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Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions
1. | Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava |
2. | Université Paris 13, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications, 99, avenue J.-B. Clément, 93430 Villetaneuse |
References:
[1] |
H. Amann, Linear parabolic problems involving measures,, Rev. R. Acad. Cien. Exactas Fís. Nat. Serie A. Mat., 95 (2001), 85.
|
[2] |
J. M. Arrieta, On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions,, Proc. Amer. Math. Soc., 136 (2008), 151.
doi: 10.1090/S0002-9939-07-08980-0. |
[3] |
F. Andreu, J. M. Mazón, J. Toledo and J. D. Rossi, Porous medium equation with absorption and a nonlinear boundary condition,, Nonlinear Analysis, 49 (2002), 541.
doi: 10.1016/S0362-546X(01)00122-5. |
[4] |
M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term,, in, (1998), 189.
|
[5] |
T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires,, Comm. Partial Differential Equations, 9 (1984), 955.
|
[6] |
M. Chipot, M. Chlebík, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\R^n_+$ with a nonlinear boundary condition,, J. Math. Anal. Appl., 223 (1998), 429.
doi: 10.1006/jmaa.1998.5958. |
[7] |
M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions,, Acta Math. Univ. Comenianae (N.S.), 60 (1991), 35.
|
[8] |
M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions,, Advances Differ. Equations, 1 (1996), 91.
|
[9] |
M. Chipot and P. Quittner, Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions,, J. Dynam. Differential Equations, 16 (2004), 91.
doi: 10.1023/B:JODY.0000041282.14930.7a. |
[10] |
M. Chlebík and M. Fila, On the blow-up rate for the heat equation with a nonlinear boundary condition,, Math. Methods Appl. Sci., 23 (2000), 1323.
doi: 10.1002/1099-1476(200010)23:15<1323::AID-MMA167>3.0.CO;2-W. |
[11] |
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 (N.S.), 63 (1994), 169.
|
[12] |
M. Fila, J. Filo and G. M. Lieberman, Blow-up on the boundary for the heat equation,, Calc. Var. Partial Differential Equations, 10 (2000), 85.
|
[13] |
M. Fila and J.-S. Guo, Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition,, Nonlinear Analysis, 48 (2002), 995.
doi: 10.1016/S0362-546X(00)00229-7. |
[14] |
M. Fila and P. Quittner, The blow-up rate for the heat equation with a nonlinear boundary condition,, Math. Methods Appl. Sci., 14 (1991), 197.
doi: 10.1002/mma.1670140304. |
[15] |
M. Fila and Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains,, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 473.
doi: 10.1007/PL00001459. |
[16] |
M. Fila, Ph. Souplet and F. B. Weissler, Linear and nonlinear heat equations in $L^p_\delta$ spaces and universal bounds for global solutions,, Math. Ann., 320 (2001), 87.
doi: 10.1007/PL00004471. |
[17] |
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.
doi: 10.1512/iumj.1985.34.34025. |
[18] |
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. |
[19] |
Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415.
doi: 10.1007/BF01211756. |
[20] |
Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables,, Indiana Univ. Math. J., 36 (1987), 1.
doi: 10.1512/iumj.1987.36.36001. |
[21] |
Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equation with subcritical nonlinearity,, Indiana Univ. Math. J., 53 (2004), 483.
doi: 10.1512/iumj.2004.53.2401. |
[22] |
Y. Giga, S. Matsui and S. Sasayama, On blow-up rate for sign-changing solutions in a convex domain,, Math. Methods Appl. Sc., 27 (2004), 1771.
doi: 10.1002/mma.562. |
[23] |
B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differ. Integral Equations, 7 (1994), 301.
|
[24] |
B. Hu, Nondegeneracy and single-point-blowup for solution of the heat equation with a nonlinear boundary condition,, J. Math. Sci. Univ. Tokyo, 1 (1994), 251.
|
[25] |
B. Hu, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition,, Differ. Integral Equations, 9 (1996), 891.
|
[26] |
B. Hu and H.-M. Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346 (1994), 117.
doi: 10.2307/2154944. |
[27] |
Y.-Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27.
doi: 10.1007/BF02786551. |
[28] |
P. Poláčik and P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation,, Nonlinear Analysis, 64 (2006), 1679.
doi: 10.1016/j.na.2005.07.016. |
[29] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations,, Indiana Univ. Math. J., 56 (2007), 879.
doi: 10.1512/iumj.2007.56.2911. |
[30] |
F. Quirós, J. D. Rossi and J. L. Vázquez, Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions,, Comm. Partial Differ. Equations, 27 (2002), 395.
|
[31] |
P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195.
|
[32] |
P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem,, Math. Ann., 320 (2001), 299.
doi: 10.1007/PL00004475. |
[33] |
P. Quittner and A. Rodríguez-Bernal, Complete and energy blow-up in parabolic problems with nonlinear boundary conditions,, Nonlinear Analysis, 62 (2005), 863.
doi: 10.1016/j.na.2005.03.099. |
[34] |
P. Quittner and Ph. Souplet, Bounds of global solutions of parabolic problems with nonlinear boundary conditions,, Indiana Univ. Math. J., 52 (2003), 875.
doi: 10.1512/iumj.2003.52.2353. |
[35] |
P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).
|
[36] |
P. Quittner and Ph. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts,, Proc. 8th AIMS Intern. Conf., (2010). Google Scholar |
[37] |
P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations,, J. Differential Equations, 196 (2004), 316.
doi: 10.1016/j.jde.2003.10.007. |
[38] |
A. Rodríguez-Bernal and A. Tajdine, Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up,, J. Differ. Equations, 169 (2001), 332.
doi: 10.1006/jdeq.2000.3903. |
[39] |
F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations, 55 (1984), 204.
doi: 10.1016/0022-0396(84)90081-0. |
[40] |
F. B. Weissler, An $L^\infty$ blow-up estimate for a nonlinear heat equation,, Comm. Pure Appl. Math., 38 (1985), 291.
doi: 10.1002/cpa.3160380303. |
show all references
References:
[1] |
H. Amann, Linear parabolic problems involving measures,, Rev. R. Acad. Cien. Exactas Fís. Nat. Serie A. Mat., 95 (2001), 85.
|
[2] |
J. M. Arrieta, On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions,, Proc. Amer. Math. Soc., 136 (2008), 151.
doi: 10.1090/S0002-9939-07-08980-0. |
[3] |
F. Andreu, J. M. Mazón, J. Toledo and J. D. Rossi, Porous medium equation with absorption and a nonlinear boundary condition,, Nonlinear Analysis, 49 (2002), 541.
doi: 10.1016/S0362-546X(01)00122-5. |
[4] |
M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term,, in, (1998), 189.
|
[5] |
T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires,, Comm. Partial Differential Equations, 9 (1984), 955.
|
[6] |
M. Chipot, M. Chlebík, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\R^n_+$ with a nonlinear boundary condition,, J. Math. Anal. Appl., 223 (1998), 429.
doi: 10.1006/jmaa.1998.5958. |
[7] |
M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions,, Acta Math. Univ. Comenianae (N.S.), 60 (1991), 35.
|
[8] |
M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions,, Advances Differ. Equations, 1 (1996), 91.
|
[9] |
M. Chipot and P. Quittner, Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions,, J. Dynam. Differential Equations, 16 (2004), 91.
doi: 10.1023/B:JODY.0000041282.14930.7a. |
[10] |
M. Chlebík and M. Fila, On the blow-up rate for the heat equation with a nonlinear boundary condition,, Math. Methods Appl. Sci., 23 (2000), 1323.
doi: 10.1002/1099-1476(200010)23:15<1323::AID-MMA167>3.0.CO;2-W. |
[11] |
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 (N.S.), 63 (1994), 169.
|
[12] |
M. Fila, J. Filo and G. M. Lieberman, Blow-up on the boundary for the heat equation,, Calc. Var. Partial Differential Equations, 10 (2000), 85.
|
[13] |
M. Fila and J.-S. Guo, Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition,, Nonlinear Analysis, 48 (2002), 995.
doi: 10.1016/S0362-546X(00)00229-7. |
[14] |
M. Fila and P. Quittner, The blow-up rate for the heat equation with a nonlinear boundary condition,, Math. Methods Appl. Sci., 14 (1991), 197.
doi: 10.1002/mma.1670140304. |
[15] |
M. Fila and Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains,, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 473.
doi: 10.1007/PL00001459. |
[16] |
M. Fila, Ph. Souplet and F. B. Weissler, Linear and nonlinear heat equations in $L^p_\delta$ spaces and universal bounds for global solutions,, Math. Ann., 320 (2001), 87.
doi: 10.1007/PL00004471. |
[17] |
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.
doi: 10.1512/iumj.1985.34.34025. |
[18] |
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. |
[19] |
Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415.
doi: 10.1007/BF01211756. |
[20] |
Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables,, Indiana Univ. Math. J., 36 (1987), 1.
doi: 10.1512/iumj.1987.36.36001. |
[21] |
Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equation with subcritical nonlinearity,, Indiana Univ. Math. J., 53 (2004), 483.
doi: 10.1512/iumj.2004.53.2401. |
[22] |
Y. Giga, S. Matsui and S. Sasayama, On blow-up rate for sign-changing solutions in a convex domain,, Math. Methods Appl. Sc., 27 (2004), 1771.
doi: 10.1002/mma.562. |
[23] |
B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differ. Integral Equations, 7 (1994), 301.
|
[24] |
B. Hu, Nondegeneracy and single-point-blowup for solution of the heat equation with a nonlinear boundary condition,, J. Math. Sci. Univ. Tokyo, 1 (1994), 251.
|
[25] |
B. Hu, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition,, Differ. Integral Equations, 9 (1996), 891.
|
[26] |
B. Hu and H.-M. Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346 (1994), 117.
doi: 10.2307/2154944. |
[27] |
Y.-Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27.
doi: 10.1007/BF02786551. |
[28] |
P. Poláčik and P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation,, Nonlinear Analysis, 64 (2006), 1679.
doi: 10.1016/j.na.2005.07.016. |
[29] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations,, Indiana Univ. Math. J., 56 (2007), 879.
doi: 10.1512/iumj.2007.56.2911. |
[30] |
F. Quirós, J. D. Rossi and J. L. Vázquez, Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions,, Comm. Partial Differ. Equations, 27 (2002), 395.
|
[31] |
P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195.
|
[32] |
P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem,, Math. Ann., 320 (2001), 299.
doi: 10.1007/PL00004475. |
[33] |
P. Quittner and A. Rodríguez-Bernal, Complete and energy blow-up in parabolic problems with nonlinear boundary conditions,, Nonlinear Analysis, 62 (2005), 863.
doi: 10.1016/j.na.2005.03.099. |
[34] |
P. Quittner and Ph. Souplet, Bounds of global solutions of parabolic problems with nonlinear boundary conditions,, Indiana Univ. Math. J., 52 (2003), 875.
doi: 10.1512/iumj.2003.52.2353. |
[35] |
P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).
|
[36] |
P. Quittner and Ph. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts,, Proc. 8th AIMS Intern. Conf., (2010). Google Scholar |
[37] |
P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations,, J. Differential Equations, 196 (2004), 316.
doi: 10.1016/j.jde.2003.10.007. |
[38] |
A. Rodríguez-Bernal and A. Tajdine, Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up,, J. Differ. Equations, 169 (2001), 332.
doi: 10.1006/jdeq.2000.3903. |
[39] |
F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations, 55 (1984), 204.
doi: 10.1016/0022-0396(84)90081-0. |
[40] |
F. B. Weissler, An $L^\infty$ blow-up estimate for a nonlinear heat equation,, Comm. Pure Appl. Math., 38 (1985), 291.
doi: 10.1002/cpa.3160380303. |
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