We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.
J.M.Arrieta, P.Quittner and A.Rodríguez-Bernal, Parabolic problems with nonlinear dynamical boundary conditions and singular initial data, Differential Integral Equations.14(12) (2001), 1487–1510.
2.
C.Bandle, J.von Below and W.Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti Accad Naz Lincei Rend Lincei Mat Appl.17(1) (2006), 35–67. doi:10.4171/RLM/453.
3.
J.Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math Z.210(3) (1992), 413–439. doi:10.1007/BF02571805.
4.
J.Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm Partial Differential Equations.18(7–8) (1993), 1309–1364. doi:10.1080/03605309308820976.
5.
J.Escher, On the qualitative behaviour of some semilinear parabolic problems, Differential Integral Equations.8(2) (1995), 247–267.
6.
J.Escher, Stable equilibria to elliptic equations in unbounded domains with nonlinear dynamic boundary conditions, Analysis (Munich)20(4) (2000), 325–351.
7.
M.Fila, K.Ishige and T.Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun Pure Appl Anal.11(3) (2012), 1285–1301. doi:10.3934/cpaa.2012.11.1285.
8.
M.Fila, K.Ishige and T.Kawakami, Large-time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition, Adv Differential Equations.18(1–2) (2013), 69–100.
9.
M.Fila, K.Ishige and T.Kawakami, Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition, Calc Var Partial Differential Equations.54(2) (2015), 2059–2078. doi:10.1007/s00526-015-0856-8.
10.
M.Fila, K.Ishige and T.Kawakami, Minimal solutions of a semilinear elliptic equation with a dynamical boundary condition, J Math Pures Appl (9)105(6) (2016), 788–809. doi:10.1016/j.matpur.2015.11.014.
11.
M.Fila, K.Ishige and T.Kawakami, An exterior nonlinear elliptic problem with a dynamical boundary condition, Rev Mat Complut.30(2) (2017), 281–312. doi:10.1007/s13163-017-0225-6.
12.
M.Fila, K.Ishige and T.Kawakami, The large diffusion limit for the heat equation with a dynamical boundary condition, 2018 June, preprint arXiv:1806.06308.
13.
M.Fila and P.Poláčik, Global nonexistence without blow-up for an evolution problem, Math Z.232(3) (1999), 531–545. doi:10.1007/PL00004770.
M.Fila and P.Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition, in: Topics in Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., Vol. 35, Birkhäuser, Basel, 1999, pp. 251–272. doi:10.1007/978-3-0348-8765-6_12.
16.
A.Fiscella and E.Vitillaro, Local Hadamard well-posedness and blow-up for reaction-diffusion equations with non-linear dynamical boundary conditions, Discrete Contin Dyn Syst.33(11–12) (2013), 5015–5047. doi:10.3934/dcds.2013.33.5015.
17.
C.G.Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J Differ Equations253(1) (2012), 126–166. doi:10.1016/j.jde.2012.02.010.
18.
C.G.Gal, The role of surface diffusion in dynamic boundary conditions: Where do we stand?, Milan J Math.83(2) (2015), 237–278. doi:10.1007/s00032-015-0242-1.
19.
C.G.Gal and M.Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc Lond Math Soc (3)108(6) (2014), 1351–1380. doi:10.1112/plms/pdt057.
20.
C.G.Gal and J.L.Shomberg, Hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions, Q Appl Math.73(1) (2015), 93–129. doi:10.1090/S0033-569X-2015-01363-5.
21.
Y.Giga and N.Hamamuki, On a dynamic boundary condition for singular degenerate parabolic equations in a half space, NoDEA Nonlinear Differential Equations Appl.25(6) (2018), 25. doi:10.1007/s00030-018-0542-6.
22.
Y.Giga, F.Onoue and K.Takasao, A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions, Hokkaido Univ Preprint Series in Math.1117 (2018), 1:45.
23.
T.Hintermann, Evolution equations with dynamic boundary conditions, Proc Roy Soc Edinburgh Sect A.113(1–2) (1989), 43–60. doi:10.1017/S0308210500023945.
24.
M.Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math J.21(2) (1992), 221–229. doi:10.14492/hokmj/1381413677.
25.
M.Kirane, E.Nabana and S.I.Pohozaev, Nonexistence of global solutions to an elliptic equation with a dynamical boundary condition, Bol Soc Parana Mat (3)22(2) (2004), 9–16.
26.
M.Kirane, E.Nabana and S.I.Pokhozhaev, On the absence of solutions for elliptic systems with dynamic boundary conditions, Differ Uravn38(6) (2002), 768–774, 862.
27.
J.L.Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
28.
A.F.M.ter Elst, M.Meyries and J.Rehberg, Parabolic equations with dynamical boundary conditions and source terms on interfaces, Ann Mat Pura Appl (4)193(5) (2014), 1295–1318. doi:10.1007/s10231-013-0329-7.
29.
J.L.Vázquez and E.Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Comm Partial Differential Equations.33(4–6) (2008), 561–612. doi:10.1080/03605300801970960.
30.
E.Vitillaro, Global existence for the heat equation with nonlinear dynamical boundary conditions, Proc Roy Soc Edinburgh Sect A.135(1) (2005), 175–207. doi:10.1017/S0308210500003838.
31.
E.Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions, Proc London Math Soc (3)93(2) (2006), 418–446. doi:10.1112/S0024611506015875.
32.
J.von Below and C.De Coster, A qualitative theory for parabolic problems under dynamical boundary conditions, J Inequal Appl.5(5) (2000), 467–486.
33.
J.von Below and G.Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Comm Partial Differential Equations.28(1–2) (2003), 223–247. doi:10.1081/PDE-120019380.
34.
J.von Below and G.Pincet Mailly, Blow up for some nonlinear parabolic problems with convection under dynamical boundary conditions. Discrete Contin Dyn Syst. 2007; (Dynamical systems and differential equations. Proceedings of the 6th AIMS International Conference, suppl.): 1031–1041.
35.
J.von Below, G.Pincet Mailly and J.F.Rault, Growth order and blow up points for the parabolic Burgers’ equation under dynamical boundary conditions, Discrete Contin Dyn Syst Ser S.6(3) (2013), 825–836.
36.
H.Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv Math Sci Appl.17(1) (2007), 67–88.
37.
Z.Yin, Global existence for elliptic equations with dynamic boundary conditions, Arch Math (Basel)81(5) (2003), 567–574. doi:10.1007/s00013-003-0104-x.
