solutions of Reation-diffusion equation

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1、GLOBALEXISTENCEFORSEMILINEARREACTION-DIFFUSIONSYSTEMSONEVOLVINGDOMAINSCHANDRASEKHARVENKATARAMAN,OMARLAKKIS,ANDANOTIDAMADZVAMUSEAbstract.Wepresentglobalexistenceresultsforsolutionsofreaction-diusionsystemsonevolvingdomains.Globalexistenceresultsforaclassofreaction-diusionsystemsonx

2、eddomainsareextendedtothesamesystemsposedonspatiallylinearisotropicallyevolvingdomains.Theresultsholdwithoutanyassumptionsonthesignofthegrowthrate.Theanalysisisvalidformanysystemsthatcommonlyariseinthetheoryofpatternformation.Wepresentnumericalresultsillustratingourtheoreticalndings

3、.1.IntroductionSincetheirseminalintroductionbyTuring[1952],reaction-diusionsystems(RDS's)havecon-stitutedastandardframeworkforthemathematicalmodellingofpatternformationinchemistryandbiology.Recentadvancesinmathematicalmodellinganddevelopmentalbiologyidentifytheimportantroleofdomaine

4、volutionascentralintheformationofpatterns,bothempiricallyKondoandAsai[1995]andcomputationallyComanici[2008];Crampinetal.[1999];Madzva-museandMaini[2007].Inthisrespect,manynumericalstudies,suchasBarrassetal.[2006];Madzvamuse[2006],ofRDS'sonevolvingdomainsareavailable.Yet,fundamentalma

5、thematicalquestionssuchasexistenceandregularityofsolutionsofRDS'sonevolvingdomainsremainsanimportantopenquestionKelkelandSurulescu[2009].NumerousstudiesonthestabilityofsolutionsofRDS'sonxeddomainsareavailable,forexample,Hollisetal.[1987];Rothe[1984];WeiandWinter[2008],butverylittlel

6、iteratureregardingthestabilityofsolutionsofRDS'sonevolvingdomains.Madzvamuseetal.[2010]providesalinearstabilityanalysisofRDS'soncontinuouslyevolvingdomains,andLabadie[2008]examinesthestabilityofsolutionsofRDS'sonmonotonicallygrowingsurfaces.Ourdiscussionherediersfromallthesestudiesi

7、nthatwefocusonplanarevolvingdomainsandweshowexistence,uniquenessandstability,foranentireclassofRDS'sonevolvingdomainsindependentlyoftherateofevolution.InthisarticleweprovethestabilityofsolutionsofRDS'sonaparticular,butfundamentallyimportant,classoftime-evolvingdomains:thatofboundedsp

8、atiallylinea