Research Article Blow-Up Phenomena for Nonlinear Reaction ...downloads.hindawi.com/journals/jfs/2016/8107657.pdf · reaction,nonlineardiusion,nonlinearconvection,andnon-linear boundary

Research ArticleBlow-Up Phenomena for Nonlinear Reaction-DiffusionEquations under Nonlinear Boundary Conditions

Juntang Ding

School of Mathematical Sciences Shanxi University Taiyuan 030006 China

Correspondence should be addressed to Juntang Ding djuntangsxueducn

Received 20 November 2015 Accepted 6 March 2016

Academic Editor Leszek Olszowy

Copyright copy 2016 Juntang DingThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinearboundary conditions (119892(119906))

119905= nabla sdot (119886(119906)nabla119906) + 119891(119906) in Ω times (0 119879) 120597119906120597119899 = 119887(119909 119906 119905) on 120597Ω times (0 119879) 119906(119909 0) = 119906

0(119909) gt 0 in Ω

where Ω sub R119873 (119873 ge 2) is a bounded domain with smooth boundary 120597Ω We obtain the conditions under which the solutionseither exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles Moreover theupper estimates of the ldquoblow-up timerdquo the ldquoblow-up raterdquo and the global solutions are also given

1 Introduction

During the past few decades the blow-up phenomena forthe nonlinear reaction-diffusion equations have been studiedby a large number of authors and the reader is referred to[1ndash8] and the references therein In this paper we considerthe following nonlinear reaction-diffusion problem undernonlinear conditions

(119892 (119906))119905= nabla sdot (119886 (119906) nabla119906) + 119891 (119906) in Ω times (0 119879)

120597119906

120597119899= 119887 (119909 119906 119905) on 120597Ω times (0 119879)

119906 (119909 0) = 1199060(119909) gt 0 in Ω

(1)

where Ω sub R119873 (119873 ge 2) is a bounded domain withsmooth boundary 120597Ω 120597119906120597119899 represents the outward normalderivative on 120597Ω and 119879 is the maximal existence time of119906 Set R

+fl (0 +infin) We assume throughout the paper

that 119886(119904) is a positive 1198622(R+) function 119892(119904) is a 1198622(R

+)

function 1198921015840(119904) gt 0 for any 119904 isin R+ 119891(119904) is a nonnegative

1198621(R+) function 119887(119909 119904 119905) is a nonnegative1198621(120597ΩtimesR

+timesR+)

function and 1199060(119909) is a positive1198622(R

+) function and satisfies

the compatibility conditions Under the above assumptionsthe local existence and uniqueness of classical solution ofproblem (1) were established by Amann [9] Furthermore itfollows frommaximum principle [10] and regularity theorem

[11] that the solution 119906(119909 119905) is positive and 119906(119909 119905) isin 1198623(Ω times

(0 119879)) cap 1198622(Ω times (0 119879))Many authors have investigated blow-up and global

solutions of nonlinear reaction-diffusion equations undernonlinear boundary conditions and have obtained a lot ofinteresting results (see eg [12ndash20]) Tomy knowledge somespecial cases of (1) have been studied Zhang [21] consideredthe following problem

(119892 (119906))119905= Δ119906 + 119891 (119906) in Ω times (0 119879)

120597119906

120597119899= 119887 (119906) on 120597Ω times (0 119879)

119906 (119909 0) = 1199060(119909) gt 0 in Ω

(2)

where Ω sub R119873 (119873 ge 2) is a bounded domain with smoothboundary 120597Ω By constructing auxiliary functions and usingmaximum principles the existence of blow-up and globalsolutions were obtained under appropriate assumptions onthe functions 119887 119891 119892 and 119906

0 Zhang et al [22] dealt with the

following problem

(119892 (119906))119905= nabla sdot (119886 (119906) nabla119906) + 119891 (119906) in Ω times (0 119879)

120597119906

120597119899= 119887 (119906) on 120597Ω times (0 119879)

119906 (119909 0) = 1199060(119909) gt 0 in Ω

(3)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 8107657 7 pageshttpdxdoiorg10115520168107657

2 Journal of Function Spaces

where Ω sub R119873 (119873 ge 2) is a bounded domain with smoothboundary 120597Ω Some conditions on nonlinearities and theinitial data were given to ensure that 119906(119909 119905) exists globallyor blows up at some finite time 119879 In addition the upperestimates of the global solution the ldquoblow-up timerdquo and theldquoblow-up raterdquo were also established

In this paper we study reaction-diffusion problem (1) It iswell known that 119891(119906) 119892(119906) 119886(119906) and 119887(119909 119906 119905) are nonlinearreaction nonlinear diffusion nonlinear convection and non-linear boundary flux respectively What interactions amongthe four nonlinear mechanisms result in the blow-up andglobal solutions of (1) is investigated in this work We notethat the boundary flux function 119887(119909 119906 119905) depends not onlyon the concentration variable 119906 but also on the space variable119909 and the time variable 119905 Hence it seems that the methodsof [21 22] are not applicable for problem (1) In this paperby constructing completely different auxiliary functions fromthose in [21 22] and technically using maximum principleswe obtain the existence theorems of the blow-up and globalsolution Moreover the upper estimates of ldquoblow-up timerdquoldquoblow-up raterdquo and global solution are also given Ourresults can be seen as the extension and supplement of thoseobtained in [21 22]

The present work is organized as follows In Section 2 wedeal with the blow-up solution of (1) Section 3 is devotedto the global solution of (1) As applications of the obtainedresults some examples are presented in Section 4

2 Blow-Up Solution

In this section we discuss what interactions among the fournonlinear mechanisms of (1) result in the blow-up solutionOur main result in this section is the following theorem

Theorem 1 Let 119906(119909 119905) be a solution of problem (1) Assumethat the following conditions (i)ndash(iv) are satisfied

(i) For 119904 isin R+

119886 (119904) minus 1198861015840(119904) ge 0

1198911015840(119904) minus 119891 (119904) ge 0

(119886 (119904)

1198921015840 (119904))

1015840

ge 0

(4)

(ii) For (119909 119904 119905) isin 120597Ω times R+times R+

119887119904(119909 119904 119905) minus 119887 (119909 119904 119905) le 0

119886 (119904) 119887 (119909 119904 119905) minus (119886 (119904) 119887 (119909 119904 119905))119904le 0

119887119905(119909 119904 119905) ge 0

(5)

(iii) Consider the following

120572 fl min119863

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

gt 0 (6)

(iv) Consider the following

int+infin

1198720

119886 (119904)

e119904d119904 lt +infin 119872

0fl max119863

1199060(119909) (7)

Then 119906(119909 119905) blows up in a finite time 119879 and

119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905))

(8)

where

119867(119910) fl int+infin

119910

119886 (119904)

e119904d119904 119910 gt 0 (9)

and 119867minus1 is the inverse function of 119867

Proof Introduce an auxiliary function

119875 (119909 119905) fl minuseminus119906119906119905+ 120572

1

119886 (119906) (10)

and then we have

nabla119875 = eminus119906119906119905nabla119906 minus eminus119906nabla119906

119905minus 120572

1198861015840

1198862nabla119906 (11)

Δ119875 = minuseminus119906119906119905|nabla119906|2+ 2eminus119906nabla119906 sdot nabla119906

119905+ eminus119906119906

119905Δ119906

minus eminus119906Δ119906119905+ (2120572

(1198861015840)2

1198863minus 120572

11988610158401015840

1198862)|nabla119906|

2

minus 1205721198861015840

1198862Δ119906

(12)

119875119905= eminus119906 (119906

119905)2

minus eminus119906 (119906119905)119905minus 120572

1198861015840

1198862119906119905

= eminus119906 (119906119905)2

minus eminus119906 ( 119886

1198921015840Δ119906 +

1198861015840

1198921015840|nabla119906|2+

119891

1198921015840)119905

minus 1205721198861015840

1198862119906119905

= eminus119906 (119906119905)2

+ (11988611989210158401015840

(1198921015840)2minus

1198861015840

1198921015840) eminus119906119906

119905Δ119906

minus119886

1198921015840eminus119906Δ119906

119905+ (

119886101584011989210158401015840

(1198921015840)2minus

11988610158401015840

1198921015840) eminus119906119906

119905|nabla119906|2

minus 21198861015840

1198921015840eminus119906 (nabla119906 sdot nabla119906

119905) + (

11989111989210158401015840

(1198921015840)2minus

1198911015840

1198921015840) eminus119906119906

119905

minus 1205721198861015840

1198862119906119905

(13)

Journal of Function Spaces 3

It follows from (12) and (13) that

119886

1198921015840Δ119875 minus 119875

119905= (

11988610158401015840

1198921015840minus

119886

1198921015840minus

119886101584011989210158401015840

(1198921015840)2) eminus119906119906

119905|nabla119906|2

+ (2119886

1198921015840+ 2

1198861015840

1198921015840) eminus119906 (nabla119906 sdot nabla119906

119905)

+ (1198861015840

1198921015840+

119886

1198921015840minus

11988611989210158401015840

(1198921015840)2) eminus119906119906

119905Δ119906

+ (2120572(1198861015840)2

11988621198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2

minus 1205721198861015840

1198861198921015840Δ119906 minus eminus119906 (119906

119905)2

+ (1198911015840

1198921015840minus

11989111989210158401015840

(1198921015840)2) eminus119906119906

119905+ 120572

1198861015840

1198862119906119905

(14)

The first equation of (1) implies

Δ119906 =1198921015840

119886119906119905minus

1198861015840

119886|nabla119906|2minus

119891

119886 (15)

Inserting (15) into (14) we obtain

119886

1198921015840Δ119875 minus 119875

119905= (

11988610158401015840

1198921015840minus

119886

1198921015840minus

(1198861015840)2

1198861198921015840minus

1198861015840

1198921015840) eminus119906119906

119905|nabla119906|2

+ (2119886

1198921015840+ 2

1198861015840


119905)

+1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ (1198911015840

1198921015840minus

1198861015840119891

1198861198921015840minus

119891

1198921015840) eminus119906119906

119905

+ (3120572(1198861015840)2

11988621198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2+ 120572

1198861015840119891

11988621198921015840

(16)

It follows from (11) that

nabla119906119905= minuse119906nabla119875 + 119906

119905nabla119906 minus 120572

1198861015840

1198862e119906nabla119906 (17)

Substituting (17) into (16) we get

119886

1198921015840Δ119875 + 2

119886 + 1198861015840

1198921015840nabla119906 sdot nabla119875 minus 119875

119905

= (11988610158401015840

1198921015840+

119886

1198921015840minus

(1198861015840)2

1198861198921015840+

1198861015840

1198921015840) eminus119906119906

119905|nabla119906|2

+ (120572(1198861015840)2

11988621198921015840minus 2120572

1198861015840

1198861198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2

+1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ (1198911015840

1198921015840minus

1198861015840119891

1198861198921015840minus

119891

1198921015840) eminus119906119906

119905+ 120572

1198861015840119891

11988621198921015840

(18)

By (10) we have

119906119905= minuse119906119875 + 120572e119906 1

119886 (19)

Now we insert (19) into (18) to deduce

119886

1198921015840Δ119875 + 2

1198861015840 + 119886

1198921015840nabla119906 sdot nabla119875

+119886

1198921015840[1 +

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

minus119891

119886119875

minus 119875119905

= 1205721

1198861198921015840(119886 minus 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ 1205721

1198861198921015840(1198911015840minus 119891)

(20)

Assumptions (4) ensure that the right side in equality (20) isnonnegative that is

119886

1198921015840Δ119875 + 2

1198861015840 + 119886

1198921015840nabla119906 sdot nabla119875

+119886

1198921015840[1 +

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

minus119891

119886119875

minus 119875119905ge 0 in Ω times (0 119879)

(21)


Next it follows from (1) and (10) that

120597119875

120597119899= eminus119906119906

119905

120597119906

120597119899minus eminus119906

120597119906119905

120597119899minus 120572

1198861015840

1198862120597119906

120597119899

= eminus119906119906119905119887 minus eminus119906 (120597119906

120597119899)119905

minus 1205721198861015840

1198862119887

= eminus119906119906119905119887 minus eminus119906 (119887 (119909 119906 119905))

119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) eminus119906119906

119905minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) (minus119875 + 120572

1

119886) minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887119906minus 119887) 119875 + 120572

119886119887 minus (119886119887)119906

1198862minus eminus119906119887

119905

on 120597Ω times (0 119879)

(22)

We note that (6) implies

maxΩ

119875 (119909 0) = maxΩ

minuseminus1199060 (1199060)119905+ 120572

1

119886 (1199060)

= maxΩ

minusnabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)

e11990601198921015840 (1199060)

+ 1205721

119886 (1199060)

= maxΩ

1

119886 (1199060)(120572

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

) = 0

(23)

There (21)ndash(23) assumption (5) and the maximum principle[10] imply that the maximum of the function 119875 in Ω times [0T)

is zero In fact if the function 119875 takes a positive maximum atpoint (119909

0 1199050) isin 120597Ω times (0 119879) then we have

119875 (1199090 1199050) gt 0

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)gt 0 (24)

Using assumption (5) and the fact that119875(1199090 1199050) gt 0 it follows

from (22) that

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)le 0 (25)

which contradicts the second inequality in (24) Hence themaximum of the function 119875 in Ω times [0 119879) is zero Now wehave

119875 le 0 in Ω times [0 119879) (26)

that is

119886 (119906)

e119906119906119905ge 120572 (27)

At the point 1199090

isin Ω where 1199060(1199090) = 119872

0 integrating

inequality (27) from 0 to 119905 we arrive at

int119905

0

119886 (119906)

e119906119906119905d119905 = int

119906(1199090119905)

1198720

119886 (119904)

e119904d119904 ge 120572119905 (28)

Inequality (28) and assumption (7) imply that 119906(119909 119905) blowsup in finite time 119905 = 119879 Now we let 119905 rarr 119879 in (28) to deduce

119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 (29)

For each fixed 119909 isin Ω integrating inequality (27) from 119905 to119904 (0 lt 119905 lt 119904 lt 119879) we get

119867(119906 (119909 119905)) ge 119867 (119906 (119909 119905)) minus 119867 (119906 (119909 119904))

= int+infin

119906(119909119905)

119886 (119904)

e119904d119904 minus int

+infin

119906(119909119904)

119886 (119904)

e119904d119904

= int119906(119909119904)

119906(119909119905)

119886 (119904)

e119904d119904 = int

119904

119905

119886 (119906)

e119906119906119905d119905

ge 120572 (119904 minus 119905)

(30)

In the above inequality letting 119904 rarr 119879 we obtain

119867(119906 (119909 119905)) ge 120572 (119879 minus 119905) (31)

We note that 119867 is a strictly decreasing function Hence

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) (32)

The proof is complete

3 Global Solution

In this section we study what interactions among the fournonlinear mechanisms of (1) result in the global solution of(1) The main results of this section are formulated in thefollowing theorem

Theorem 2 Let 119906(119909 119905) be a solution of problem (1) Assumethat the following conditions (i)ndash(iv) are fulfilled


119886 (119904) + 1198861015840(119904) le 0

119891 (119904) + 1198911015840(119904) le 0

(119886 (119904)

1198921015840 (119904))

1015840

le 0

(33)


119887 (119909 119904 119905) + 119887119904(119909 119904 119905) le 0

119886 (119904) 119887 (119909 119904 119905) + (119886 (119904) 119887 (119909 119904 119905))119904le 0

119887119905(119909 119904 119905) le 0

(34)



120573 fl maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

gt 0 (35)


int+infin

1198980

119886 (119904)

eminus119904d119904 = +infin 119898

0fl minΩ

1199060(119909) (36)

Then 119906(119909 119905) must be a global solution and

119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (37)

where

119870(119910) fl int119910

1198980

119886 (119904)

eminus119904d119904 119910 ge 119898

0 (38)

and 119870 is the inverse function of 119870

Proof We consider an auxiliary function

119876 (119909 119905) fl minuse119906119906119905+ 120573

1

119886 (119906) (39)

Using the same reasoning process as that of (11)ndash(20) weobtain

119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876

minus 119876119905

= 1205731

1198861198921015840(119886 + 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

e119906 (119906119905)2

+ 1205731

1198861198921015840(1198911015840+ 119891)

(40)

Assumptions (33) imply that the right side of (40) is nonpos-itive that is

119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876

minus 119876119905le 0 in Ω times (0 119879)

(41)

By (1) and (39) we get

120597119876

120597119899= minuse119906119906

119905

120597119906

120597119899minus e119906

120597119906119905

120597119899minus 120573

1198861015840

1198862120597119906

120597119899

= minuse119906119906119905119887 minus e119906 (120597119906

120597119899)119905

minus 1205731198861015840

1198862119887

= minuse119906119906119905119887 minus e119906 (119887 (119909 119906 119905))

119905minus 120573

1198861015840

1198862119887

= minus (119887 + 119887119906) e119906119906119905minus e119906119887119905minus 120573

1198861015840

1198862119887

= (119887 + 119887119906) (119876 minus 120573

1

119886) minus e119906119887

119905minus 120573

1198861015840

1198862119887

= (119887 + 119887119906) 119876 minus 120573

119886119887 + (119886119887)119906

1198862minus e119906119887119905

on 120597Ω times (0 119879)

(42)

Assumption (35) implies

minΩ

119876 (119909 0) = minΩ

minuse1199060 (1199060)119905+ 120573

1

119886 (1199060)

= minΩ


0) nabla1199060) + 119891 (119906

0)

eminus11990601198921015840 (1199060)

+ 1205731

119886 (1199060)

= minΩ

1

119886 (1199060)(120573

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

) = 0

(43)

It follows from (41)ndash(43) (34) and the maximum principlethat the minimum of 119876 in Ω times [0 119879) is zero Hence we havethe following inequality

119876 ge 0 in Ω times [0 119879) (44)

that is

119886 (119906)

eminus119906119906119905le 120573 (45)

For each fixed 119909 isin Ω we integrate (45) from 0 to 119905 to deduce

1

120573int119905

0

119886 (119906)

eminus119906119906119905d119905 =

1

120573int119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904 le 119905 (46)

Inequality (46) and assumption (36) imply that 119906 must be aglobal solution Furthermore it follows from (45) that

119870 (119906 (119909 119905)) minus 119870 (1199060(119909))

= int119906(119909119905)

1198980

119886 (119904)

eminus119904d119904 minus int

1199060(119909)

1198980

119886 (119904)

eminus119904d119904 = int

119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904

= int119905

0

119886 (119906)

eminus119906119906119905d119905 le 120573119905

(47)


Since 119870 is a strictly increasing function we obtain

119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (48)


4 Applications

When 119886(119906) = 1 and 119887(119909 119906 119905) = 119887(119906) the results ofTheorems 1ndash2 still hold In this sense our results extend andsupplement those obtained in [21 22]

In the following we give a few examples to demonstratethe applications of Theorems 1ndash2

Example 3 Let 119906 be a solution of the following problem

(119906

2+ e1199062)

119905

= nabla sdot (e1199062nabla119906) + e119906 in Ω times (0 119879)

120597119906

120597119899= e(34)(119906minus2) + e119906minus2+119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(49)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923

lt 1 Wenote

119886 (119906) = e2119906

119892 (119906) =119906

2+ e1199062

119891 (119906) = e119906

119887 (119909 119906 119905) = e(34)(119906minus2) + e119906minus2+119905|119909|2

(50)

In order to calculate the constant 120572 we set

120596 = |119909|2 (51)

and then 0 le 120596 le 1 and

120572 = minΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

= minΩ

2 [6 + 2 |119909|2+ e(12)(1+|119909|

2)]

e(12)(1+|119909|2) + 1

= min0le120596le1

2 [6 + 2120596 + e(12)(1+120596)]

e(12)(1+120596) + 1 =

2 (8 + e)1 + e

(52)

We can check that (4) (5) and (7) hold It follows fromTheorem 1 that 119906(119909 119905) blows up in a finite time 119879 and

119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 =

1 + e2 (8 + e)

int+infin

2

eminus1199042d119904

=1 + e

e (8 + e)

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) = 2 ln 1 + e(8 + e) (119879 minus 119905)

(53)


(119906e119906)119905= nabla sdot (

eminus119906

1 + 119906nabla119906) + eminus119906 in Ω times (0 119879)

120597119906

120597119899= eminus2119906 + eminus119906minus119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(54)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923lt 1 Now

we have

119886 (119906) =eminus119906

1 + 119906

119892 (119906) = 119906e119906

119891 (119906) = eminus119906

119887 (119909 119906 119905) = eminus2119906 + eminus119906minus119905|119909|2

(55)

Setting

120596 = |119909|2 (56)

we get 0 le 120596 le 1 and

120573 = maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

= maxΩ

16 minus 2 |119909|2minus 3 |119909|

4

e2(1+|119909|2) (2 + |119909|2)4

= max0le120596le1

16 minus 2120596 minus 31205962

e2(1+120596) (2 + 120596)4 = eminus2

(57)

We can also check that (33) (34) and (36) hold HenceTheorem 2 implies 119906 must be a global solution and

119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) = (2 + |119909|

2) e119905eminus2

minus 1 (58)

Remark 5 When the functions 119886 119887 119891 and 119892 are all exponen-tial functions Theorems 1 and 2 can be used When not all ofthem are exponential functionsTheorems 1 and 2 can be usedin some special cases

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61473180)

References

[1] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007


[2] MMarras and S Vernier Piro ldquoOn global existence and boundsfor blow-up time in nonlinear parabolic problems with timedependent coefficientsrdquo Discrete and Continuous DynamicalSystems vol 2013 pp 535ndash544 2013

[3] L E Payne G A Philippin and P W Schaefer ldquoBlow-upphenomena for some nonlinear parabolic problemsrdquoNonlinearAnalysis Theory Methods amp Applications vol 69 no 10 pp3495ndash3502 2008

[4] S H Chen ldquoGlobal existence and nonexistence for some degen-erate and quasilinear parabolic systemsrdquo Journal of DifferentialEquations vol 245 no 4 pp 1112ndash1136 2008

[5] S H Chen and K MacDonald ldquoGlobal and blowup solutionsfor general quasilinear parabolic systemsrdquo Nonlinear AnalysisReal World Applications vol 14 no 1 pp 423ndash433 2013

[6] J T Ding and H J Hu ldquoBlow-up and global solutions for aclass of nonlinear reaction diffusion equations under Dirichletboundary conditionsrdquo Journal of Mathematical Analysis andApplications vol 433 no 2 Article ID 19743 pp 1718ndash17352016

[7] J T Ding and M Wang ldquoBlow-up solutions global existenceand exponential decay estimates for second order parabolicproblemsrdquo Boundary Value Problems vol 2015 no 168 16pages 2015

[8] L L Zhang ldquoBlow-up of solutions for a class of nonlinearparabolic equationsrdquo Zeitschrift fur Analysis und Ihre Anwen-dungen vol 25 no 4 pp 479ndash486 2006

[9] H Amann ldquoQuasilinear parabolic systems under nonlinearboundary conditionsrdquo Archive for Rational Mechanics andAnalysis vol 92 no 2 pp 153ndash192 1986

[10] R SperbMaximumPrinciples andTheirApplications AcademicPress New York NY USA 1981

[11] A Friedman Partial Differential Equation of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[12] J Harada ldquoNon self-similar blow-up solutions to the heat equa-tion with nonlinear boundary conditionsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 102 no 2 pp 36ndash83 2014

[13] J Harada and K Mihara ldquoBlow-up rate for radially symmetricsolutions of some parabolic equations with nonlinear boundaryconditionsrdquo Journal of Differential Equations vol 253 no 5 pp1647ndash1663 2012

[14] Y Liu ldquoLower bounds for the blow-up time in a non-local reac-tion diffusion problem under nonlinear boundary conditionsrdquoMathematical and ComputerModelling vol 57 no 3-4 pp 926ndash931 2013

[15] Y Liu S G Luo and Y H Ye ldquoBlow-up phenomena for aparabolic problemwith a gradient nonlinearity under nonlinearboundary conditionsrdquo Computers ampMathematics with Applica-tions vol 65 no 8 pp 1194ndash1199 2013

[16] F S Li and J L Li ldquoGlobal existence and blow-up phenom-ena for nonlinear divergence form parabolic equations withinhomogeneous Neumann boundary conditionsrdquo Journal ofMathematical Analysis and Applications vol 385 no 2 pp1005ndash1014 2012

[17] F Liang ldquoBlow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary conditionrdquoApplied Mathematics and Computation vol 218 no 8 pp3993ndash3999 2011

[18] L E Payne G A Philippin and S Vernier Piro ldquoBlow-up phenomena for a semilinear heat equation with nonlinearboundary condition IIrdquoNonlinear Analysis Theory Methods ampApplications vol 73 no 4 pp 971ndash978 2010

[19] J Ding X Gao and S Li ldquoGlobal existence and blow-up prob-lems for reaction diffusion model with multiple nonlinearitiesrdquoJournal of Mathematical Analysis and Applications vol 343 no1 pp 159ndash169 2008

[20] Y W Qi M X Wang and Z J Wang ldquoExistence and non-existence of global solutions of diffusion systems with nonlin-ear boundary conditionsrdquo Proceedings of the Royal Society ofEdinburghmdashSection A Mathematics vol 134 no 6 pp 1199ndash1217 2004

[21] H L Zhang ldquoBlow-up solutions and global solutions for non-linear parabolic problemsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 69 no 12 pp 4567ndash4574 2008

[22] L L Zhang N Zhang and L X Li ldquoBlow-up solutions andglobal existence for a kind of quasilinear reaction-diffusionequationsrdquo Zeitschrift fur Analysis und ihre Anwendungen vol33 no 3 pp 247ndash258 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where Ω sub R119873 (119873 ge 2) is a bounded domain with smoothboundary 120597Ω Some conditions on nonlinearities and theinitial data were given to ensure that 119906(119909 119905) exists globallyor blows up at some finite time 119879 In addition the upperestimates of the global solution the ldquoblow-up timerdquo and theldquoblow-up raterdquo were also established

In this paper we study reaction-diffusion problem (1) It iswell known that 119891(119906) 119892(119906) 119886(119906) and 119887(119909 119906 119905) are nonlinearreaction nonlinear diffusion nonlinear convection and non-linear boundary flux respectively What interactions amongthe four nonlinear mechanisms result in the blow-up andglobal solutions of (1) is investigated in this work We notethat the boundary flux function 119887(119909 119906 119905) depends not onlyon the concentration variable 119906 but also on the space variable119909 and the time variable 119905 Hence it seems that the methodsof [21 22] are not applicable for problem (1) In this paperby constructing completely different auxiliary functions fromthose in [21 22] and technically using maximum principleswe obtain the existence theorems of the blow-up and globalsolution Moreover the upper estimates of ldquoblow-up timerdquoldquoblow-up raterdquo and global solution are also given Ourresults can be seen as the extension and supplement of thoseobtained in [21 22]

The present work is organized as follows In Section 2 wedeal with the blow-up solution of (1) Section 3 is devotedto the global solution of (1) As applications of the obtainedresults some examples are presented in Section 4

2 Blow-Up Solution

In this section we discuss what interactions among the fournonlinear mechanisms of (1) result in the blow-up solutionOur main result in this section is the following theorem

Theorem 1 Let 119906(119909 119905) be a solution of problem (1) Assumethat the following conditions (i)ndash(iv) are satisfied


119886 (119904) minus 1198861015840(119904) ge 0

1198911015840(119904) minus 119891 (119904) ge 0

(119886 (119904)

1198921015840 (119904))

1015840

ge 0

(4)


119887119904(119909 119904 119905) minus 119887 (119909 119904 119905) le 0

119886 (119904) 119887 (119909 119904 119905) minus (119886 (119904) 119887 (119909 119904 119905))119904le 0

119887119905(119909 119904 119905) ge 0

(5)


120572 fl min119863

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

gt 0 (6)


int+infin

1198720

119886 (119904)

e119904d119904 lt +infin 119872

0fl max119863

1199060(119909) (7)

Then 119906(119909 119905) blows up in a finite time 119879 and

119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905))

(8)

where

119867(119910) fl int+infin

119910

119886 (119904)

e119904d119904 119910 gt 0 (9)

and 119867minus1 is the inverse function of 119867

Proof Introduce an auxiliary function

119875 (119909 119905) fl minuseminus119906119906119905+ 120572

1

119886 (119906) (10)

and then we have

nabla119875 = eminus119906119906119905nabla119906 minus eminus119906nabla119906

119905minus 120572

1198861015840

1198862nabla119906 (11)

Δ119875 = minuseminus119906119906119905|nabla119906|2+ 2eminus119906nabla119906 sdot nabla119906

119905+ eminus119906119906

119905Δ119906

minus eminus119906Δ119906119905+ (2120572

(1198861015840)2

1198863minus 120572

11988610158401015840

1198862)|nabla119906|

2

minus 1205721198861015840

1198862Δ119906

(12)

119875119905= eminus119906 (119906

119905)2

minus eminus119906 (119906119905)119905minus 120572

1198861015840

1198862119906119905

= eminus119906 (119906119905)2

minus eminus119906 ( 119886

1198921015840Δ119906 +

1198861015840

1198921015840|nabla119906|2+

119891

1198921015840)119905

minus 1205721198861015840

1198862119906119905

= eminus119906 (119906119905)2

+ (11988611989210158401015840

(1198921015840)2minus

1198861015840

1198921015840) eminus119906119906

119905Δ119906

minus119886

1198921015840eminus119906Δ119906

119905+ (

119886101584011989210158401015840

(1198921015840)2minus

11988610158401015840

1198921015840) eminus119906119906

119905|nabla119906|2

minus 21198861015840

1198921015840eminus119906 (nabla119906 sdot nabla119906

119905) + (

11989111989210158401015840

(1198921015840)2minus

1198911015840

1198921015840) eminus119906119906

119905

minus 1205721198861015840

1198862119906119905

(13)



119886

1198921015840Δ119875 minus 119875

119905= (

11988610158401015840

1198921015840minus

119886

1198921015840minus

119886101584011989210158401015840

(1198921015840)2) eminus119906119906

119905|nabla119906|2

+ (2119886

1198921015840+ 2

1198861015840


119905)

+ (1198861015840

1198921015840+

119886

1198921015840minus

11988611989210158401015840

(1198921015840)2) eminus119906119906

119905Δ119906

+ (2120572(1198861015840)2

11988621198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2

minus 1205721198861015840

1198861198921015840Δ119906 minus eminus119906 (119906

119905)2

+ (1198911015840

1198921015840minus

11989111989210158401015840

(1198921015840)2) eminus119906119906

119905+ 120572

1198861015840

1198862119906119905

(14)


Δ119906 =1198921015840

119886119906119905minus

1198861015840


119891

119886 (15)


119886

1198921015840Δ119875 minus 119875

119905= (

11988610158401015840

1198921015840minus

119886

1198921015840minus

(1198861015840)2

1198861198921015840minus

1198861015840

1198921015840) eminus119906119906

119905|nabla119906|2

+ (2119886

1198921015840+ 2

1198861015840


119905)

+1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ (1198911015840

1198921015840minus

1198861015840119891

1198861198921015840minus

119891

1198921015840) eminus119906119906

119905

+ (3120572(1198861015840)2

11988621198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2+ 120572

1198861015840119891

11988621198921015840

(16)



119905nabla119906 minus 120572

1198861015840

1198862e119906nabla119906 (17)


119886

1198921015840Δ119875 + 2

119886 + 1198861015840


119905

= (11988610158401015840

1198921015840+

119886

1198921015840minus

(1198861015840)2

1198861198921015840+

1198861015840

1198921015840) eminus119906119906

119905|nabla119906|2

+ (120572(1198861015840)2

11988621198921015840minus 2120572

1198861015840

1198861198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2

+1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ (1198911015840

1198921015840minus

1198861015840119891

1198861198921015840minus

119891

1198921015840) eminus119906119906

119905+ 120572

1198861015840119891

11988621198921015840

(18)

By (10) we have

119906119905= minuse119906119875 + 120572e119906 1

119886 (19)


119886

1198921015840Δ119875 + 2

1198861015840 + 119886

1198921015840nabla119906 sdot nabla119875

+119886

1198921015840[1 +

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

minus119891

119886119875

minus 119875119905

= 1205721

1198861198921015840(119886 minus 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ 1205721

1198861198921015840(1198911015840minus 119891)

(20)


119886

1198921015840Δ119875 + 2

1198861015840 + 119886

1198921015840nabla119906 sdot nabla119875

+119886

1198921015840[1 +

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

minus119891

119886119875


(21)



120597119875

120597119899= eminus119906119906

119905

120597119906

120597119899minus eminus119906

120597119906119905

120597119899minus 120572

1198861015840

1198862120597119906

120597119899


120597119899)119905

minus 1205721198861015840

1198862119887


119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) eminus119906119906

119905minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) (minus119875 + 120572

1

119886) minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887119906minus 119887) 119875 + 120572

119886119887 minus (119886119887)119906

1198862minus eminus119906119887

119905

on 120597Ω times (0 119879)

(22)


maxΩ

119875 (119909 0) = maxΩ

minuseminus1199060 (1199060)119905+ 120572

1

119886 (1199060)

= maxΩ


0) nabla1199060) + 119891 (119906

0)

e11990601198921015840 (1199060)

+ 1205721

119886 (1199060)

= maxΩ

1

119886 (1199060)(120572

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

) = 0

(23)




119875 (1199090 1199050) gt 0

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)gt 0 (24)


from (22) that

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)le 0 (25)


119875 le 0 in Ω times [0 119879) (26)

that is

119886 (119906)

e119906119906119905ge 120572 (27)


isin Ω where 1199060(1199090) = 119872

0 integrating


int119905

0

119886 (119906)

e119906119906119905d119905 = int

119906(1199090119905)

1198720

119886 (119904)

e119904d119904 ge 120572119905 (28)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 (29)


119867(119906 (119909 119905)) ge 119867 (119906 (119909 119905)) minus 119867 (119906 (119909 119904))

= int+infin

119906(119909119905)

119886 (119904)


+infin

119906(119909119904)

119886 (119904)

e119904d119904

= int119906(119909119904)

119906(119909119905)

119886 (119904)

e119904d119904 = int

119904

119905

119886 (119906)

e119906119906119905d119905

ge 120572 (119904 minus 119905)

(30)


119867(119906 (119909 119905)) ge 120572 (119879 minus 119905) (31)


119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) (32)


3 Global Solution




119886 (119904) + 1198861015840(119904) le 0

119891 (119904) + 1198911015840(119904) le 0

(119886 (119904)

1198921015840 (119904))

1015840

le 0

(33)


119887 (119909 119904 119905) + 119887119904(119909 119904 119905) le 0

119886 (119904) 119887 (119909 119904 119905) + (119886 (119904) 119887 (119909 119904 119905))119904le 0

119887119905(119909 119904 119905) le 0

(34)



120573 fl maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

gt 0 (35)


int+infin

1198980

119886 (119904)

eminus119904d119904 = +infin 119898

0fl minΩ

1199060(119909) (36)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (37)

where

119870(119910) fl int119910

1198980

119886 (119904)

eminus119904d119904 119910 ge 119898

0 (38)



119876 (119909 119905) fl minuse119906119906119905+ 120573

1

119886 (119906) (39)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876

minus 119876119905

= 1205731

1198861198921015840(119886 + 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

e119906 (119906119905)2

+ 1205731

1198861198921015840(1198911015840+ 119891)

(40)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876


(41)


120597119876

120597119899= minuse119906119906

119905

120597119906

120597119899minus e119906

120597119906119905

120597119899minus 120573

1198861015840

1198862120597119906

120597119899

= minuse119906119906119905119887 minus e119906 (120597119906

120597119899)119905

minus 1205731198861015840

1198862119887

= minuse119906119906119905119887 minus e119906 (119887 (119909 119906 119905))

119905minus 120573

1198861015840

1198862119887


1198861015840

1198862119887

= (119887 + 119887119906) (119876 minus 120573

1

119886) minus e119906119887

119905minus 120573

1198861015840

1198862119887

= (119887 + 119887119906) 119876 minus 120573

119886119887 + (119886119887)119906

1198862minus e119906119887119905

on 120597Ω times (0 119879)

(42)


minΩ

119876 (119909 0) = minΩ

minuse1199060 (1199060)119905+ 120573

1

119886 (1199060)

= minΩ


0) nabla1199060) + 119891 (119906

0)

eminus11990601198921015840 (1199060)

+ 1205731

119886 (1199060)

= minΩ

1

119886 (1199060)(120573

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

) = 0

(43)


119876 ge 0 in Ω times [0 119879) (44)

that is

119886 (119906)

eminus119906119906119905le 120573 (45)


1

120573int119905

0

119886 (119906)

eminus119906119906119905d119905 =

1

120573int119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904 le 119905 (46)


119870 (119906 (119909 119905)) minus 119870 (1199060(119909))

= int119906(119909119905)

1198980

119886 (119904)


1199060(119909)

1198980

119886 (119904)


119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904

= int119905

0

119886 (119906)

eminus119906119906119905d119905 le 120573119905

(47)



119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (48)


4 Applications




(119906

2+ e1199062)

119905


120597119906

120597119899= e(34)(119906minus2) + e119906minus2+119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(49)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923

lt 1 Wenote

119886 (119906) = e2119906

119892 (119906) =119906

2+ e1199062

119891 (119906) = e119906

119887 (119909 119906 119905) = e(34)(119906minus2) + e119906minus2+119905|119909|2

(50)


120596 = |119909|2 (51)


120572 = minΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

= minΩ

2 [6 + 2 |119909|2+ e(12)(1+|119909|

2)]

e(12)(1+|119909|2) + 1

= min0le120596le1

2 [6 + 2120596 + e(12)(1+120596)]

e(12)(1+120596) + 1 =

2 (8 + e)1 + e

(52)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 =

1 + e2 (8 + e)

int+infin

2

eminus1199042d119904

=1 + e

e (8 + e)

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) = 2 ln 1 + e(8 + e) (119879 minus 119905)

(53)


(119906e119906)119905= nabla sdot (

eminus119906


120597119906


2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(54)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923lt 1 Now

we have

119886 (119906) =eminus119906

1 + 119906

119892 (119906) = 119906e119906

119891 (119906) = eminus119906


(55)

Setting

120596 = |119909|2 (56)


120573 = maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

= maxΩ

16 minus 2 |119909|2minus 3 |119909|

4

e2(1+|119909|2) (2 + |119909|2)4

= max0le120596le1

16 minus 2120596 minus 31205962

e2(1+120596) (2 + 120596)4 = eminus2

(57)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) = (2 + |119909|

2) e119905eminus2

minus 1 (58)


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886

1198921015840Δ119875 minus 119875

119905= (

11988610158401015840

1198921015840minus

119886

1198921015840minus

119886101584011989210158401015840

(1198921015840)2) eminus119906119906

119905|nabla119906|2

+ (2119886

1198921015840+ 2

1198861015840


119905)

+ (1198861015840

1198921015840+

119886

1198921015840minus

11988611989210158401015840

(1198921015840)2) eminus119906119906

119905Δ119906

+ (2120572(1198861015840)2

11988621198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2

minus 1205721198861015840

1198861198921015840Δ119906 minus eminus119906 (119906

119905)2

+ (1198911015840

1198921015840minus

11989111989210158401015840

(1198921015840)2) eminus119906119906

119905+ 120572

1198861015840

1198862119906119905

(14)


Δ119906 =1198921015840

119886119906119905minus

1198861015840


119891

119886 (15)


119886

1198921015840Δ119875 minus 119875

119905= (

11988610158401015840

1198921015840minus

119886

1198921015840minus

(1198861015840)2

1198861198921015840minus

1198861015840

1198921015840) eminus119906119906

119905|nabla119906|2

+ (2119886

1198921015840+ 2

1198861015840


119905)

+1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ (1198911015840

1198921015840minus

1198861015840119891

1198861198921015840minus

119891

1198921015840) eminus119906119906

119905

+ (3120572(1198861015840)2

11988621198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2+ 120572

1198861015840119891

11988621198921015840

(16)



119905nabla119906 minus 120572

1198861015840

1198862e119906nabla119906 (17)


119886

1198921015840Δ119875 + 2

119886 + 1198861015840


119905

= (11988610158401015840

1198921015840+

119886

1198921015840minus

(1198861015840)2

1198861198921015840+

1198861015840

1198921015840) eminus119906119906

119905|nabla119906|2

+ (120572(1198861015840)2

11988621198921015840minus 2120572

1198861015840

1198861198921015840minus 120572

11988610158401015840

1198861198921015840) |nabla119906|

2

+1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ (1198911015840

1198921015840minus

1198861015840119891

1198861198921015840minus

119891

1198921015840) eminus119906119906

119905+ 120572

1198861015840119891

11988621198921015840

(18)

By (10) we have

119906119905= minuse119906119875 + 120572e119906 1

119886 (19)


119886

1198921015840Δ119875 + 2

1198861015840 + 119886

1198921015840nabla119906 sdot nabla119875

+119886

1198921015840[1 +

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

minus119891

119886119875

minus 119875119905

= 1205721

1198861198921015840(119886 minus 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

eminus119906 (119906119905)2

+ 1205721

1198861198921015840(1198911015840minus 119891)

(20)


119886

1198921015840Δ119875 + 2

1198861015840 + 119886

1198921015840nabla119906 sdot nabla119875

+119886

1198921015840[1 +

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

minus119891

119886119875


(21)



120597119875

120597119899= eminus119906119906

119905

120597119906

120597119899minus eminus119906

120597119906119905

120597119899minus 120572

1198861015840

1198862120597119906

120597119899


120597119899)119905

minus 1205721198861015840

1198862119887


119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) eminus119906119906

119905minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) (minus119875 + 120572

1

119886) minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887119906minus 119887) 119875 + 120572

119886119887 minus (119886119887)119906

1198862minus eminus119906119887

119905

on 120597Ω times (0 119879)

(22)


maxΩ

119875 (119909 0) = maxΩ

minuseminus1199060 (1199060)119905+ 120572

1

119886 (1199060)

= maxΩ


0) nabla1199060) + 119891 (119906

0)

e11990601198921015840 (1199060)

+ 1205721

119886 (1199060)

= maxΩ

1

119886 (1199060)(120572

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

) = 0

(23)




119875 (1199090 1199050) gt 0

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)gt 0 (24)


from (22) that

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)le 0 (25)


119875 le 0 in Ω times [0 119879) (26)

that is

119886 (119906)

e119906119906119905ge 120572 (27)


isin Ω where 1199060(1199090) = 119872

0 integrating


int119905

0

119886 (119906)

e119906119906119905d119905 = int

119906(1199090119905)

1198720

119886 (119904)

e119904d119904 ge 120572119905 (28)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 (29)


119867(119906 (119909 119905)) ge 119867 (119906 (119909 119905)) minus 119867 (119906 (119909 119904))

= int+infin

119906(119909119905)

119886 (119904)


+infin

119906(119909119904)

119886 (119904)

e119904d119904

= int119906(119909119904)

119906(119909119905)

119886 (119904)

e119904d119904 = int

119904

119905

119886 (119906)

e119906119906119905d119905

ge 120572 (119904 minus 119905)

(30)


119867(119906 (119909 119905)) ge 120572 (119879 minus 119905) (31)


119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) (32)


3 Global Solution




119886 (119904) + 1198861015840(119904) le 0

119891 (119904) + 1198911015840(119904) le 0

(119886 (119904)

1198921015840 (119904))

1015840

le 0

(33)


119887 (119909 119904 119905) + 119887119904(119909 119904 119905) le 0

119886 (119904) 119887 (119909 119904 119905) + (119886 (119904) 119887 (119909 119904 119905))119904le 0

119887119905(119909 119904 119905) le 0

(34)



120573 fl maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

gt 0 (35)


int+infin

1198980

119886 (119904)

eminus119904d119904 = +infin 119898

0fl minΩ

1199060(119909) (36)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (37)

where

119870(119910) fl int119910

1198980

119886 (119904)

eminus119904d119904 119910 ge 119898

0 (38)



119876 (119909 119905) fl minuse119906119906119905+ 120573

1

119886 (119906) (39)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876

minus 119876119905

= 1205731

1198861198921015840(119886 + 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

e119906 (119906119905)2

+ 1205731

1198861198921015840(1198911015840+ 119891)

(40)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876


(41)


120597119876

120597119899= minuse119906119906

119905

120597119906

120597119899minus e119906

120597119906119905

120597119899minus 120573

1198861015840

1198862120597119906

120597119899

= minuse119906119906119905119887 minus e119906 (120597119906

120597119899)119905

minus 1205731198861015840

1198862119887

= minuse119906119906119905119887 minus e119906 (119887 (119909 119906 119905))

119905minus 120573

1198861015840

1198862119887


1198861015840

1198862119887

= (119887 + 119887119906) (119876 minus 120573

1

119886) minus e119906119887

119905minus 120573

1198861015840

1198862119887

= (119887 + 119887119906) 119876 minus 120573

119886119887 + (119886119887)119906

1198862minus e119906119887119905

on 120597Ω times (0 119879)

(42)


minΩ

119876 (119909 0) = minΩ

minuse1199060 (1199060)119905+ 120573

1

119886 (1199060)

= minΩ


0) nabla1199060) + 119891 (119906

0)

eminus11990601198921015840 (1199060)

+ 1205731

119886 (1199060)

= minΩ

1

119886 (1199060)(120573

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

) = 0

(43)


119876 ge 0 in Ω times [0 119879) (44)

that is

119886 (119906)

eminus119906119906119905le 120573 (45)


1

120573int119905

0

119886 (119906)

eminus119906119906119905d119905 =

1

120573int119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904 le 119905 (46)


119870 (119906 (119909 119905)) minus 119870 (1199060(119909))

= int119906(119909119905)

1198980

119886 (119904)


1199060(119909)

1198980

119886 (119904)


119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904

= int119905

0

119886 (119906)

eminus119906119906119905d119905 le 120573119905

(47)



119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (48)


4 Applications




(119906

2+ e1199062)

119905


120597119906

120597119899= e(34)(119906minus2) + e119906minus2+119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(49)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923

lt 1 Wenote

119886 (119906) = e2119906

119892 (119906) =119906

2+ e1199062

119891 (119906) = e119906

119887 (119909 119906 119905) = e(34)(119906minus2) + e119906minus2+119905|119909|2

(50)


120596 = |119909|2 (51)


120572 = minΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

= minΩ

2 [6 + 2 |119909|2+ e(12)(1+|119909|

2)]

e(12)(1+|119909|2) + 1

= min0le120596le1

2 [6 + 2120596 + e(12)(1+120596)]

e(12)(1+120596) + 1 =

2 (8 + e)1 + e

(52)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 =

1 + e2 (8 + e)

int+infin

2

eminus1199042d119904

=1 + e

e (8 + e)

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) = 2 ln 1 + e(8 + e) (119879 minus 119905)

(53)


(119906e119906)119905= nabla sdot (

eminus119906


120597119906


2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(54)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923lt 1 Now

we have

119886 (119906) =eminus119906

1 + 119906

119892 (119906) = 119906e119906

119891 (119906) = eminus119906


(55)

Setting

120596 = |119909|2 (56)


120573 = maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

= maxΩ

16 minus 2 |119909|2minus 3 |119909|

4

e2(1+|119909|2) (2 + |119909|2)4

= max0le120596le1

16 minus 2120596 minus 31205962

e2(1+120596) (2 + 120596)4 = eminus2

(57)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) = (2 + |119909|

2) e119905eminus2

minus 1 (58)


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119875

120597119899= eminus119906119906

119905

120597119906

120597119899minus eminus119906

120597119906119905

120597119899minus 120572

1198861015840

1198862120597119906

120597119899


120597119899)119905

minus 1205721198861015840

1198862119887


119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) eminus119906119906

119905minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887 minus 119887119906) (minus119875 + 120572

1

119886) minus eminus119906119887

119905minus 120572

1198861015840

1198862119887

= (119887119906minus 119887) 119875 + 120572

119886119887 minus (119886119887)119906

1198862minus eminus119906119887

119905

on 120597Ω times (0 119879)

(22)


maxΩ

119875 (119909 0) = maxΩ

minuseminus1199060 (1199060)119905+ 120572

1

119886 (1199060)

= maxΩ


0) nabla1199060) + 119891 (119906

0)

e11990601198921015840 (1199060)

+ 1205721

119886 (1199060)

= maxΩ

1

119886 (1199060)(120572

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

) = 0

(23)




119875 (1199090 1199050) gt 0

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)gt 0 (24)


from (22) that

120597119875

120597119899

10038161003816100381610038161003816100381610038161003816(1199090 1199050)le 0 (25)


119875 le 0 in Ω times [0 119879) (26)

that is

119886 (119906)

e119906119906119905ge 120572 (27)


isin Ω where 1199060(1199090) = 119872

0 integrating


int119905

0

119886 (119906)

e119906119906119905d119905 = int

119906(1199090119905)

1198720

119886 (119904)

e119904d119904 ge 120572119905 (28)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 (29)


119867(119906 (119909 119905)) ge 119867 (119906 (119909 119905)) minus 119867 (119906 (119909 119904))

= int+infin

119906(119909119905)

119886 (119904)


+infin

119906(119909119904)

119886 (119904)

e119904d119904

= int119906(119909119904)

119906(119909119905)

119886 (119904)

e119904d119904 = int

119904

119905

119886 (119906)

e119906119906119905d119905

ge 120572 (119904 minus 119905)

(30)


119867(119906 (119909 119905)) ge 120572 (119879 minus 119905) (31)


119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) (32)


3 Global Solution




119886 (119904) + 1198861015840(119904) le 0

119891 (119904) + 1198911015840(119904) le 0

(119886 (119904)

1198921015840 (119904))

1015840

le 0

(33)


119887 (119909 119904 119905) + 119887119904(119909 119904 119905) le 0

119886 (119904) 119887 (119909 119904 119905) + (119886 (119904) 119887 (119909 119904 119905))119904le 0

119887119905(119909 119904 119905) le 0

(34)



120573 fl maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

gt 0 (35)


int+infin

1198980

119886 (119904)

eminus119904d119904 = +infin 119898

0fl minΩ

1199060(119909) (36)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (37)

where

119870(119910) fl int119910

1198980

119886 (119904)

eminus119904d119904 119910 ge 119898

0 (38)



119876 (119909 119905) fl minuse119906119906119905+ 120573

1

119886 (119906) (39)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876

minus 119876119905

= 1205731

1198861198921015840(119886 + 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

e119906 (119906119905)2

+ 1205731

1198861198921015840(1198911015840+ 119891)

(40)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876


(41)


120597119876

120597119899= minuse119906119906

119905

120597119906

120597119899minus e119906

120597119906119905

120597119899minus 120573

1198861015840

1198862120597119906

120597119899

= minuse119906119906119905119887 minus e119906 (120597119906

120597119899)119905

minus 1205731198861015840

1198862119887

= minuse119906119906119905119887 minus e119906 (119887 (119909 119906 119905))

119905minus 120573

1198861015840

1198862119887


1198861015840

1198862119887

= (119887 + 119887119906) (119876 minus 120573

1

119886) minus e119906119887

119905minus 120573

1198861015840

1198862119887

= (119887 + 119887119906) 119876 minus 120573

119886119887 + (119886119887)119906

1198862minus e119906119887119905

on 120597Ω times (0 119879)

(42)


minΩ

119876 (119909 0) = minΩ

minuse1199060 (1199060)119905+ 120573

1

119886 (1199060)

= minΩ


0) nabla1199060) + 119891 (119906

0)

eminus11990601198921015840 (1199060)

+ 1205731

119886 (1199060)

= minΩ

1

119886 (1199060)(120573

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

) = 0

(43)


119876 ge 0 in Ω times [0 119879) (44)

that is

119886 (119906)

eminus119906119906119905le 120573 (45)


1

120573int119905

0

119886 (119906)

eminus119906119906119905d119905 =

1

120573int119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904 le 119905 (46)


119870 (119906 (119909 119905)) minus 119870 (1199060(119909))

= int119906(119909119905)

1198980

119886 (119904)


1199060(119909)

1198980

119886 (119904)


119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904

= int119905

0

119886 (119906)

eminus119906119906119905d119905 le 120573119905

(47)



119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (48)


4 Applications




(119906

2+ e1199062)

119905


120597119906

120597119899= e(34)(119906minus2) + e119906minus2+119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(49)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923

lt 1 Wenote

119886 (119906) = e2119906

119892 (119906) =119906

2+ e1199062

119891 (119906) = e119906

119887 (119909 119906 119905) = e(34)(119906minus2) + e119906minus2+119905|119909|2

(50)


120596 = |119909|2 (51)


120572 = minΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

= minΩ

2 [6 + 2 |119909|2+ e(12)(1+|119909|

2)]

e(12)(1+|119909|2) + 1

= min0le120596le1

2 [6 + 2120596 + e(12)(1+120596)]

e(12)(1+120596) + 1 =

2 (8 + e)1 + e

(52)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 =

1 + e2 (8 + e)

int+infin

2

eminus1199042d119904

=1 + e

e (8 + e)

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) = 2 ln 1 + e(8 + e) (119879 minus 119905)

(53)


(119906e119906)119905= nabla sdot (

eminus119906


120597119906


2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(54)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923lt 1 Now

we have

119886 (119906) =eminus119906

1 + 119906

119892 (119906) = 119906e119906

119891 (119906) = eminus119906


(55)

Setting

120596 = |119909|2 (56)


120573 = maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

= maxΩ

16 minus 2 |119909|2minus 3 |119909|

4

e2(1+|119909|2) (2 + |119909|2)4

= max0le120596le1

16 minus 2120596 minus 31205962

e2(1+120596) (2 + 120596)4 = eminus2

(57)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) = (2 + |119909|

2) e119905eminus2

minus 1 (58)


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573 fl maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

gt 0 (35)


int+infin

1198980

119886 (119904)

eminus119904d119904 = +infin 119898

0fl minΩ

1199060(119909) (36)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (37)

where

119870(119910) fl int119910

1198980

119886 (119904)

eminus119904d119904 119910 ge 119898

0 (38)



119876 (119909 119905) fl minuse119906119906119905+ 120573

1

119886 (119906) (39)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876

minus 119876119905

= 1205731

1198861198921015840(119886 + 119886

1015840) |nabla119906|

2+

1198921015840

119886(

119886

1198921015840)

1015840

e119906 (119906119905)2

+ 1205731

1198861198921015840(1198911015840+ 119891)

(40)


119886

1198921015840Δ119876 + 2

1198861015840 minus 119886

1198921015840nabla119906 sdot nabla119876

+119886

1198921015840[1 minus

1198861015840

119886+ (

1198861015840

119886)

1015840

] |nabla119906|2+ (

119891

119886)

1015840

+119891

119886119876


(41)


120597119876

120597119899= minuse119906119906

119905

120597119906

120597119899minus e119906

120597119906119905

120597119899minus 120573

1198861015840

1198862120597119906

120597119899

= minuse119906119906119905119887 minus e119906 (120597119906

120597119899)119905

minus 1205731198861015840

1198862119887

= minuse119906119906119905119887 minus e119906 (119887 (119909 119906 119905))

119905minus 120573

1198861015840

1198862119887


1198861015840

1198862119887

= (119887 + 119887119906) (119876 minus 120573

1

119886) minus e119906119887

119905minus 120573

1198861015840

1198862119887

= (119887 + 119887119906) 119876 minus 120573

119886119887 + (119886119887)119906

1198862minus e119906119887119905

on 120597Ω times (0 119879)

(42)


minΩ

119876 (119909 0) = minΩ

minuse1199060 (1199060)119905+ 120573

1

119886 (1199060)

= minΩ


0) nabla1199060) + 119891 (119906

0)

eminus11990601198921015840 (1199060)

+ 1205731

119886 (1199060)

= minΩ

1

119886 (1199060)(120573

minus119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

) = 0

(43)


119876 ge 0 in Ω times [0 119879) (44)

that is

119886 (119906)

eminus119906119906119905le 120573 (45)


1

120573int119905

0

119886 (119906)

eminus119906119906119905d119905 =

1

120573int119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904 le 119905 (46)


119870 (119906 (119909 119905)) minus 119870 (1199060(119909))

= int119906(119909119905)

1198980

119886 (119904)


1199060(119909)

1198980

119886 (119904)


119906(119909119905)

1199060(119909)

119886 (119904)

eminus119904d119904

= int119905

0

119886 (119906)

eminus119906119906119905d119905 le 120573119905

(47)



119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (48)


4 Applications




(119906

2+ e1199062)

119905


120597119906

120597119899= e(34)(119906minus2) + e119906minus2+119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(49)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923

lt 1 Wenote

119886 (119906) = e2119906

119892 (119906) =119906

2+ e1199062

119891 (119906) = e119906

119887 (119909 119906 119905) = e(34)(119906minus2) + e119906minus2+119905|119909|2

(50)


120596 = |119909|2 (51)


120572 = minΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

= minΩ

2 [6 + 2 |119909|2+ e(12)(1+|119909|

2)]

e(12)(1+|119909|2) + 1

= min0le120596le1

2 [6 + 2120596 + e(12)(1+120596)]

e(12)(1+120596) + 1 =

2 (8 + e)1 + e

(52)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 =

1 + e2 (8 + e)

int+infin

2

eminus1199042d119904

=1 + e

e (8 + e)

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) = 2 ln 1 + e(8 + e) (119879 minus 119905)

(53)


(119906e119906)119905= nabla sdot (

eminus119906


120597119906


2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(54)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923lt 1 Now

we have

119886 (119906) =eminus119906

1 + 119906

119892 (119906) = 119906e119906

119891 (119906) = eminus119906


(55)

Setting

120596 = |119909|2 (56)


120573 = maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

= maxΩ

16 minus 2 |119909|2minus 3 |119909|

4

e2(1+|119909|2) (2 + |119909|2)4

= max0le120596le1

16 minus 2120596 minus 31205962

e2(1+120596) (2 + 120596)4 = eminus2

(57)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) = (2 + |119909|

2) e119905eminus2

minus 1 (58)


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) (48)


4 Applications




(119906

2+ e1199062)

119905


120597119906

120597119899= e(34)(119906minus2) + e119906minus2+119905|119909|

2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(49)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923

lt 1 Wenote

119886 (119906) = e2119906

119892 (119906) =119906

2+ e1199062

119891 (119906) = e119906

119887 (119909 119906 119905) = e(34)(119906minus2) + e119906minus2+119905|119909|2

(50)


120596 = |119909|2 (51)


120572 = minΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

e11990601198921015840 (1199060)

= minΩ

2 [6 + 2 |119909|2+ e(12)(1+|119909|

2)]

e(12)(1+|119909|2) + 1

= min0le120596le1

2 [6 + 2120596 + e(12)(1+120596)]

e(12)(1+120596) + 1 =

2 (8 + e)1 + e

(52)


119879 le1

120572int+infin

1198720

119886 (119904)

e119904d119904 =

1 + e2 (8 + e)

int+infin

2

eminus1199042d119904

=1 + e

e (8 + e)

119906 (119909 119905) le 119867minus1

(120572 (119879 minus 119905)) = 2 ln 1 + e(8 + e) (119879 minus 119905)

(53)


(119906e119906)119905= nabla sdot (

eminus119906


120597119906


2

on 120597Ω times (0 119879)

119906 (119909 0) = 1 + |119909|2 in Ω

(54)

where Ω = 119909 = (1199091 1199092 1199093) | |119909|2 = 1199092

1+ 11990922+ 11990923lt 1 Now

we have

119886 (119906) =eminus119906

1 + 119906

119892 (119906) = 119906e119906

119891 (119906) = eminus119906


(55)

Setting

120596 = |119909|2 (56)


120573 = maxΩ

119886 (1199060) [nabla sdot (119886 (119906

0) nabla1199060) + 119891 (119906

0)]

eminus11990601198921015840 (1199060)

= maxΩ

16 minus 2 |119909|2minus 3 |119909|

4

e2(1+|119909|2) (2 + |119909|2)4

= max0le120596le1

16 minus 2120596 minus 31205962

e2(1+120596) (2 + 120596)4 = eminus2

(57)


119906 (119909 119905) le 119870minus1

(120573119905 + 119870 (1199060(119909))) = (2 + |119909|

2) e119905eminus2

minus 1 (58)


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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