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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

Marek Fila 1, , Kazuhiro Ishige 2, and  Tatsuki Kawakami 3,

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.
Keywords: dynamical boundary conditions, Poisson kernel., Laplace equation.
Mathematics Subject Classification: Primary: 47J35, 35R11; Secondary: 35B40, 35J6.
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), 321-328.  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), 169-192.  Google Scholar

[3]

J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math. Z., 210 (1992), 413-439. 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), 173-183.  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), 494-521. 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), 1325-1333. 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 "Topics in Nonlinear Analysis," Progr. Nonlinear Differential Equations Appl., 35, Birkhäuser Verlag, Basel, 1999, 251-272.  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), 109-124.  Google Scholar

[9]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N. J. 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), 125-146. 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), 1099-1120.  Google Scholar

[12]

B. Hu and H.-M. Yin, On critical exponents for the heat equation with a nonlinear boundary condition, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 707-732.  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), 683-711.  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), 2673-2708. 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), 1351-1371. 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), 429-457. 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), 320-329. 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), 221-229.  Google Scholar

[20]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288. 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), 663-675. doi: 10.1007/s002080050055.  Google Scholar

[22]

P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[23]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.  Google Scholar

[24]

E. Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions, Proc. London Math. Soc., 93 (2006), 418-446. 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), 29-40.  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), 321-328.  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), 169-192.  Google Scholar

[3]

J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math. Z., 210 (1992), 413-439. 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), 173-183.  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), 494-521. 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), 1325-1333. 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 "Topics in Nonlinear Analysis," Progr. Nonlinear Differential Equations Appl., 35, Birkhäuser Verlag, Basel, 1999, 251-272.  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), 109-124.  Google Scholar

[9]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N. J. 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), 125-146. 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), 1099-1120.  Google Scholar

[12]

B. Hu and H.-M. Yin, On critical exponents for the heat equation with a nonlinear boundary condition, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 707-732.  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), 683-711.  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), 2673-2708. 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), 1351-1371. 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), 429-457. 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), 320-329. 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), 221-229.  Google Scholar

[20]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288. 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), 663-675. doi: 10.1007/s002080050055.  Google Scholar

[22]

P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[23]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.  Google Scholar

[24]

E. Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions, Proc. London Math. Soc., 93 (2006), 418-446. 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), 29-40.  Google Scholar

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