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May  2016, 36(5): 2627-2652. doi: 10.3934/dcds.2016.36.2627

## Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

 1 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

Received  April 2015 Revised  August 2015 Published  October 2015

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$ and the lower blow-up estimates of the solutions.
Citation: Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
##### References:
 [1] R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar [2] D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1991), 363.   Google Scholar [3] D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources,, Manuscripta Math., 77 (1992), 127.  doi: 10.1007/BF02567050.  Google Scholar [4] J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities,, J. Differential Equations, 156 (1999), 376.  doi: 10.1006/jdeq.1998.3612.  Google Scholar [5] J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds,, Comm. Partial Differential Equations, 25 (2000), 1.  doi: 10.1080/03605300008821506.  Google Scholar [6] J. M. 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Levine, On critical exponents for a system of heat equations coupled in the boundary conditions,, Acta Math. Univ. Comenianae, 63 (1994), 169.   Google Scholar [12] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation,, Indiana Univ. Math. J., 32 (1983), 83.  doi: 10.1512/iumj.1983.32.32008.  Google Scholar [13] E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar [14] J. Fernández Bonder and J. D. Rossi, Life span for solutions of the heat equation with a nonlinear boundary condition,, Tsukuba J. Math., 25 (2001), 215.   Google Scholar [15] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions,, Comm. Math. Univ. Carol. 30 (1989), 30 (1989), 479.   Google Scholar [16] 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 [17] J. Filo and J. Kačur, Local existence of general nonlinear parabolic systems,, Nonlinear Anal., 24 (1995), 1597.  doi: 10.1016/0362-546X(94)00093-W.  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] M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions,, Progr. Nonlinear Differential Equations Appl., (2010).  doi: 10.1007/978-0-8176-4651-6.  Google Scholar [20] J.-S. Guo and B. Hu, Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary,, J. Math. Anal. Appl., 269 (2002), 28.  doi: 10.1016/S0022-247X(02)00002-1.  Google Scholar [21] J. Harada, Single point blow-up solutions to the heat equation with nonlinear boundary conditions,, Differ. Equ. Appl., 5 (2013), 271.  doi: 10.7153/dea-05-17.  Google Scholar [22] B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301.   Google Scholar [23] 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 [24] B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 891.   Google Scholar [25] 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: 10.1090/S0002-9947-1994-1270664-3.  Google Scholar [26] K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation,, SIAM J. Math. Anal., 27 (1996), 1235.  doi: 10.1137/S0036141094270370.  Google Scholar [27] 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.  Google Scholar [28] 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.  Google Scholar [29] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, American Mathematical Society Translations, (1968).   Google Scholar [30] T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem,, Trans. Amer. Math. Soc., 333 (1992), 365.  doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar [31] Y. Maekawa and Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces,, Differential Integral Equations, 19 (2006), 369.   Google Scholar [32] M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations,, Nonlinear Anal., 10 (1986), 299.  doi: 10.1016/0362-546X(86)90005-2.  Google Scholar [33] P. Quittner, Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure,, Math. Ann., (2015), 1.  doi: 10.1007/s00208-015-1219-7.  Google Scholar [34] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts, (2007).   Google Scholar [35] P. Quittner and P. Souplet, Blow-up rate of solutions of parabolic problems with nonlinear boundary conditions,, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 671.  doi: 10.3934/dcdss.2012.5.671.  Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar [2] D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1991), 363.   Google Scholar [3] D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources,, Manuscripta Math., 77 (1992), 127.  doi: 10.1007/BF02567050.  Google Scholar [4] J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities,, J. Differential Equations, 156 (1999), 376.  doi: 10.1006/jdeq.1998.3612.  Google Scholar [5] J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds,, Comm. Partial Differential Equations, 25 (2000), 1.  doi: 10.1080/03605300008821506.  Google Scholar [6] J. M. Arrieta and A. Rodríguez-Bernal, Non well posedness of parabolic equations with supercritical nonlinearities,, Commun. Contemp. Math., 6 (2004), 733.  doi: 10.1142/S0219199704001495.  Google Scholar [7] J. M. Arrieta, A. Rodríguez-Bernal, J. W. Cholewa and T. Dlotko, Linear parabolic equations in locally uniform spaces,, Math. Models Methods Appl. Sci., 14 (2004), 253.  doi: 10.1142/S0218202504003234.  Google Scholar [8] M. Chlebík and M. Fila, From critical exponents to blow-up rates for parabolic problems,, Rend. Mat. Appl., 19 (1999), 449.   Google Scholar [9] 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 [10] M. Chlebík and M. Fila, Some recent results on blow-up on the boundary for the heat equation,, in Evolution Equations: Existence, (2000), 61.   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, 63 (1994), 169.   Google Scholar [12] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation,, Indiana Univ. Math. J., 32 (1983), 83.  doi: 10.1512/iumj.1983.32.32008.  Google Scholar [13] E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar [14] J. Fernández Bonder and J. D. Rossi, Life span for solutions of the heat equation with a nonlinear boundary condition,, Tsukuba J. Math., 25 (2001), 215.   Google Scholar [15] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions,, Comm. Math. Univ. Carol. 30 (1989), 30 (1989), 479.   Google Scholar [16] 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 [17] J. Filo and J. Kačur, Local existence of general nonlinear parabolic systems,, Nonlinear Anal., 24 (1995), 1597.  doi: 10.1016/0362-546X(94)00093-W.  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] M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions,, Progr. Nonlinear Differential Equations Appl., (2010).  doi: 10.1007/978-0-8176-4651-6.  Google Scholar [20] J.-S. Guo and B. Hu, Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary,, J. Math. Anal. Appl., 269 (2002), 28.  doi: 10.1016/S0022-247X(02)00002-1.  Google Scholar [21] J. Harada, Single point blow-up solutions to the heat equation with nonlinear boundary conditions,, Differ. Equ. Appl., 5 (2013), 271.  doi: 10.7153/dea-05-17.  Google Scholar [22] B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301.   Google Scholar [23] 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 [24] B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 891.   Google Scholar [25] 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: 10.1090/S0002-9947-1994-1270664-3.  Google Scholar [26] K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation,, SIAM J. Math. Anal., 27 (1996), 1235.  doi: 10.1137/S0036141094270370.  Google Scholar [27] 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.  Google Scholar [28] 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.  Google Scholar [29] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, American Mathematical Society Translations, (1968).   Google Scholar [30] T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem,, Trans. Amer. Math. Soc., 333 (1992), 365.  doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar [31] Y. Maekawa and Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces,, Differential Integral Equations, 19 (2006), 369.   Google Scholar [32] M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations,, Nonlinear Anal., 10 (1986), 299.  doi: 10.1016/0362-546X(86)90005-2.  Google Scholar [33] P. Quittner, Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure,, Math. Ann., (2015), 1.  doi: 10.1007/s00208-015-1219-7.  Google Scholar [34] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts, (2007).   Google Scholar [35] P. Quittner and P. Souplet, Blow-up rate of solutions of parabolic problems with nonlinear boundary conditions,, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 671.  doi: 10.3934/dcdss.2012.5.671.  Google Scholar
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