# American Institute of Mathematical Sciences

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

 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.
Citation: Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and 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), 85-119. [2] J. M. Arrieta, On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions, Proc. Amer. Math. Soc., 136 (2008), 151-160. 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), 541-563. 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 "Equations aux Dérivées Partielles et Applications," articles dédiés à Jacques-Louis Lions, Gauthier-Villars, éd. Sci. Méd. Elsevier, Paris, 1998, 189-198. [5] T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations, 9 (1984), 955-978. [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), 429-471. 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), 35-103. [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), 91-110. [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), 91-138. 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), 1323-1330. 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), 169-192. [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), 85-99. [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), 995-1002. 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), 197-205. 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), 473-480. 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), 87-113. 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), 425-447. 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), 125-146. doi: 10.1007/BF02762700. [19] Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421. doi: 10.1007/BF01211756. [20] Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40. 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), 483-514. 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), 1771-1782. 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), 301-313. [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), 251-276. [25] B. Hu, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differ. Integral Equations, 9 (1996), 891-901. [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), 117-135. 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), 27-87. 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), 1679-1689. 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), 879-908. 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), 395-424. [31] P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195-203. [32] P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann., 320 (2001), 299-305. 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), 863-875. 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), 875-900. 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, Birkhäuser Verlag, Basel 2007. [36] P. Quittner and Ph. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts, Proc. 8th AIMS Intern. Conf., Dresden 2010, to appear. [37] P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, J. Differential Equations, 196 (2004), 316-339. 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), 332-372. 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), 204-224. 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), 291-295. doi: 10.1002/cpa.3160380303.

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##### References:
 [1] H. Amann, Linear parabolic problems involving measures, Rev. R. Acad. Cien. Exactas Fís. Nat. Serie A. Mat., 95 (2001), 85-119. [2] J. M. Arrieta, On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions, Proc. Amer. Math. Soc., 136 (2008), 151-160. 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), 541-563. 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 "Equations aux Dérivées Partielles et Applications," articles dédiés à Jacques-Louis Lions, Gauthier-Villars, éd. Sci. Méd. Elsevier, Paris, 1998, 189-198. [5] T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations, 9 (1984), 955-978. [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), 429-471. 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), 35-103. [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), 91-110. [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), 91-138. 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), 1323-1330. 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), 169-192. [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), 85-99. [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), 995-1002. 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), 197-205. 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), 473-480. 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), 87-113. 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), 425-447. 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), 125-146. doi: 10.1007/BF02762700. [19] Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421. doi: 10.1007/BF01211756. [20] Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40. 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), 483-514. 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), 1771-1782. 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), 301-313. [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), 251-276. [25] B. Hu, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differ. Integral Equations, 9 (1996), 891-901. [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), 117-135. 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), 27-87. 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), 1679-1689. 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), 879-908. 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), 395-424. [31] P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195-203. [32] P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann., 320 (2001), 299-305. 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), 863-875. 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), 875-900. 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, Birkhäuser Verlag, Basel 2007. [36] P. Quittner and Ph. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts, Proc. 8th AIMS Intern. Conf., Dresden 2010, to appear. [37] P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, J. Differential Equations, 196 (2004), 316-339. 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), 332-372. 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), 204-224. 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), 291-295. doi: 10.1002/cpa.3160380303.
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