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1、THEPOROUSMEDIUMEQUATIONMathematicaltheorybyJUANLUISVAZQUEZ¶Dpto.deMatem¶aticasUniv.Auton¶omadeMadrid28049Madrid,SPAIN
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9、-iTomywifeMariluzPrefaceTheHeatEquationisoneofthethreeclassicallinearpartialdi®erentialequationsofsecondorderthatformthebasisofanyelementaryintrod
10、uctiontotheareaofPartialDi®erentialEquations.Itssuccessindescribingtheprocessofthermalpropagationhasknownaper-manentpopularitysinceFourier'sessayTh¶eorieAnalytiquedelaChaleurwaspublishedin1822,[237],andhasmotivatedthecontinuousgrowthofmathematicsintheformofFourieranalysi
11、s,spectraltheory,settheory,operatortheory,andsoon.Lateron,itcontributedtothedevelopmentofmeasuretheoryandprobability,amongothertopics.ThehighregardoftheHeatEquationhasnotbeenisolated.Anumberofrelatedequa-tionshavebeenproposedbothbyappliedscientistsandpuremathematiciansas
12、objectsofstudy.Ina¯rstextensionofthe¯eld,thetheoryoflinearparabolicequationswasdevel-oped,withconstantandthenvariablecoe±cients.Thelineartheoryenjoyedmuchprogress,butitwassoonobservedthatmostoftheequationsmodellingphysicalphenomenawithoutexcessivesimpli¯cationarenonlinea
13、r.However,themathematicaldi±cultiesofbuildingtheoriesfornonlinearversionsofthethreeclassicalpartialdi®erentialequations(Laplace'sequation,heatequationandwaveequation)madeitimpossibletomakesigni¯cantprogressuntilthe20thcenturywaswelladvanced.Andthisobservationappliestooth
14、erimportantnonlinearPDEsorsystemsofPDEs,liketheNavier-Stokesequations.ThegreatdevelopmentofFunctionalAnalysisinthedecadesfromthe1930'stothe1960'smadeitpossibleforthe¯rsttimetostartbuildingtheoriesforthesenonlinearPDEswithfullmathematicalrigor.Thishappenedinparticularinth
15、eareaofparabolicequationswherethetheoryofLinearandQuasilinearParabolicEquationsindivergenceformreachedadegreeofmaturityre°ectedforinstanceintheclassicalbooksofLadyzhenskayaetal.[357]andFriedman[239].Theaimofthepresenttextistoprovideasystematicpresentationofthemathematica
16、ltheoryofthenonlinearheatequation(PME)@u=¢(um);m>1;tusuallycalledthePorousMediumEquation(shortly,PME),p
