Citation: |
[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. |
[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. |
[3] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math. Z., 210 (1992), 413-439.doi: 10.1007/BF02571805. |
[4] |
J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions, Lecture Notes Pure Appl. Math., 155 (1994), 173-183. |
[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. |
[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. |
[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. |
[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. |
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N. J. 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), 125-146.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), 1099-1120. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
K. Ishige and T. Kawakami, Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic problems, preprint. |
[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. |
[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. |
[20] |
H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.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), 663-675.doi: 10.1007/s002080050055. |
[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. |
[23] |
S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51. |
[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. |
[25] |
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. |