Global Existence of Solutions of the Gierer-Meinhardt System with Mixed Boundary Conditions

Transcript

1 pplied Mahemaics hp:// ISSN Online: ISSN Pin: Global Exisence of Soluions of he Giee-Meinhad Sysem wih Mixed Bounday Condiions Kwadwo nwi-fodjou Maius Nkashama Depamen of Mahemaics Ealham College Richmond IN US Depamen of Mahemaics Univesiy of labama a Bimingham Bimingham L US ow o cie his pape: nwi-fodjou K. Nkashama M. (7) Global Exisence of Soluions of he Giee-Meinhad Sysem wih Mixed Bounday Condiions. pplied Mahemaics hps://doi.og/.436/am Received: pil 3 7 cceped: June 6 7 Published: June 9 7 bsac We sudy he global (in ime) exisence of nonnegaive soluions of he Giee-Meinhad sysem wih mixed bounday condiions. In he eseach he Robin bounday Neumann bounday condiions wee used on he acivao he inhibio condiions especively. Based on he pioi esimaes of soluions he consideable esuls wee obained. Keywods Copyigh 7 by auhos Scienific Reseach Publishing Inc. This wok is licensed unde he Ceaive Commons ibuion Inenaional License (CC BY 4.). hp://ceaivecommons.og/licenses/by/4./ Open ccess civao-inhibio Sysem Giee-Meinhad Sysem Robin Neumann Bounday Condiions Global Exisence. Inoducion Biological spaial paen fomaion is one aea in applied mahemaics undegoing vivid invesigaions in ecen yeas. Mos models involved in biological phenomena ae of he geneal eacion-diffusion ype consideed by Tuing []. The disincive aibue of Tuing s appoach was he ole of auocaalysis in coexisence wih laeal inhibiion. These sudies led o he assumpion of he exisence of wo chemical subsances known as he acivao he inhibio [] [3]. One of he famous sudied models in biological spaial paen fomaion is he Giee-Meinhad sysem which has eceived numeous aenion has been exensively sudied [4] [5] [6]. The Giee-Meinhad sysem was used o model he head fomaion of a small fesh-wae animal called hyda [4]. We conside an acivao concenaion an inhibio concenaion saisfying he acivao-inhibio sysem given by DOI:.436/am June 9 7

2 K. nwi-fodjou M. Nkashama p + in T + b D + in T s + a on ν ν ( x) ( x) > ( x) ( x) in whee a b Laplace o diffusion opeao in ( T ) () N is a bounded smooh domain; is he N ; ( x) ν is he uni oue nomal a x ν : ν is he diecional deivaive in he diecion of he veco ν. We assume ha he eacion exponens ( p s ) > > > saisfy p < <. () s + The diffusion consans ae > D > fo he acivao inhibio especively. The ime elaxaion consan > was mahemaically ino- duced due o is usefulness on he sabiliy of he sysem. The consan b povides addiional suppo o he inhibio may be hough of as a measue of he effeciveness of he inhibio in suppessing he poducion of he acivao ha of is own. In [7] he aio in he middle of () is called ne self-acivaion index since i compaes how songly he acivao acivaes he poducion of iself wih how songly i acivaes ha of he inhibio. On he ohe h hey call he aio on he igh h side of () ne coss-inhibiion index since i compaes how songly he inhibio suppesses he poducion of he acivao wih ha of iself. Fo he he ineualiy in () we expec he poducion of he acivao o be seveely suppessed by he inhibio. In [4] some biological applicaions such as modeling of skeleal limb developmen Robin bounday condiions ae moe ealisic since he Neumann bounday condiions. compaaive numeical sudy of a eacion-diffusion sysem was made in [8] wih a ange of diffeen bounday condiions i evealed ha ceain ypes of bounday condiions seleced a paicula paen modes a he expense of ohes. I was shown ha he obusness of ceain paens could be gealy enhanced he auhos showed a possible applicaion o skeleal paen of limb. Special case was consideed fo he Neumann bounday condiion (i.e. a ) in [5] [9]. In [5] Masuda Takahashi poved he global soluions of he special case of () wih b exiss fo > povided in addiion o () one has ( p ) < ( N + ) we noe he sic ineualiy hee. In [9] Jiang impoved he ne self-acivaion index noed in [5] o ( p ) < showed ha he soluions exiss globally in ime. In his pape we conside he Robin bounday condiion (a ) on he acivao Neumann bounday condiion on he inhibio sudy he global (in ime) exisence of soluions fo he Giee-Meinhad sysem in (). The heoem lemmas in his cuen manuscip ae inspied by [9]. We esablish he global (in ime) exisence of () by poving he heoem below: 858

3 K. nwi-fodjou M. Nkashama Theoem. Suppose is a smooh bounded domain wih a smooh N p bounday in. ssume ha < min s+. Le W > max { N }. Then evey soluion ( ( x ) ( x )) of () exiss globally in ime.. Poof of Theoem The local exisence uniueness of () is sad moe deails can be found in [] []. pioi-esimaes need o be asceain in ode o pove global in ime exisence of soluions. Le ( ) be a soluion of () in [ T ). We wan o asceain ha is bounded away fom zeo. Le u inf x T hen Lemma. Poof. Le hen ( ) bu hus [ ) x u inf x > x u u e fo all < T. x saisfies x x u e (3) u( e ) D + ( e s + u ) D u( e ) u( e s ) D u( e ) + u( e s + ) D + u( e ) u( e s + ) D + D s + s D in [ T) x u x ν ν ν ( ) ( ) e ( ) on ν [ T ) on [ T ). ν 859

4 K. nwi-fodjou M. Nkashama ddiionally a fom (3) inf x x u x x x So ( x) fo any x. ence fom maximum pinciple ( x ) in [ T ) x u e inf ( ) ( ) e [ ) u x u T x Lemma. Fo any wo consans > le Define ( x ) ( x ) h d x T < hus p < min s+. Suppose ( )( + ) ( + D) ε D (4) hen ee whee > h ϑ h + Cu h. (5) ϑ ( p) ( s+ )( p ) s+ ( p) > ( ) ϑ Poof. Le > p+ ( p) ϑ C d h dx dx d x d + p D dx + s b p x d dx+ dx b ( + ) + D x d + + dx dx s p + D dx x d x d dx dx s b + (6) (7) 86

5 K. nwi-fodjou M. Nkashama Bu x d νds ( ) dx ddiionally x d a ds ε ( ) dx x d a ds ( ) dx x d ( + ) D ν D D x d ds+ dx D x d + D ( + ) D ( ) ds dx D x d + ( + ) D D dx x d + + now we have h dx a ds dx D ( + ) D + + x d dx p + + dx d. x + + s+ dx a ds dx + + D ( + ) D dx + p + + dx d. x + s p + dx a ds dx dx s D + ( ) dx+ + dx + D d x. dx 86

6 K. nwi-fodjou M. Nkashama We deduce fom above a uadaic euaion involving fix > choose we have heefoe he uadaic fom involving is non-posiive since is deeminan Thus ( )( + ) ( + D) D. Le us in he ineualiy above D D + 4 ( ) ( + ). + p s+ h dx a ds dx dx We have + p + dx dx dx + + s+ + + p + h + dx d. x + + s+ p < ( p ) > ( p )( s+ ) p < > s+ we choose > sufficienly small such ha Now we wie p > p s+ s + > +. b + p + p + p ( ) ( )( + ) ( + ) p p s s p + p ( p) ( p )( s ) s ( p ) + + ( p )( s+ ) s+ + p ( p) + p p+ ( p) p ( s ) + + ( s ) + p+ p+ ϑ ϑ + p p + p+ p+ p+ p p p ( ) p s ( + s+ ) 86

7 K. nwi-fodjou M. Nkashama bu hus + p p p+ p+ ϑ p+ p+ p+ + p+ ϑ p+ p+ p+ p+ ϑ p+ ( ϑ) p+ ( + ) p+ p ( ) p + ϑ + ( + ) ϑ + p + p+ p+ p+ ( + s+ ) p+ p+ ( ϑ) p+ ( + ) whee ϑ ae defined by (6) (7) p+ ( ϑ) p+ ( + s+ ) p+ ϑ p+ + + s+ p+ p p ϑ s+ ( + b) p+ p+ ϑ + u + s+ p+ p+ ϑ + u + s+ by Young s ineualiy we obain p+ p+ p+ ϑ + u + s+ p+ ( p) + p ϑ + u + + s+ ( + b) p+ ( p p ) + + ϑ u. + s+ ( + b) 863

8 K. nwi-fodjou M. Nkashama Theefoe p+ ( p p ) + + ϑ + s+ dx dx u dx ( + b) bu by ölde s ineualiy Thus ϑ ϑ ϑ ϑ ϑ dx ( dx) d x h. dx b + p + ( + ) p+ ( p) + s+ dx u h Cu h ϑ ϑ ϑ. Finally h h Cu h + ϑ. Remak. The condiion in (4) is ue fo any whee D K max D < K Lemma 3. Le < θ > ζ > on funcion. Le h h be a nonnegaive funcion on [ ) diffeenial ineualiy T be an inegable T saisfying he h θh + ζh < T. (8) Then h κ < T (9) whee κ is he maximal oo of he algebaic euaion Moeove if T we have ( ζ ) x G x h. lim sup h () whee κ is he maximal oo of he algebaic euaion κ ( ζ ). x G x Poof. h θh + ζh 864

9 K. nwi-fodjou M. Nkashama e h + θh ζh h + θh e ζh θ θ d e h ζ e h d θ θ d e d χ e ζ ( χ) h ( χ) d χ d θχ θχ h e h h e ζ χ h χ dχ θ θχ + e h h e ζ χ h χ dχ θ θχ θ θ( χ) h e h + e ζ χ h χ d χ. () Le ( χ ) h sup h < χ< << T θ( χ) G ζ : sup e ζ χ d χ in paicula a T we obain now ha G θ( χ) ζ : lim sup e ζ χ dχ ( ) ( ζ ) h h + G h () Noice ha he uaniy G ( ζ ) is finie hence (9) follows fom (). s in () we asceain lim sup h G ( ζ ) lim sup h hus () follows since G ( ζ ) is finie. The nex Lemma follows afe applying Lemma 3 o (5). Lemma 4. Fo any > such ha < all condiions in Lemma C T C T such ha hold ue. Then hee exiss a consan h C( T) fo all [ T). (3) Poof. Fo sufficienly small > such ha < wih > we obain D( )( + ) ( + D) < heefoe we deduce fom Lemma ha h saisfies h h Cu + h ϑ. Since < 865

10 K. nwi-fodjou M. Nkashama fo all [ T) u u e hen fom Lemma 3 (3) is ue fo > sufficienly small such ha <. Since is bounded away fom zeo hen (3) is ue fo any >. Fom Lemma Lemma 4 we deduce he Coollay below. Coollay. Le all ohe assumpions in Theoem Lemma Lemma 3 Lemma 4 hold ue. Define g g p + b hen hee exis posiive consan C ( T) s such ha g j C T j L fo all < T. Poof. The poof o his Coollay follows fom Lemma 3 Lemma Conclusion In his pape we have sudied he Giee-Meinhad sysem wih Robin bounday condiions Neumann bounday condiions on he acivao inhibio especively. Global exisence of soluions have been obained unde he mixed bounday condiions using a pioi esimaes of soluions. Refeences [] Tuing.M. (95) The chemical basis of mophogenesis. Philosophical Tansacions of he Royal Sociey of London Seies B hps://doi.og/.98/sb.95. [] Meinhad. (98) Models of Biological Paen Fomaion. cademic Pess London New Yok. [3] Meinhad. (998) The lgoihmic Beauy of Sea Shells. nd Ediion Spinge Belin eidelbeg. hps://doi.og/.7/ [4] Giee. Meinhad. (97) Theoy of Biological Paen Fomaion. Kybeneik (Belin) hps://doi.og/.7/bf8934 [5] Masuda K. Takahashi K. (987) Reacion-Diffusion Sysems in he Giee-Meinhad Theoy of Biological Paen Fomaion. Japan Jounal of pplied Mahemaics hps://doi.og/.7/bf [6] Maini P.K. Wei J. Wine M. (7) Sabiliy of Spikes in he Shadow Giee-Meinhad Sysem wih Robin Bounday Condiions. Chaos hps://doi.og/.7/bf [7] Ni W.-M. Suzuki K. Takagi I. (6) The Dynamics of a Kineic civao-inhibio Sysems. The Jounal of Diffeenial Euaions hps://doi.og/.6/j.jde.6.3. [8] Dillon R. Maini P.K. Ohme.G. (994) Paen Fomaion in Genealized 866

11 K. nwi-fodjou M. Nkashama Tuing Sysems. I. Seady-Sae Paens in Sysems wih Mixed Bounday Condiions. The Jounal of Mahemaical Biology hps://doi.og/.7/bf665 [9] Jiang. (6) Global Exisence of Soluions of an civao-inhibio Sysems. Discee Coninuous Dynamical Sysems hps://doi.og/.3934/dcds [] eny D. (98) Geomeic Theoy of Semilinea Paabolic Euaions. Spinge- Velag New Yok. hps://doi.og/.7/bfb89647 [] Pazy. (983) Semigoups of Linea Opeaos pplicaions o Paial Diffeenial Euaions. Spinge-Velag New Yok. hps://doi.og/.7/ Submi o ecommend nex manuscip o SCIRP we will povide bes sevice fo you: cceping pe-submission inuiies hough Facebook LinkedIn Twie ec. wide selecion of jounals (inclusive of 9 subjecs moe han jounals) Poviding 4-hou high-ualiy sevice Use-fiendly online submission sysem Fai swif pee-eview sysem Efficien ypeseing poofeading pocedue Display of he esul of downloads visis as well as he numbe of cied aicles Maximum disseminaion of you eseach wok Submi you manuscip a: hp://papesubmission.scip.og/ O conac am@scip.og 867