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The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition

  • * Corresponding author: Johannes Lankeit

    * Corresponding author: Johannes Lankeit
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  • We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

    Mathematics Subject Classification: Primary: 35K05, 35B40.

    Citation:

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  • [1] M. FilaK. Ishige and T. Kawakami, An exterior nonlinear elliptic problem with a dynamical boundary condition, Rev. Mat. Complut., 30 (2017), 281-312.  doi: 10.1007/s13163-017-0225-6.
    [2] M. Fila, K. Ishige and T. Kawakami, The large diffusion limit for the heat equation with a dynamical boundary condition, to appear in Commun. Contemp. Math.. doi: 10.1142/S0219199720500030.
    [3] M. FilaK. IshigeT. Kawakami and J. Lankeit, Rate of convergence in the large diffusion limit for the heat equation with a dynamical boundary condition, Asymptot. Anal., 114 (2019), 37-57.  doi: 10.3233/ASY-181517.
    [4] Y. Fujishima, T. Kawakami and Y. Sire, Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients, Calc. Var. Partial Differential Equations, 58 (2019), 25pp. doi: 10.1007/s00526-019-1525-0.
    [5] C. G. Gal, The role of surface diffusion in dynamic boundary conditions: Where do we stand?, Milan J. Math., 83 (2015), 237-278.  doi: 10.1007/s00032-015-0242-1.
    [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.
    [7] A. Grigor'yan and L. Saloff-Coste, Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math., 55 (2002), 93-133.  doi: 10.1002/cpa.10014.
    [8] K. Ishige and Y. Kabeya, Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball, J. Math. Soc. Japan, 59 (2007), 861-898.  doi: 10.2969/jmsj/05930861.
    [9] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.
    [10] J. von Below and C. De Coster, A qualitative theory for parabolic problems under dynamical boundary conditions, J. Inequal. Appl., 5 (2000), 467-486.  doi: 10.1155/S1025583400000266.
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