for 119905 = 5 119905 =10 119905 = 20 and 119905 = 50

With 119888 = 01 Figure 4(a) shows that 119906119899ℎ decays exponen-tially as

1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)

This is also confirmed by Figure 4(b) that shows the ratio119910 = 1198921(119905)119890minus01119905 is less than one and remains decreasing for alarge value of 119879Application 2 The exponent function 119901ℎ(119909 119910) = (15)lceil119909rceil2 +25 in Figure 5 satisfies the required conditions (1)ndash(3) as

(i) 1199011 = 25 1199012 = 75

(ii) |119901(119909 119910) minus 119901(1199090 1199100)| = (15)|lceil119909rceil2 minus lceil1199090rceil2| leminus(20radic2log(1120575))log|119909 minus 119910| for |119909 minus 119910| lt 120575 with0 lt 120575 lt 1Figure 6 shows the numerical solutions for 119905 = 5 119905 = 10119905 = 20 and 119905 = 50In this case the solution 119906119899ℎ has a polynomial decay With119888 = 1 Figure 7(a) shows that

1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)

This is also confirmed by Figure 7(b) which shows that theratio 119910 = 1198922(119905)(1 + 119905)minus211 remains less than one anddecreasing until 119879

We conclude that the numerical results in above applica-tions verify Proposition 9

Conflicts of Interest

The authors declare that they have no conflicts of interest


03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

(a) Solid line 119910 = (1 + 119905)minus211 dashed 119910 = 1198922(119905)0 10 20 30 40 50 60 70 80 90 100

1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211

Figure 7 Polynomial decay


Acknowledgments

The authors acknowledge King Fahd University of Petroleumand Minerals for its support This work is sponsored byKFUPM under Project no FT 161004

References

[1] M K Alaoui S A Messaoudi and H B Khenous ldquoA blow-up result for nonlinear generalized heat equationrdquo ComputersampMathematics with Applications An International Journal vol68 no 12 part A pp 1723ndash1732 2014

[2] J P Pinasco ldquoBlow-up for parabolic and hyperbolic problemswith variable exponentsrdquo Nonlinear Analysis Theory Methodsamp Applications An International Multidisciplinary Journal vol71 no 3-4 pp 1094ndash1099 2009

[3] L Shangerganesh A Gurusamy and K Balachandran ldquoWeaksolutions for nonlinear parabolic equations with variable expo-nentsrdquo Communications in Mathematics vol 25 no 1 pp 55ndash70 2017

[4] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[5] E Acerbi and G Mingione ldquoRegularity results for stationaryelectro-rheological fluidsrdquo Archive for Rational Mechanics andAnalysis vol 164 no 3 pp 213ndash259 2002

[6] Y Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006

[7] L Diening F Ettwein and M Ruzicka ldquoC1120572-regularity forelectrorheological fluids in two dimensionsrdquo Nonlinear Differ-ential Equations and Applications NoDEA vol 14 no 1-2 pp207ndash217 2007

[8] F Ettwein and M Ruzicka ldquoExistence of local strong solutionsfor motions of electrorheological fluids in three dimensionsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 53 no 3-4 pp 595ndash604 2007

[9] Z Guo Q Liu J Sun and B Wu ldquoReaction-diffusion systemswith 119901(119909)-growth for image denoisingrdquo Nonlinear AnalysisReal World Applications vol 12 no 5 pp 2904ndash2918 2011

[10] L Diening P Harjulehto P Hasto and M Ruzicka ldquoLebesgueand Sobolev spaces with variable exponentsrdquo Lecture Notes inMathematics vol 2017 pp 1ndash518 2011

[11] M Ruzicka Electrorheological Fluids Modeling and Mathemat-ical Theory vol 1748 of Lecture Notes in Mathematics SpringerBerlin Germany 2000

[12] M Bendahmane P Wittbold and A Zimmermann ldquoRenor-malized solutions for a nonlinear parabolic equation with vari-able exponents and 1198711-datardquo Journal of Differential Equationsvol 249 no 6 pp 1483ndash1515 2010

[13] G Akagi and K Matsuura ldquoWell-posedness and large-timebehaviors of solutions for a parabolic equation involving 119901(119909)-Laplacianrdquo Discrete and Continuous Dynamical Systems - SeriesA Dynamical systems differential equations and applications8th AIMS Conference Suppl Vol I pp 22ndash31 2011

[14] D E Edmunds and J Rakosnik ldquoSobolev embeddings withvariable exponentrdquo StudiaMathematica vol 143 no 3 pp 267ndash293 2000

[15] D E Edmunds and J Rakosnik ldquoSobolev embeddings withvariable exponent IIrdquoMathematische Nachrichten vol 246247pp 53ndash67 2002

[16] X Fan and D Zhao ldquoOn the spaces 119871119901(119909)(Ω) and 119882119898119901(119909)(Ω)rdquoJournal of Mathematical Analysis and Applications vol 263 no2 pp 424ndash446 2001

[17] L Simon Application of Monotone Type Operators to NonlinearPDEs Prime Rate Kft Budapest Hungary 2013

[18] A Quarteroni and A Valli Numerical approximation of partialdifferential equations vol 23 of Springer Series in ComputationalMathematics Springer-Verlag Berlin Germany 1994

[19] D Li and J Wang ldquoUnconditionally optimal error analysisof Crank-Nicolson Galerkin FEMs for a strongly nonlinearparabolic systemrdquo Journal of Scientific Computing vol 72 no2 pp 892ndash915 2017

[20] J Leray and J-L Lions ldquoQuelques resulatats de Visick sur lesproblemes elliptiques nonlineaires par les methodes de Minty-Browderrdquo Bulletin de la Societe Mathematique de France vol 93pp 97ndash107 1965

[21] F Hecht Freefem++ manual http httpwwwfreefemorg

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


here that most of the results deal with blow-up and globalnonexistence Let us mention some of these works Forinstance Alaoui et al [1] considered the following nonlinearheat equation

119906119905 (119909 119905) minus div (|nabla119906|119898(119909)minus2 nabla119906) = 119906 |119906|119901(119909)minus2 + 119891 (4)

in a bounded domain in Ω sub R119899 (119899 ge 1) with a smoothboundary 120597Ω Under appropriate conditions on the exponentfunctions 119898 119901 and for 119891 = 0 they showed that any solutionwith nontrivial initial datumblows up in finite timeThey alsogave a two-dimensional numerical example to illustrate theirresult Pinasco [2] studied the following problem

119906119905 minus Δ119906 = 119891 (119906) in Ω times [0 119879)119906 (119909 119905) = 0 on 120597Ω times [0 119879)119906 (119909 0) = 1199060 (119909) in Ω

(5)

whereΩ sub R119899 is a bounded domain with a smooth boundary120597Ω and the source term is of the following form

119891 (119906) = 119886 (119909) 119906119901(119909)or 119891 (119906) = 119886 (119909) int

Ω119906119902(119910) (119910 119905) 119889119910 (6)

with 119901 119902 Ω rarr (1infin) and the continuous function 119886Ω rarr R being given functions satisfying specific conditionsThey established the local existence of positive solutions andproved that solutions with initial data sufficiently large blowup in finite time Parabolic problems with sources like theones in (5) appear in several branches of appliedmathematicsand they have been used to model chemical reactions heattransfer or population dynamics

Recently Shangerganesh et al [3] studied the followingfourth-order degenerate parabolic equation

119906119905 + div (|nablaΔ119906|119901(119909)minus2 nablaΔ119906) = 119891 minus div119892 (7)

in a bounded domainΩ sub R119899 (119899 ge 1) with a smooth bound-ary 120597Ω and proved the existence and uniqueness of weaksolutions of (7) by using the difference and variationmethodsunder suitable assumptions on 119891 119892 and the exponents 119901

Equation (P) is a nonlinear diffusion equation which hasbeen used to study image restoration and electrorheologicalfluids (see [4ndash11]) In particular Bendahmane et al [12]proved the well-posedness of a solution for 1198711-data Akagiand Matsuura [13] gave the well-posedness for 1198712 initialdatum and discussed the long-time behaviour of the solutionusing the subdifferential calculus approach In our paper wegive an alternative proof of the well-posedness of (P)which issimpler than that in [13] using a theory of nonlinear evolutionequations In addition we give a numerical example in 2D toillustrate the decay result obtained in [13]

This paper consists of three sections in addition to theintroduction In Section 2 we recall the definitions of thevariable-exponent Lebesgue spaces 119871119901(sdot)(Ω) the Sobolevspaces1198821119901(sdot)(Ω) as well as some of their propertiesWe also

state without proof a proposition to be used in the proof ofour main result In Section 3 we state and prove the well-posedness of solution to our problem In Section 4 we givea numerical verification of the decay result

2 Preliminaries

We present some preliminary facts about the Lebesgue andSobolev spaceswith variable exponents (see [1 14ndash16]) Let119901 Ω rarr [1infin] be a measurable function where Ω is a domainofR119899We define the Lebesgue space with a variable-exponent119901(sdot) by

119871119901(sdot) (Ω) fl 119906 Ω997888rarr R measurable in Ω 984858119871119901(sdot)(Ω) (120582119906)lt infin for some 120582 gt 0

(8)

where

984858119871119901(sdot)(Ω) (119906) = intΩ|119906 (119909)|119901(119909) 119889119909 (9)

is called a modular Equipped with the Luxembourg-typenorm

119906119871119901(sdot)(Ω) fl inf 120582 gt 0 984858119871119901(sdot)(Ω) (119906120582) le 1 (10)

119871119901(sdot)(Ω) is a Banach space (see [10])

Lemma 1 (Holderrsquos inequality [10]) Let 119901 119902 119904 ge 1 be meas-urable functions defined on Ω such that

1119904 (119910) =

1119901 (119910) +

1119902 (119910) (11)

for ae 119910 isin Ω If 119891 isin 119871119901(sdot)(Ω) and 119892 isin 119871119902(sdot)(Ω) then 119891119892 isin119871119904(sdot)(Ω) and

10038171003817100381710038171198911198921003817100381710038171003817119904(sdot) le 2 10038171003817100381710038171198911003817100381710038171003817119901(sdot) 10038171003817100381710038171198921003817100381710038171003817119902(sdot) (12)

Lemma 2 (see [10]) Let 119901 be a measurable function on ΩThen

(a) 119891119901(sdot) le 1 if and only if 984858119901(sdot)(119891) le 1(b) for 119891 isin 119871119901(sdot)(Ω) if 119891119901(sdot) le 1 then 984858119901(sdot)(119891) le 119891119901(sdot)

and if 119891119901(sdot) ge 1 then 119891119901(sdot) le 984858119901(sdot)(119891)(c) 119891119901(sdot) le 1 + 984858119901(sdot)(119891)

Lemma 3 (see [10]) If (1) holds then

min 1199061199011119901(sdot) 1199061199012119901(sdot) le 984858119901(sdot) (119906)le max 1199061199011119901(sdot) 1199061199012119901(sdot)

(13)

for any 119906 isin 119871119901(sdot)(Ω)







essinf119909isinΩ

(119901lowast (119909) minus 119902 (119909)) gt 0

with 119901lowast (119909) =

119899119901 (119909)esssup119909isinΩ


(16)







120582 997891997888rarr ⟨119860 (119906 + 120582V) 119908⟩ (18)





119906119905 + 119860 (119906) = 119891119906 (sdot 0) = 1199060 (20)


119906 isin 119871119901 ((0 119879) 119881) 119908119894119905ℎ 119906119905 isin 1198711199011015840 ((0 119879) 1198811015840) (21)





(22)


1199061198821119901(sdot)(Ω) = nabla119906119901(sdot) (23)





(25)








119860 (119906)1198811015840 le 2 (1 + 984858119901(sdot) (nabla119906)) (29)






(30)








⟨119860 (119906) 119906⟩ ge 1199061199011119881 (34)



Therefore we have





119892 (120582119896) = intΩ


Since










on 120597119876 = 120597Ω times (0 119879)119906 (119909 119910 0) = 1199060 (119909 119910) in Ω

(119875lowast)













119905119899 = 119899120591 120591 = 119879119873 (45)

and denote by

119906119899 (119909 119910) fl 119906 (119909 119910 119905119899) 119899 = 0 1 119873 (46)


119906119899+1119905 fl119906119899+1 minus 119906119899

120591 (47)



(48)



119905 isin [119905119894minus1 119905119894] 119894 = 1 2 119899(49)


minus500

minus400

minus300

minus200

minus100

000

100

200

300

400

500

minus500

minus400

minus300

minus200

minus100

000

000

100

200

300400

500

Y

Z

X




intΩℎ


(50)







119879 = 100120591 = 01ℎ = 01




1

095

09

085

08

075

07

065

06

055

05

045

04

035

03

025

02

015

01

005

13888e minus 011


0082556

0078429

0074301

0070173

0066045

0061917

0057789

0053662

0049534

0045406

0041278

003715

0033023

0028895

0024767

0020639

0016511

0012383

00082556

00041278

28278e minus 038

(a) 1199065ℎ

0031551

0029974

0028396

0026819

0025241

0023664

0022086

0020508

0018931

0017353

0015776

0014198

0012621

0011043

00094654

00078879

00063103

00047327

00031551

00015776

13558e minus 038

(b) 11990610ℎ

22716e minus 039

00051812

00049221

0004663

0004404

00041449

00038859

00036268

00033678

00031087

00028496

00025906

00023315

00020725

00018134

00015543

00012953

00010362

000077717

000051812

000025906

(c) 11990620ℎ

10126e minus 041


(d) 11990650ℎ



0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0


1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905


75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25






1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)


(i) 1199011 = 25 1199012 = 75


1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)






03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018














essinf119909isinΩ

(119901lowast (119909) minus 119902 (119909)) gt 0

with 119901lowast (119909) =

119899119901 (119909)esssup119909isinΩ


(16)







120582 997891997888rarr ⟨119860 (119906 + 120582V) 119908⟩ (18)





119906119905 + 119860 (119906) = 119891119906 (sdot 0) = 1199060 (20)


119906 isin 119871119901 ((0 119879) 119881) 119908119894119905ℎ 119906119905 isin 1198711199011015840 ((0 119879) 1198811015840) (21)





(22)


1199061198821119901(sdot)(Ω) = nabla119906119901(sdot) (23)





(25)








119860 (119906)1198811015840 le 2 (1 + 984858119901(sdot) (nabla119906)) (29)






(30)








⟨119860 (119906) 119906⟩ ge 1199061199011119881 (34)



Therefore we have





119892 (120582119896) = intΩ


Since










on 120597119876 = 120597Ω times (0 119879)119906 (119909 119910 0) = 1199060 (119909 119910) in Ω

(119875lowast)













119905119899 = 119899120591 120591 = 119879119873 (45)

and denote by

119906119899 (119909 119910) fl 119906 (119909 119910 119905119899) 119899 = 0 1 119873 (46)


119906119899+1119905 fl119906119899+1 minus 119906119899

120591 (47)



(48)



119905 isin [119905119894minus1 119905119894] 119894 = 1 2 119899(49)


minus500

minus400

minus300

minus200

minus100

000

100

200

300

400

500

minus500

minus400

minus300

minus200

minus100

000

000

100

200

300400

500

Y

Z

X




intΩℎ


(50)







119879 = 100120591 = 01ℎ = 01




1

095

09

085

08

075

07

065

06

055

05

045

04

035

03

025

02

015

01

005

13888e minus 011


0082556

0078429

0074301

0070173

0066045

0061917

0057789

0053662

0049534

0045406

0041278

003715

0033023

0028895

0024767

0020639

0016511

0012383

00082556

00041278

28278e minus 038

(a) 1199065ℎ

0031551

0029974

0028396

0026819

0025241

0023664

0022086

0020508

0018931

0017353

0015776

0014198

0012621

0011043

00094654

00078879

00063103

00047327

00031551

00015776

13558e minus 038

(b) 11990610ℎ

22716e minus 039

00051812

00049221

0004663

0004404

00041449

00038859

00036268

00033678

00031087

00028496

00025906

00023315

00020725

00018134

00015543

00012953

00010362

000077717

000051812

000025906

(c) 11990620ℎ

10126e minus 041


(d) 11990650ℎ



0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0


1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905


75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25






1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)


(i) 1199011 = 25 1199012 = 75


1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)






03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018










119860 (119906)1198811015840 le 2 (1 + 984858119901(sdot) (nabla119906)) (29)






(30)








⟨119860 (119906) 119906⟩ ge 1199061199011119881 (34)



Therefore we have





119892 (120582119896) = intΩ


Since










on 120597119876 = 120597Ω times (0 119879)119906 (119909 119910 0) = 1199060 (119909 119910) in Ω

(119875lowast)













119905119899 = 119899120591 120591 = 119879119873 (45)

and denote by

119906119899 (119909 119910) fl 119906 (119909 119910 119905119899) 119899 = 0 1 119873 (46)


119906119899+1119905 fl119906119899+1 minus 119906119899

120591 (47)



(48)



119905 isin [119905119894minus1 119905119894] 119894 = 1 2 119899(49)


minus500

minus400

minus300

minus200

minus100

000

100

200

300

400

500

minus500

minus400

minus300

minus200

minus100

000

000

100

200

300400

500

Y

Z

X




intΩℎ


(50)







119879 = 100120591 = 01ℎ = 01




1

095

09

085

08

075

07

065

06

055

05

045

04

035

03

025

02

015

01

005

13888e minus 011


0082556

0078429

0074301

0070173

0066045

0061917

0057789

0053662

0049534

0045406

0041278

003715

0033023

0028895

0024767

0020639

0016511

0012383

00082556

00041278

28278e minus 038

(a) 1199065ℎ

0031551

0029974

0028396

0026819

0025241

0023664

0022086

0020508

0018931

0017353

0015776

0014198

0012621

0011043

00094654

00078879

00063103

00047327

00031551

00015776

13558e minus 038

(b) 11990610ℎ

22716e minus 039

00051812

00049221

0004663

0004404

00041449

00038859

00036268

00033678

00031087

00028496

00025906

00023315

00020725

00018134

00015543

00012953

00010362

000077717

000051812

000025906

(c) 11990620ℎ

10126e minus 041


(d) 11990650ℎ



0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0


1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905


75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25






1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)


(i) 1199011 = 25 1199012 = 75


1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)






03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018













119905119899 = 119899120591 120591 = 119879119873 (45)

and denote by

119906119899 (119909 119910) fl 119906 (119909 119910 119905119899) 119899 = 0 1 119873 (46)


119906119899+1119905 fl119906119899+1 minus 119906119899

120591 (47)



(48)



119905 isin [119905119894minus1 119905119894] 119894 = 1 2 119899(49)


minus500

minus400

minus300

minus200

minus100

000

100

200

300

400

500

minus500

minus400

minus300

minus200

minus100

000

000

100

200

300400

500

Y

Z

X




intΩℎ


(50)







119879 = 100120591 = 01ℎ = 01




1

095

09

085

08

075

07

065

06

055

05

045

04

035

03

025

02

015

01

005

13888e minus 011


0082556

0078429

0074301

0070173

0066045

0061917

0057789

0053662

0049534

0045406

0041278

003715

0033023

0028895

0024767

0020639

0016511

0012383

00082556

00041278

28278e minus 038

(a) 1199065ℎ

0031551

0029974

0028396

0026819

0025241

0023664

0022086

0020508

0018931

0017353

0015776

0014198

0012621

0011043

00094654

00078879

00063103

00047327

00031551

00015776

13558e minus 038

(b) 11990610ℎ

22716e minus 039

00051812

00049221

0004663

0004404

00041449

00038859

00036268

00033678

00031087

00028496

00025906

00023315

00020725

00018134

00015543

00012953

00010362

000077717

000051812

000025906

(c) 11990620ℎ

10126e minus 041


(d) 11990650ℎ



0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0


1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905


75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25






1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)


(i) 1199011 = 25 1199012 = 75


1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)






03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









1

095

09

085

08

075

07

065

06

055

05

045

04

035

03

025

02

015

01

005

13888e minus 011


0082556

0078429

0074301

0070173

0066045

0061917

0057789

0053662

0049534

0045406

0041278

003715

0033023

0028895

0024767

0020639

0016511

0012383

00082556

00041278

28278e minus 038

(a) 1199065ℎ

0031551

0029974

0028396

0026819

0025241

0023664

0022086

0020508

0018931

0017353

0015776

0014198

0012621

0011043

00094654

00078879

00063103

00047327

00031551

00015776

13558e minus 038

(b) 11990610ℎ

22716e minus 039

00051812

00049221

0004663

0004404

00041449

00038859

00036268

00033678

00031087

00028496

00025906

00023315

00020725

00018134

00015543

00012953

00010362

000077717

000051812

000025906

(c) 11990620ℎ

10126e minus 041


(d) 11990650ℎ



0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0


1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905


75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25






1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)


(i) 1199011 = 25 1199012 = 75


1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)






03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02

01

0


1

09

08

07

06

05

04

03

02

01

0

(b) 119910 = 1198921(119905)119890minus01119905


75

725

7

675

65

625

6

575

55

525

5

475

45

425

4

375

35

325

3

275

25






1198921 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le 119890minus01119905 0 le 119905 le 119879 (54)


(i) 1199011 = 25 1199012 = 75


1198922 (119905) =1003817100381710038171003817119906119899ℎ100381710038171003817100381710038171003817100381710038171199060ℎ1003817100381710038171003817 le (1 + 119905)minus211 0 le 119905 le 119879 (55)






03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









03065902910902755802600802445802290702135701980701825601670601515501360501205501050400895400740370058533004302900275260012022minus00034815

(a) 1199065ℎ

02523102396402269702143020164018897017630163630150960138290125620112960100290087618007494900622810049612003694300242750011606minus00010623

(b) 11990610ℎ

020635019593018551017509016467015425014383013341012299011257010215009172700813060070885006046500500440039623002920200187820008361minus00020598

(c) 11990620ℎ

01616401533901451401368801286301203801121201038700956170087363007911007085600626030054350046096003784300295890021336001308200048291minus00034243

(d) 11990650ℎ


0 10 20 30 40 50 60 70 80 90 100

1

09

08

07

06

05

04

03

02


1

095

09

085

08

075

07

065

(b) 119910 = 1198922(119905)(1 + 119905)minus211



Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Well-Posedness and Numerical Study for Solutions of a ...downloads.hindawi.com/journals/ijde/2018/9754567.pdf · 0.0038859 0.0036268 0.0033678 0.0031087 0.0028496 0.0025906 0.0023315