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

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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

Kazuhiro Ishige ^1, and Ryuichi Sato ^1,

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

Received April 2015 Revised August 2015 Published October 2015

Abstract
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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.

Keywords: heat equation, blow-up time, blow-up rate., Nonlinear boundary condition, uniformly local $L^{r}$ spaces.

Mathematics Subject Classification: Primary: 35B44, 35K55, 35K6.

Citation: Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627

References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, 1975.
[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), 363-441.
[3]	D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math., 77 (1992), 127-159. doi: 10.1007/BF02567050.
[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), 376-406. doi: 10.1006/jdeq.1998.3612.
[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), 1-37. doi: 10.1080/03605300008821506.
[6]	J. M. Arrieta and A. Rodríguez-Bernal, Non well posedness of parabolic equations with supercritical nonlinearities, Commun. Contemp. Math., 6 (2004), 733-764. doi: 10.1142/S0219199704001495.
[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), 253-293. doi: 10.1142/S0218202504003234.
[8]	M. Chlebík and M. Fila, From critical exponents to blow-up rates for parabolic problems, Rend. Mat. Appl., 19 (1999), 449-470.
[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), 1323-1330. doi: 10.1002/1099-1476(200010)23:15<1323::AID-MMA167>3.0.CO;2-W.
[10]	M. Chlebík and M. Fila, Some recent results on blow-up on the boundary for the heat equation, in Evolution Equations: Existence, Regularity and Singularities, Banach Center Publ., 52, Polish Acad. Sci., Warsaw, 2000, 61-71.
[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), 169-192.
[12]	E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008.
[13]	E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[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), 215-220.
[15]	M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comm. Math. Univ. Carol. 30 (1989), 479-484.
[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), 197-205. doi: 10.1002/mma.1670140304.
[17]	J. Filo and J. Kačur, Local existence of general nonlinear parabolic systems, Nonlinear Anal., 24 (1995), 1597-1618. doi: 10.1016/0362-546X(94)00093-W.
[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]	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., 79, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.
[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), 28-49. doi: 10.1016/S0022-247X(02)00002-1.
[21]	J. Harada, Single point blow-up solutions to the heat equation with nonlinear boundary conditions, Differ. Equ. Appl., 5 (2013), 271-295. doi: 10.7153/dea-05-17.
[22]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313.
[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), 251-276.
[24]	B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Differential Integral Equations, 9 (1996), 891-901.
[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), 117-135. doi: 10.1090/S0002-9947-1994-1270664-3.
[26]	K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation, SIAM J. Math. Anal., 27 (1996), 1235-1260. doi: 10.1137/S0036141094270370.
[27]	K. Ishige and T. Kawakami, Global solutions of the heat equation with a nonlinear boundary condition, Calc. Var. Partial Differential Equations, 39 (2010), 429-457. doi: 10.1007/s00526-010-0316-4.
[28]	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.
[29]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society Translations, Vol. 23, American Mathematical Society, Providence, RI, 1968.
[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), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.
[31]	Y. Maekawa and Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces, Differential Integral Equations, 19 (2006), 369-400.
[32]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal., 10 (1986), 299-314. doi: 10.1016/0362-546X(86)90005-2.
[33]	P. Quittner, Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure, Math. Ann., (2015), 1-24. doi: 10.1007/s00208-015-1219-7.
[34]	P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.
[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), 671-681. doi: 10.3934/dcdss.2012.5.671.

show all references

References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, 1975.
[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), 363-441.
[3]	D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math., 77 (1992), 127-159. doi: 10.1007/BF02567050.
[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), 376-406. doi: 10.1006/jdeq.1998.3612.
[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), 1-37. doi: 10.1080/03605300008821506.
[6]	J. M. Arrieta and A. Rodríguez-Bernal, Non well posedness of parabolic equations with supercritical nonlinearities, Commun. Contemp. Math., 6 (2004), 733-764. doi: 10.1142/S0219199704001495.
[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), 253-293. doi: 10.1142/S0218202504003234.
[8]	M. Chlebík and M. Fila, From critical exponents to blow-up rates for parabolic problems, Rend. Mat. Appl., 19 (1999), 449-470.
[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), 1323-1330. doi: 10.1002/1099-1476(200010)23:15<1323::AID-MMA167>3.0.CO;2-W.
[10]	M. Chlebík and M. Fila, Some recent results on blow-up on the boundary for the heat equation, in Evolution Equations: Existence, Regularity and Singularities, Banach Center Publ., 52, Polish Acad. Sci., Warsaw, 2000, 61-71.
[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), 169-192.
[12]	E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008.
[13]	E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[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), 215-220.
[15]	M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comm. Math. Univ. Carol. 30 (1989), 479-484.
[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), 197-205. doi: 10.1002/mma.1670140304.
[17]	J. Filo and J. Kačur, Local existence of general nonlinear parabolic systems, Nonlinear Anal., 24 (1995), 1597-1618. doi: 10.1016/0362-546X(94)00093-W.
[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]	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., 79, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.
[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), 28-49. doi: 10.1016/S0022-247X(02)00002-1.
[21]	J. Harada, Single point blow-up solutions to the heat equation with nonlinear boundary conditions, Differ. Equ. Appl., 5 (2013), 271-295. doi: 10.7153/dea-05-17.
[22]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313.
[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), 251-276.
[24]	B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Differential Integral Equations, 9 (1996), 891-901.
[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), 117-135. doi: 10.1090/S0002-9947-1994-1270664-3.
[26]	K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation, SIAM J. Math. Anal., 27 (1996), 1235-1260. doi: 10.1137/S0036141094270370.
[27]	K. Ishige and T. Kawakami, Global solutions of the heat equation with a nonlinear boundary condition, Calc. Var. Partial Differential Equations, 39 (2010), 429-457. doi: 10.1007/s00526-010-0316-4.
[28]	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.
[29]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society Translations, Vol. 23, American Mathematical Society, Providence, RI, 1968.
[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), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.
[31]	Y. Maekawa and Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces, Differential Integral Equations, 19 (2006), 369-400.
[32]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal., 10 (1986), 299-314. doi: 10.1016/0362-546X(86)90005-2.
[33]	P. Quittner, Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure, Math. Ann., (2015), 1-24. doi: 10.1007/s00208-015-1219-7.
[34]	P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.
[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), 671-681. doi: 10.3934/dcdss.2012.5.671.

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