Variedades inerciales para una ecuación diferencial parcial en espacios de Sobolev con peso
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Date
2024
Authors
Reyes Carrera, Pedro Gustavo
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Publisher
Universidad Nacional de Trujillo
Abstract
En elpresentetrabajo,sedemuestralaexistenciadeunavariedadinercialparauna
ecuaci´on diferencialparcialenespaciodeSobolevconpeso.Seusaronresultadosdelan´alisis
funcional enespaciosdeHilbertconoperadoresautoadjuntosnoacotados;analiz´andose la
ecuaci´on diferencialparcial,
𝑢𝑡 + 𝐴𝑢 + 𝐹(𝑢) = 0, (1)
siendo 𝐴 un operadorpositivonoacotadoautoadjuntoydisipativoenunespaciodeSobolev
con pesoen 𝐻, 𝐹 es elt´erminonolinealconlapropiedaddeLipschitzlocaleneldominiode
𝐷(𝐴) = 𝐻. Alrealizarelan´alisis delaecuaci´on (1)seobtuvieronlossiguientesresultados:
i) Para 𝜆 una barreraespectraldelaecuaci´on (1)talque 𝜆 >𝜆0, paraalg´un 𝜆0 y 𝑃𝜆𝐻
de dimensi´on finitaseconcluyequeGr(𝑄) = {𝑢 + 𝑄(𝑢) : 𝑢 ∈ 𝑃𝜆𝐻} es unavariedad
Lipschitzianadedimensi´on finitasatisfaciendolassiguientespropiedades:
a) Gr(𝑄) es invarianteparaelsemigrupo {𝑆(𝑡)}𝑡≥0.
b) Gr(𝑄) atrae exponencialmentetodaslas ´orbitas delaecuaci´on deevoluci´on (1).
ii) Si 𝜆 ∉ 𝜎(𝐴), setieneunavariedadinercialparalaecuaci´on deevoluci´on nolineal
𝑢𝑡 + 𝐴𝑢 + 𝐹(𝑢) = 0.
Finalmenteseconcluyeque:Si 𝜆 es unabarreraespectralpara(1)talque 𝜆 >𝜆0,
𝑃𝜆𝐻 es dedimensi´on finitay 𝜆 ∉ 𝜎(𝐴), entonces,lafunci´on Gr(𝑄) es unavariedadinercial
para (1).
In thepresentwork,theexistenceofaninertialmanifoldisdemonstratedfora partialdifferentialequationinSobolevspacewithweight.Themethodologyoffunctional analysisinHilbertspacewithunboundedself-adjointoperatorswasused;analyzingthe partialdifferentialequation 𝑢𝑡 + 𝐴𝑢 + 𝐹(𝑢) = 0, (1) where 𝐴 is aself-adjointanddissipativeunboundedpositiveoperatoronaSobolevspace with weighton 𝐻, 𝐹 is thenonlineartermwiththelocalLipschitzpropertyinthedomainof 𝐷(𝐴) = 𝐻. Whenperformingtheanalysisofequation(1)thefollowingresultswereobtained: i) For 𝜆 spectral barrierofequation(1)suchthat 𝜆 >𝜆0, forsome 𝜆0 and 𝑃𝜆𝐻 of finite dimension itfollowsthatGr(𝑄) = {𝑢 + 𝑄(𝑢) : 𝑢 ∈ 𝑃𝜆𝐻} is afinite-dimensional Lipschitzianmanifoldsatisfyingthefollowingproperties: a) Gr(𝑄) is invariantforthesemigroup {𝑆(𝑡)}𝑡≥0. b) Gr(𝑄) exponentiallyattractsalltheorbitsoftheevolutionequation(1). ii) If 𝜆 ∉ 𝜎(𝐴), wehaveaninertialmanifoldforthenonlinearevolutionequation 𝑢𝑡 + 𝐴𝑢 + 𝐹(𝑢) = 0. Finallyitisconcludedthat:Let 𝜆 be aspectralbarrierfor(1)suchthat 𝜆 >𝜆0, 𝑃𝜆𝐻 is offinitedimensionand 𝜆 ∉ 𝜎(𝐴). Then,thefunctionGr(𝑄) is aninertialmanifoldfor(1).
In thepresentwork,theexistenceofaninertialmanifoldisdemonstratedfora partialdifferentialequationinSobolevspacewithweight.Themethodologyoffunctional analysisinHilbertspacewithunboundedself-adjointoperatorswasused;analyzingthe partialdifferentialequation 𝑢𝑡 + 𝐴𝑢 + 𝐹(𝑢) = 0, (1) where 𝐴 is aself-adjointanddissipativeunboundedpositiveoperatoronaSobolevspace with weighton 𝐻, 𝐹 is thenonlineartermwiththelocalLipschitzpropertyinthedomainof 𝐷(𝐴) = 𝐻. Whenperformingtheanalysisofequation(1)thefollowingresultswereobtained: i) For 𝜆 spectral barrierofequation(1)suchthat 𝜆 >𝜆0, forsome 𝜆0 and 𝑃𝜆𝐻 of finite dimension itfollowsthatGr(𝑄) = {𝑢 + 𝑄(𝑢) : 𝑢 ∈ 𝑃𝜆𝐻} is afinite-dimensional Lipschitzianmanifoldsatisfyingthefollowingproperties: a) Gr(𝑄) is invariantforthesemigroup {𝑆(𝑡)}𝑡≥0. b) Gr(𝑄) exponentiallyattractsalltheorbitsoftheevolutionequation(1). ii) If 𝜆 ∉ 𝜎(𝐴), wehaveaninertialmanifoldforthenonlinearevolutionequation 𝑢𝑡 + 𝐴𝑢 + 𝐹(𝑢) = 0. Finallyitisconcludedthat:Let 𝜆 be aspectralbarrierfor(1)suchthat 𝜆 >𝜆0, 𝑃𝜆𝐻 is offinitedimensionand 𝜆 ∉ 𝜎(𝐴). Then,thefunctionGr(𝑄) is aninertialmanifoldfor(1).
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