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Israel Journal of Mathematics

, Volume 229, Issue 1, pp 219–254 | Cite as

Well-posedness for degenerate third order equations with delay and applications to inverse problems

  • J. Alberto Conejero
  • Carlos LizamaEmail author
  • Marina Murillo-Arcila
  • Juan B. Seoane-Sepúlveda
Article
  • 39 Downloads

Abstract

In this paper, we study well-posedness for the following third-order in time equation with delay
$$\left( {0.1} \right)\;\alpha \left( {Mu'} \right)''\left( t \right) + \left( {Nu'} \right)'\left( t \right) = \beta Au\left( t \right) + \gamma Bu{'(t)} + Gu{'_t}+F{u_t} + f\left( t \right),\;t \in \left[ {0,2\pi } \right]$$
where α, β, γ are real numbers, A and B are linear operators defined on a Banach space X with domains D(A) and D(B) such that
$$D(A)\cap{D(B)}\subset{D(M)}\cap{D(N)};$$
u(t)is the state function taking values in X and ut: (−∞, 0] → X defined as ut(θ) = u(t+θ) for θ < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue–Bochner spaces \(L^p(\mathbb{T}, X)\), periodic Besov spaces \(B^s_{p,q}(\mathbb{T}, X)\) and periodic Triebel–Lizorkin spaces \(F^s_{p,q}(\mathbb{T}, X)\). A novel application to an inverse problem is given.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • J. Alberto Conejero
    • 1
  • Carlos Lizama
    • 2
    Email author
  • Marina Murillo-Arcila
    • 3
  • Juan B. Seoane-Sepúlveda
    • 4
  1. 1.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValènciaSpain
  2. 2.Departamento de Matemática y Ciencia de la Computación, Facultad de CienciasUniversidad de Santiago de ChileSantiagoChile
  3. 3.Institut de Matemàtiques i Aplicacions de Castelló (IMAC)Universitat Jaume ICastellóSpain
  4. 4.Instituto de Matemática Interdisplinar (IMI), Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

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