Global Journal of Science Frontier Research: F
Mathematics and Decision Sciences
Volume 14 Issue 2 Version 1.0 Year 2014
Type : Double Blind Peer Reviewed International Research Journal
Publisher: Global Journals Inc. (USA)
Online ISSN: 2249-4626 & Print ISSN: 0975-5896
Global Existence and Uniqueness of the Weak Solution in
Keller Segel Model
By C. Messikh, A. Guesmia & S. Saadi
University of Badji Moktar, Algeria
Abstract- This paper deals with the global existence, uniqueness and boundedness of the weak
solution for the chemotaxis system (P) defined as
#
The system (P) is under homogeneous Dirichlet boundary conditions in a convex smooth
bounded domain with smooth boundary ,9/ Based on Galerkin’s
method, Lax -Milgran’s and maximum principle, a prove of the existence and
uniqueness of a global solution for the system (P) is determined. Moreover we show that the
unique solution is positive.
Keywords and Phrases: chemotaxis, global existence, boundedness, positive solution.
GJSFR-F Classification : MSC 2010: 35K58, 45G05, 65C35, 82C22, 82C31,82C80, 92C17
GlobalExistenceandUniquenessoftheWeakSolutioninKellerSegelModel
:
Strictly as per the compliance and regulations of
© 2014. C. Messikh, A. Guesmia & S. Saadi. This is a research/review paper, distributed under the terms of the Creative
Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non
commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Year
C. Messikh ɲ, A. Guesmia ʍ & S. Saadi ʌ
The system (P) is under homogeneous Dirichlet boundary conditions in a convex smooth bounded domain
with smooth boundary ,9/ Based on Galerkin ’s method, Lax -Milgran’s Theorem
and maximum principle, a prove of the existence and uniqueness of a global solution for the system (P) is determined. Moreover we show that the unique solution is positive.
Keywords and Phrases: chemotaxis, global existence, boundedness, positive solution.
I.
I.
Introduction
308:?,C4> 4> ,9 48;:=?,9? 80,9> 1:= .077@7,= .:88@94.,?4:9 ? 4> ?30 49
[email protected] :1 .3084.,7 >@->?,9.0> 49 ?30 09A4=:9809? :9 ?30 8:A0809? :1 8:-470
>;0.40> &34> .,9 70,/ ?: >?=4.?7D :=409?0/ 8:A0809? := ?: ;,=?4,77D :=409?0/ ,9/
;,=?4,77D ?@8-7492 8:A0809? &30 8:A0809? ?:B,=/> , 34230= .:9.09?=,?4:9
:1 ?30 .3084.,7 >@->?,9.0 4> .,770/ ;:>4?4A0 .308:?,C4> B30=0,> ?30 8:A0809?
?:B,=/> =024:9> :1 7:B0= .3084.,7 .:9.09?=,?4:9 4> .,770/ 902,?4A0 .308:?,.?4.,7
8:A0809?
&30 .7,>>4.,7 .308:?,C4> 8:/07 J ?30 >:.,770/ 0770=I%0207 8:/07 J >D>
?08 /0F90/ 49 B,> F=>? 49?=:/@.0/ -D #,7?,6 0770= ,9/ %0207 ) ) ) ) ( , , X Issue II Version I
F ) Volume XIV
#
B30=0 )( , /09:?0> ?30 /09>4?D :1 -,.?0=4, 49 ?30 ;:>4?4:9 , ,9/
,? ?480 ( ?30 .:9.09?=,?4:9 :1 .3084.,7 >429,7 >@->?,9.0 ?30 =07,C
,?4:9 ?480 ?30 ;,=,80?0= ?30 >09>4?4A4?D :1 .077> ?: ?30 .308:,??=,.?,9? ,9/
,=0 24A09 >8::?3 1@9.?4:9> > 4? .,9 -0 >009 B309 ?30 8:/07 4>
Author ɲ: University of Badji Moktar, Laboratory LMA, Department of Mathematics, Faculty of Sciences, BP 12. 23000 El Hadjer Algeria.
e-mail: [email protected]
Author ʍ: University of Skika 20 août 1955 Department of Mathematics, Faculty of Science. e-mail: [email protected]
Author ʌ: University of Badji Moktar, Laboratory LANOS, Department of Mathematics, Faculty of Sciences, BP12. 23000 El Hadjer Algeria.
e-mail: Signor [email protected]
© 2014 Global Journals Inc. (US)
Global Journal of Science Frontier Research
47
)
*+ 0770= ,9/ %0207 94?4,?4:9 :1 >7480 8:7/ ,22=02,?4:9 A40B0/
,> ,9 49>?,-474?D &30:= 4:7:2D Abstract- This paper deals with the global existence, uniqueness and boundedness of the weak solution for the
chemotaxis system (P) defined as
2014
Global Existence and Uniqueness of the
Weak Solution in Keller Segel Model
Ref
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
.,770/ #,=,-:74.#,=,-:74. B3470 4? 4> ,9 774;?4.#,=,-:74. 8:/07 B309 &34> 8:/077492 4> A0=D >48;70 4? 0C34-4?> , ;=:1:@9/ 8,?308,?4.,7 >?=@.?@=0
,9/ 8:>?7D :97D /4809>4:9 4> @9/0=>?::/ 0>;0.4,77D .308:?,.?4. .:77,;>0
&30 ;=:;:>0/ 8:/07 3,> -009 0C?09>4A07D >?@/40/ 49 ?30 7,>? 10B D0,=> >00
*+*+*
+*+ 1:= , =0.09? >@=A0D ,=?4.70>
Year
2014
&30 #,=,-:74.#,=,-:74. 8:/07 3,> -009 49A0>?42,?0/ -D 8,9D ,@?30=> >00
1:= 0C,8;70> $01> *+ ,9/ *+ ,?6@7749 *+ 3,/ /0A07:;0/ 9@80=4.,7 80?3:/
, .:8;:>4?0 ;,=?4.702=4/ B4?3 ,/,;?4A0 ?480 >?0;;492 B34.3 ,77:B> @> ?:
=0>:7A0 ,9/ ;=:;,2,?0> >492@7,= >:7@?4:9> B309 B4?3 !0@8,99 -:@9/,=D .:9/4
?4:9
)
Global Journal of Science Frontier Research
9 ?34> ;,;0= B0 /08:9>?=,?0 ?30 27:-,7 0C4>?09.0 ,9/ @94<@090>> :1 B0,6
;:>4?4A0 >:7@?4:9 1:= ?30 0774;?4.;,=,-:74. 8:/07H> ;=:-708 /0F90/ ,>
) ) ) )
) , )
,
( , ,
)30=0 )( , 4> , 1@9.?4:9 /09:?0> ?30 /09>4?D :1 -,.?0=4, 49 ?30 ;:>4?4:9
, := 4> , -:@9/0/ .:9A0C /:8,49 B4?3 >8::?3 -:@9/,=D /09:?0> ?30 .:9.09?=,?4:9 :1 .3084.,7 >429,7 ?3,? >?48@7,?0> ?30 -,.?0=4, &30
;,=,80?0= 4> , ?480 .:9>?,9? ,9/ 4? 4> 0C;=0>>0/ :9 ?30 :90 3,9/ ?30 8:A0809?
:1 -,.?0=4, =0;=0>09?492 , =,9/:8 /4>?=4-@?4:9 >4/0 ,9/ , /0?0=8494>?4. /=41? 49
?30 /4=0.?4:9 :1 3423 .:9.09?=,?4:9> ,9/ >0.:9/7D ?30 /4K@>4:9 /02=,/,?4:9 :1
&: >48;741D ?30 >:7@?4:9 :1 ?30 >D>?08 # , /0.:8;:>4?4:9 :1 # 49?: ?B:
>@->D>?08> # ,9/ # ,=0 ,/:;?0/ ,C 472=,8H> &30:=08 4> A0=D 48;:=
?,9? ?30:=08 B34.3 B0 307; @> ?: /08:9>?=,?0 ?30 0C4>?09.0 ,9/ @94<@090>> :1
, B0,6 >:7@?4:9 1:= ?30 >D>?08 # :B0A0= ?34> ?30:=08 .,9 9:? -0 ,;;740/
/4=0.?7D -0.,@>0 4? 4> , 9:93:8:209:@> >D>?08 := ?34> =,4>:9 ,9 ,/:;?,
?4:9 :1 &=,.0 &30:=08 4> @>0/ ?: >48;741D ?30 >D>?08 # ,9/ ?:20?30= B4?3
,70=6492 80?3:/ B0 .,9 /08:9>?=,?0 ?30 0C4>?09.0 ,9/ @94<@90>> :1 , B0,6
>:7@?4:9 1:= ?30 >D>?08 # &30=01:=0 B0 3,A0 ?30 0C4>?09.0 ,9/ @94<@90>> 1:=
?30 ;=:-708 # :=0:A0= B0 >3:B ?3,? ?30 >:7@?4:9 4> ;:>4?4A0 &30 1:77:B
492 494?4,7-:@9/,=D .:9/4?4:9> :9 ) ,9/ ,>>@8;?4:9> ,=0 @>0/ ?: ;=:A0 ?30
;=:;:>0/ >:7@?4:9 :1 #
© 2014 Global Journals Inc. (US)
) ) ,9/
*+ & 47709 #,49?0= 7:-,7 0C4>?09.0 1:= , ;,=,-:74. .308:?,C4> 8:/07
B4?3 ;=0A09?4:9 :1 :A0=.=:B/492 /A ;;7 ,?3 F ) Volume XIV Issue II Version I
48
&30 774;?4.#,=,-:74. 8:/07 3,> -009 49A0>?42,?0/ -D 8,9D ,@?30=> >00 1:=
0C,8;70> $01> *+ ,9/ *+ &34> 8:/07 3,A0 -009 .,==40/ :@? B30=0 ?30 8,49
.:9.0=9 4> B30?30= ?30 >:7@?4:9 :1 8:/07 4> -:@9/0/ := -7:B@; ? 3,> -009
;=:A0/ ?3,? ?30 >:7@?4:9 >?=:927D /0;09/> :9 ?30 >;,?4,7 /4809>4:9 ? /:0>
9:? :..@= 49 :90/4809>4:9,7 ;=:-708> ,9/ 4? :..@=> .:9/4?4:9,77D 49 34230=
/4809>4:9,7 >4?@,?4:9> :=0 ;=0.4>70D >00 *
+ 49 .,>0 34230= /4809>4:9> $ 41 ?30 9:=8 :1 494?4,7 .:9/4?4:9 ) 4> >8,77 49 >;,.0 ?309 ?30=0 ,=0 27:-,7
, )
) B4?3 4> >8,77 ?309 ?30=0
B0,6 >:7@?4:9> ,9/ 41
4> -7:B @; 49 , F94?0 ?480 @? 49 ?B: /4809>4:9 >00 41 ) ?30=0 ,=0 >8::?3 >:7@?4:9> ,9/ 41 ) ?30=0 4> .=0,?4:9 :1
, >492@7,= 80,>@=0 -7:B@; 49 F94?0 ?480
Ref
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@94<@090>> :1 B0,6 >:7@?4:9 :1 ;=:-708 # B34.3 4?> A,=4,?4:9,7 1:=8@7,? 4>
24A09 -D 0<@,?4:9 ,9/ @>0 ?30 80?3:/ :1 ,70=6492 ?: >?@/D ?30 0C4>?09.0
,9/ @94<@090>> :1 B0,6 >:7@?4:9 :1 ;=:-708 # B34.3 4?> A,=4,?4:9,7 1:=8@7,?
4> 24A09 -D 0<@,?4:9 %: B0 3,A0 ?30 0C4>?09.0 ,9/ @94<@090>> :1 B0,6
>:7@?4:9 :1 ;=:-708 #
a) Existence of weak solution of the problem (P2)
&)$ " /$! $4+*/$!.%. $*' . $!) /$! +-*'!( $. *)'4 *)!
.*'0/%*) "*- )4 & :1 ?30 ;=:-708 D ,;;7D492 ?30 &30:=08 :1 ,C 472=,9 ?30 >:7@?4:9 0C4>?> ,9/ 4? 4> @94<@0 %: #
3,> @94<@0 >:7@?4:9
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# 4> ;:>4?4A0 ,> 1:77:B> @7?4;7D492 ?30 F=>? 0<@,?4:9 :1 # -D & B0 :-?,49 :?30= A,=4,?4:9,7 1:=8@7,? 1:= ;=:-708 # #
& & , © 2014 Global Journals Inc. (US)
)
I.
Existence of Weak Solution of the Problem (P)
X Issue II Version I
F ) Volume XIV
49
2$!-!
II.
Year
) * ) * ( Global Journal of Science Frontier Research
Notes
1 ?30 3D;:?30>4> 4> >,?4>F0> ,9/ @>492 ?30 ?30:=08 :1 ?=,.0 :90 .,9 F9/
, 741?492 :1 ?34> ?=,.0 B34.3 B0 /09:?0 &3@> -D /0F94?4:9 4?
A0=4F0> !:B 7::6492 1:= 3,A492 ?30 1:=8 =0/@.0>
?30 ;=:-708 # ?: 49 , :9 0%!+!&% ! .4 ) 2%/$ ) %. 2!& .*'0/%*) *" /$! +-*'!( %" ) *)'4 %"
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Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
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1 :9 ?309 %: B0 3,A0
,
Year
2014
F ) Volume XIV Issue II Version I
50
,
,
,
%49.0 ?30 >@;;:=? :1 1@9.?4:9> ,9/ 4> >0? , ,) , &34> 48;740> ?3,? ) :9 , > ?3@> B0 3,A0
, 49,77D B0 F9/ b) Existence of a weak solution of the problem (P1)
01:=0 ;=:A492 ?30 0C4>?09.0 ,9/ @94<@090>> :1 B0,6 >:7@?4:9 :1 ;=:-708 # B0 900/ ?30 1:77:B492 7088,
$$ % *- '' * /$!) ( %. *)/%)0*0. %) /$!-! !3%./. *)./)/ +*.%/%1! .0$ /$/
) * ( ) * )
Global Journal of Science Frontier Research
%% *- )4 ) ) %. $*' $!) /$!-! !3%./. *)./)/ +*.%/%1! .0$ /$/
) ) ) (
)&& 4 )0 @>0
?30 ,@.3D%3B,=E 490<@,74?D ,9/ 1:=
,9D & B4?3 $ := $ B0 :-?,49 4 ,> 1:77:B>
) * (
44
) * ) * ) * ) * ,6492 @>0 :1 ?30 0C;=0>>4:9 :1 ) ) ( -0.:80>
) ) ,
) ,
) ) ) , ) ) ) ( © 2014 Global Journals Inc. (US)
) Notes
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
49,77D -D #:49.,==0 490<@,74?D D407/>
) ) ( ) &: /08:9>?=,?0 ?30 0C4>?09.0 :1 B0,6 >:7@?4:9 :1 # A4, ?30 80?3:/ :1
,70=6492 B0 ,>>@80 + + , ,=0 >8::?3 1@9.?4:9> A0=41D492
+ 4> ,9 :=?3:2:9,7 -,>4> :1 :9>4/0= , ;:>4?4A0 49?020= # )0 B477 7::6 1:= , 1@9.?4:9 ) :1 ?30 1:=8
) ( ( +
B34.3 >,?4>F0>
) + ,9/
) + ) + ( ( ,9/
! #
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@>492 B0 :-?,490/
2014
+ 4> ,9 :=?3:9:=8,7 -,>4> :1 Year
,9/
51
X Issue II Version I
F ) Volume XIV
Notes
+ + ( ( ,9/ ! #
..:=/492 ?: >?,9/,=/ 0C4>?09.0 ?30:=D 1:= :=/49,=D /4K0=09?4,7
0<@,?4:9>
?30=0
0C4>?> , @94<@0 ,->:7@?07D .:9?49@:@> 1@9.?4:9> ( >,?4>
1D492 ,9/ %: ) :1 ?30 1:=8 >,?4>F0> ,9/ 1:= ,77
( c) Energy estimates
)0 ;=:;:>0 9:B ?: >09/ # ?: 49F94?D ,9/ >3:B , >@->0<@09.0 :1 :@= >:7@?4:9>
) :1 ?30 ,;;=:C48,?4:9 ;=:-708> ,9/ .:9A0=20> ?: , B0,6 >:7@?4:9
:1 # := ?34> B0 B477 900/ >:80 @941:=8 0>?48,?0>
&)$ )!-#4 !./%(/!. $!-! !3%./. *)./)/ !+!) %)# *)'4
*) ) .0$ /$/
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?309 =0.,77492 B0 F9/
) ) ) ) ( © 2014 Global Journals Inc. (US)
Global Journal of Science Frontier Research
)
I.
( Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
1:= ,77 ( =:8 088, ?30=0 0C4>?> .:9>?,9? >@.3 ?3,?
) ) ) (
1:= ,77 ( :9>0<@09?7D D407/> ?30 490<@,74?D
) ) 1:= ,77 ( (
&34> 48;740> ?3,?
Year
2014
) ) ) 1:= ,77 ( F ) Volume XIV Issue II Version I
52
)
Global Journal of Science Frontier Research
%: B0 3,A0
) ) 9?02=,?0 490<@,74?D 1=:8 ?: ,9/ B0 08;7:D ?30 490<@,74?D ?: F9/
) ) ( ) * ,9/ B=4?0 * * * B30=0
4C ,9D * B4?3
* '%$ + ,9/ * + ! # )0 @>0 B0 /0/@.0
1:= ,77 ( ?3,?
) * ) * ( &309 48;740>
.:9>0<@09?7D
) * ) * ) * ) * ( ) * ) %49.0 * * &3@>
) ) ,9/ ?30=01:=0
) ) ( ) ( ) d) Existence and uniqueness
!0C? B0 ;,>> ?: 7484?> ,> # ?: -@47/ , B0,6 >:7@?4:9 :1 :@= 494?4,7
-:@9/,=DA,7@0 ;=:-708 # &)$ 3%./!)! *" 2!& .*'0/%*) ) !- $4+*/$!.%. ) /$!-!
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2!&'4 %) ) )
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* ( © 2014 Global Journals Inc. (US)
Notes
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
2$!-! -! #%1!) .(**/$ "0)/%*). ! $**.! # (0'/%+'4 !,0
/%*) 4 ( .0( "*- ! ) /$!) %)/!#-/! 2%/$ -!.+!/ /* ( /*
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% !)/%/4 2! $1!
( ( -*( !(( 2! $1! ( .* /$!) %)/!#-/! 2%/$ -!.+!/ /* ( /* 5)
53
)
I.
2014
) * ) * ( ( Year
X Issue II Version I
F ) Volume XIV
/$0. © 2014 Global Journals Inc. (US)
Global Journal of Science Frontier Research
Notes
! -!'' /* 5) 0+*) +..%)# /* 2!& '%(%/. /$/
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
e)
Global solution of problem (P)
"@= 8,49 =0>@7?> 49 ?34> ;,;0= ,=0 >?,?0/ ,> 1:77:B>
&)$ % %" $!) ' .*'0/%*) ) *" +-*'!( %. #'*'
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( F ) Volume XIV Issue II Version I
) ,
)0 /0=4A,?0 ?30 0<@,?4:9 ,9/ B0 @>0 F=>?> 0<@,?4:9> :1 # ,9/ # ?:
F9/
4 )0 3,A0
) ) ( (
54
)
Global Journal of Science Frontier Research
?30=01:=0
( 44 )0 3,A0
) ) ( (
) ( ) ) , &34> 48;740> ?3,?
( )&'&*!+!&% !/ ) ) ) %.
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, ) ' ' )
) , %)! ) ) * ) References Références Referencias
*+ 77,4=0 9,7D>0 9@8M=4<@0 0? :;?484>,?4:9 /4?4:9> /0 7H0.:70 ;:7D?0.3
94<@0>
*
+ 7,9.30? ,==477: # ,@=09L:? =4?4.,7 8,>> 1:= , #,?7,6I0770=I
%0207 8:/07 B4?3 /02090=,?0 /4K@>4:9 49 34230= /4809>4:9> (# *+
# =0990= # :9>?,9?49 # ,/,9:K %.309607 % (096,?,=:
8,94 4K@>>4:9 ,??=,.?4:9 ,9/ .:77,;0 !:97490,=4?D © 2014 Global Journals Inc. (US)
Notes
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
*+
# =0990= % 0A4?:A " @/=090 #3D>4.,7 80.3,94>8> 1:= .308:
?,.?4. ;,??0=9 1:=8,?4:9 -D -,.?0=4, 4:;3D> 5 *+ :7-0,@7? ,9/ %.3804>0= &30 ?B:/4809>4:9,7 0770=%0207 8:/07
,1?0= -7:B@; 4>.=0?0 ,9/ :9?49@:@> D9,84.,7 %D>?08> *+ A,9> #,=?4,7 /4K0=09?4,7 <@,?4:9 % #=0>>
*+ & 47709 #,49?0= 7:-,7 0C4>?09.0 1:= , ;,=,-:74. .308:?,C4> 8:/07
B4?3 ;=0A09?4:9 :1 :A0=.=:B/492 /A ;;7 ,?3 *+ 0770= ,9/ %0207 94?4,?4:9 :1 >7480 8:7/ ,22=02,?4:9 A40B0/
,> ,9 49>?,-474?D &30:= 4:7:2D *+ ,?6@7749 >?@/D :1 -7:B@;> 49 ?30 0770=I%0207 8:/07 :1 .308:?,C4>
!:97490,=4?D I
*+ % #,?7,6 $,9/:8 B,76 B4?3 ;0=>4>?09.0 ,9/ 0C?0=9,7 -4,> @77
4:;3D> I
,?3
*
+
)49670= 22=02,?4:9 A> 7:-,7 /4K@>4A0 -03,A4:= 49 ?30 34230=
/4809>4:9,7 0770=%0207 8:/07 4K0=09?4,7 <@,?4:9> *+
)49670= ->09.0 :1 .:77,;>0 49 , ;,=,-:74. .308:?,C4> >D>?08 B4?3
>429,7/0;09/09? >09>4?4A4?D ,?3 !,.3= 2014
0==0=: (07,E<@0E -7:B@; 80.3,94>8 1:= , .308:?,C4>
8:/07 99 %.@:7, !:=8 %@; #4>, 7 %.4 Year
*+
55
X Issue II Version I
F ) Volume XIV
Notes
Global Journal of Science Frontier Research
)
I.
© 2014 Global Journals Inc. (US)
Global Existence and Uniqueness of the Weak Solution in Keller Segel Model
Year
2014
Notes
F ) Volume XIV Issue II Version I
56
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Global Journal of Science Frontier Research
)
© 2014 Global Journals Inc. (US)