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

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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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Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022, València, Spain

    J. Alberto Conejero

  2. Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

    Carlos Lizama

  3. Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Campus del Riu Sec s/n, 12071, Castelló, Spain

    Marina Murillo-Arcila

  4. Instituto de Matemática Interdisplinar (IMI), Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain

    Juan B. Seoane-Sepúlveda

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  1. J. Alberto Conejero
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  2. Carlos Lizama
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  3. Marina Murillo-Arcila
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  4. Juan B. Seoane-Sepúlveda
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Correspondence to Carlos Lizama.

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Conejero, J.A., Lizama, C., Murillo-Arcila, M. et al. Well-posedness for degenerate third order equations with delay and applications to inverse problems. Isr. J. Math. 229, 219–254 (2019). https://doi.org/10.1007/s11856-018-1796-8

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  • DOI: https://doi.org/10.1007/s11856-018-1796-8

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