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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 |
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.
|
[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.
|
[3] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Math. Z., 210 (1992), 413.
doi: 10.1007/BF02571805. |
[4] |
J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions,, Lecture Notes Pure Appl. Math., 155 (1994), 173.
|
[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. |
[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. |
[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.
|
[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.
|
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964).
|
[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. |
[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.
|
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[19] |
M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type,, Hokkaido Math. J., 21 (1992), 221.
|
[20] |
H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Rev., 32 (1990), 262.
doi: 10.1137/1032046. |
[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. |
[22] |
P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,", Birkh\, (2007).
|
[23] |
S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations,, Osaka J. Math., 12 (1975), 45.
|
[24] |
E. Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions,, Proc. London Math. Soc., 93 (2006), 418.
doi: 10.1112/S0024611506015875. |
[25] |
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29.
|
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.
|
[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.
|
[3] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Math. Z., 210 (1992), 413.
doi: 10.1007/BF02571805. |
[4] |
J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions,, Lecture Notes Pure Appl. Math., 155 (1994), 173.
|
[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. |
[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. |
[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.
|
[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.
|
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964).
|
[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. |
[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.
|
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[19] |
M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type,, Hokkaido Math. J., 21 (1992), 221.
|
[20] |
H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Rev., 32 (1990), 262.
doi: 10.1137/1032046. |
[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. |
[22] |
P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,", Birkh\, (2007).
|
[23] |
S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations,, Osaka J. Math., 12 (1975), 45.
|
[24] |
E. Vitillaro, On the Laplace equation with non-linear dynamical boundary conditions,, Proc. London Math. Soc., 93 (2006), 418.
doi: 10.1112/S0024611506015875. |
[25] |
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29.
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