# American Institute of Mathematical Sciences

May  2012, 11(3): 1285-1301. doi: 10.3934/cpaa.2012.11.1285

## Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition

 1 Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava 2 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578 3 Department of Applied Mathematics, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Received  November 2010 Revised  February 2011 Published  December 2011

We study the large time behavior of positive solutions for the Laplace equation on the half-space with a nonlinear dynamical boundary condition. We show the convergence to the Poisson kernel in a suitable sense provided initial data are sufficiently small.
Citation: Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
##### References:
 [1] H. Amann and M. Fila, A Fujita-type theorem for the Laplace equation with a dynamical boundary condition,, Acta Math. Univ. Comenianae, 66 (1997), 321. Google Scholar [2] K. Deng, M. Fila and H. A. Levine, On critical exponents for a system of heat equations coupled in the boundary conditions,, Acta Math. Univ. Comenianae, 63 (1994), 169. Google Scholar [3] J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Math. Z., 210 (1992), 413. doi: 10.1007/BF02571805. Google Scholar [4] J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions,, Lecture Notes Pure Appl. Math., 155 (1994), 173. Google Scholar [5] M. Fila and H. A. Levine, On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition,, J. Math. Anal. Appl., 204 (1996), 494. doi: 10.1006/jmaa.1996.0451. Google Scholar [6] M. Fila and P. Quittner, Global solutions of the Laplace equation with a nonlinear dynamical boundary condition,, Math. Meth. Appl. Sciences, 20 (1997), 1325. doi: 10.1002/(SICI)1099-1476(199710)20:15<1325::AID-MMA916>3.0.CO;2-G. Google Scholar [7] M. Fila and P. Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition,, in, 35 (1999), 251. Google Scholar [8] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109. Google Scholar [9] A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964). Google Scholar [10] V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary,, Israel J. Math., 94 (1996), 125. doi: 10.1007/BF02762700. Google Scholar [11] B. Hu and H.-M. Yin, Critical exponents for a system of heat equations coupled in a non-linear boundary condition,, Math. Methods Appl. Sci., 19 (1996), 1099. Google Scholar [12] B. Hu and H.-M. Yin, On critical exponents for the heat equation with a nonlinear boundary condition,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 13 (1996), 707. Google Scholar [13] B. Hu and H.-M. Yin, On critical exponents for the heat equation with a mixed nonlinear Dirichlet-Neumann boundary condition,, J. Math. Anal. Appl., 209 (1997), 683. Google Scholar [14] K. Ishige, M. Ishiwata and T. Kawakami, The decay of the solutions for the heat equation with a potential,, Indiana Univ. Math. J., 58 (2009), 2673. doi: 10.1512/iumj.2009.58.3771. Google Scholar [15] K. Ishige and T. Kawakami, Asymptotic behavior of solutions for some semilinear heat equations in $R^N$,, Commun. Pure Appl. Anal., 8 (2009), 1351. doi: 10.3934/cpaa.2009.8.1351. Google Scholar [16] K. Ishige and T. Kawakami, Global solutions of the heat equation with a nonlinear boundary condition,, Calc. Var. Partial Differential Equations, 39 (2010), 429. doi: 10.1007/s00526-010-0316-4. Google Scholar [17] K. Ishige and T. Kawakami, Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic problems,, preprint., (). Google Scholar [18] T. Kawakami, Global existence of solutions for the heat equation with a nonlinear boundary condition,, J. Math. Anal. Appl., 368 (2010), 320. doi: 10.1016/j.jmaa.2010.02.007. Google Scholar [19] M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type,, Hokkaido Math. J., 21 (1992), 221. Google Scholar [20] H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Rev., 32 (1990), 262. doi: 10.1137/1032046. Google Scholar [21] N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation,, Math. Ann., 307 (1997), 663. doi: 10.1007/s002080050055. Google Scholar [22] P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,", Birkh\, (2007). Google Scholar [23] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations,, Osaka J. Math., 12 (1975), 45. Google Scholar [24] E. Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions,, Proc. London Math. Soc., 93 (2006), 418. doi: 10.1112/S0024611506015875. Google Scholar [25] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29. Google Scholar

show all references

##### References:
 [1] H. Amann and M. Fila, A Fujita-type theorem for the Laplace equation with a dynamical boundary condition,, Acta Math. Univ. Comenianae, 66 (1997), 321. Google Scholar [2] K. Deng, M. Fila and H. A. Levine, On critical exponents for a system of heat equations coupled in the boundary conditions,, Acta Math. Univ. Comenianae, 63 (1994), 169. Google Scholar [3] J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Math. Z., 210 (1992), 413. doi: 10.1007/BF02571805. Google Scholar [4] J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions,, Lecture Notes Pure Appl. Math., 155 (1994), 173. Google Scholar [5] M. Fila and H. A. Levine, On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition,, J. Math. Anal. Appl., 204 (1996), 494. doi: 10.1006/jmaa.1996.0451. Google Scholar [6] M. Fila and P. Quittner, Global solutions of the Laplace equation with a nonlinear dynamical boundary condition,, Math. Meth. Appl. Sciences, 20 (1997), 1325. doi: 10.1002/(SICI)1099-1476(199710)20:15<1325::AID-MMA916>3.0.CO;2-G. Google Scholar [7] M. Fila and P. Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition,, in, 35 (1999), 251. Google Scholar [8] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109. Google Scholar [9] A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964). Google Scholar [10] V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary,, Israel J. Math., 94 (1996), 125. doi: 10.1007/BF02762700. Google Scholar [11] B. Hu and H.-M. Yin, Critical exponents for a system of heat equations coupled in a non-linear boundary condition,, Math. Methods Appl. Sci., 19 (1996), 1099. Google Scholar [12] B. Hu and H.-M. Yin, On critical exponents for the heat equation with a nonlinear boundary condition,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 13 (1996), 707. Google Scholar [13] B. Hu and H.-M. Yin, On critical exponents for the heat equation with a mixed nonlinear Dirichlet-Neumann boundary condition,, J. Math. Anal. Appl., 209 (1997), 683. Google Scholar [14] K. Ishige, M. Ishiwata and T. Kawakami, The decay of the solutions for the heat equation with a potential,, Indiana Univ. Math. J., 58 (2009), 2673. doi: 10.1512/iumj.2009.58.3771. Google Scholar [15] K. Ishige and T. Kawakami, Asymptotic behavior of solutions for some semilinear heat equations in $R^N$,, Commun. Pure Appl. Anal., 8 (2009), 1351. doi: 10.3934/cpaa.2009.8.1351. Google Scholar [16] K. Ishige and T. Kawakami, Global solutions of the heat equation with a nonlinear boundary condition,, Calc. Var. Partial Differential Equations, 39 (2010), 429. doi: 10.1007/s00526-010-0316-4. Google Scholar [17] K. Ishige and T. Kawakami, Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic problems,, preprint., (). Google Scholar [18] T. Kawakami, Global existence of solutions for the heat equation with a nonlinear boundary condition,, J. Math. Anal. Appl., 368 (2010), 320. doi: 10.1016/j.jmaa.2010.02.007. Google Scholar [19] M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type,, Hokkaido Math. J., 21 (1992), 221. Google Scholar [20] H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Rev., 32 (1990), 262. doi: 10.1137/1032046. Google Scholar [21] N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation,, Math. Ann., 307 (1997), 663. doi: 10.1007/s002080050055. Google Scholar [22] P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,", Birkh\, (2007). Google Scholar [23] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations,, Osaka J. Math., 12 (1975), 45. Google Scholar [24] E. Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions,, Proc. London Math. Soc., 93 (2006), 418. doi: 10.1112/S0024611506015875. Google Scholar [25] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29. Google Scholar
 [1] Klemens Fellner, Stefanie Sonner, Bao Quoc Tang, Do Duc Thuan. Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4055-4078. doi: 10.3934/dcdsb.2019050 [2] Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441 [3] Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259 [4] Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825 [5] Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831 [6] Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000 [7] Victor Isakov. On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data. Inverse Problems & Imaging, 2008, 2 (1) : 151-165. doi: 10.3934/ipi.2008.2.151 [8] Igor Chueshov, Björn Schmalfuss. Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 315-338. doi: 10.3934/dcds.2007.18.315 [9] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [10] Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255 [11] Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175 [12] Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034 [13] Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317 [14] Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007 [15] François Bolley, Arnaud Guillin, Xinyu Wang. Non ultracontractive heat kernel bounds by Lyapunov conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 857-870. doi: 10.3934/dcds.2015.35.857 [16] Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219 [17] Tae Gab Ha. On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. Evolution Equations & Control Theory, 2018, 7 (2) : 281-291. doi: 10.3934/eect.2018014 [18] Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763 [19] Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 [20] Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631

2018 Impact Factor: 0.925