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November  2020, 40(11): 6529-6546. doi: 10.3934/dcds.2020289

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

 1 Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia 2 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan 3 Applied Mathematics and Informatics Course, Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-cho, Otsu, Shiga 520-2194, Japan 4 Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

* Corresponding author: Johannes Lankeit

Received  April 2020 Revised  May 2020 Published  July 2020

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.

Citation: Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, Johannes Lankeit. The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6529-6546. doi: 10.3934/dcds.2020289
##### References:
 [1] M. Fila, K. 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. Fila, K. Ishige, T. 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.

show all references

##### References:
 [1] M. Fila, K. 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. Fila, K. Ishige, T. 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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