American Institute of Mathematical Sciences

Advanced
June  2012, 5(3): 671-681. doi: 10.3934/dcdss.2012.5.671

Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions

Pavol Quittner 1, and  Philippe Souplet 2,

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

Received  September 2010 Revised  October 2010 Published  October 2011

We review known and prove new results on blow-up rate of solutions of parabolic problems with nonlinear boundary conditions. We also compare these results and methods of their proofs with corresponding results and methods for the nonlinear heat equation.
Keywords: blow-up rate, nonlinear boundary conditions, a priori estimate., Parabolic.
Mathematics Subject Classification: Primary: 35K55, 35K57, 35B40; Secondary: 35B4.
Citation: Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
References:
[1]

H. Amann, Linear parabolic problems involving measures,, Rev. R. Acad. Cien. Exactas Fís. Nat. Serie A. Mat., 95 (2001), 85.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term,, in, (1998), 189.   Google Scholar

[5]

T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires,, Comm. Partial Differential Equations, 9 (1984), 955.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415.  doi: 10.1007/BF01211756.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differ. Integral Equations, 7 (1994), 301.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195.   Google Scholar

[32]

P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem,, Math. Ann., 320 (2001), 299.  doi: 10.1007/PL00004475.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term,, in, (1998), 189.   Google Scholar

[5]

T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires,, Comm. Partial Differential Equations, 9 (1984), 955.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415.  doi: 10.1007/BF01211756.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differ. Integral Equations, 7 (1994), 301.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195.   Google Scholar

[32]

P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem,, Math. Ann., 320 (2001), 299.  doi: 10.1007/PL00004475.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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