# American Institute of Mathematical Sciences

• Previous Article
Invariance properties of the Monge-Kantorovich mass transport problem
• DCDS Home
• This Issue
• Next Article
Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions
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, 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.  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), 363-441.  Google Scholar [3] D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math., 77 (1992), 127-159. 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), 376-406. 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), 1-37. 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), 733-764. 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), 253-293. 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), 449-470.  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), 1323-1330. 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, Regularity and Singularities, Banach Center Publ., 52, Polish Acad. Sci., Warsaw, 2000, 61-71.  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), 169-192.  Google Scholar [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.  Google Scholar [13] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 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), 215-220.  Google Scholar [15] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comm. Math. Univ. Carol. 30 (1989), 479-484.  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), 197-205. doi: 10.1002/mma.1670140304.  Google Scholar [17] J. Filo and J. Kačur, Local existence of general nonlinear parabolic systems, Nonlinear Anal., 24 (1995), 1597-1618. 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), 125-146. 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., 79, Birkhäuser Boston, Inc., Boston, MA, 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), 28-49. 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), 271-295. 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), 301-313.  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), 251-276.  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), 891-901.  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), 117-135. 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), 1235-1260. 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), 429-457. 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), 320-329. 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, Vol. 23, American Mathematical Society, Providence, RI, 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), 365-378. 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), 369-400.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 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), 671-681. doi: 10.3934/dcdss.2012.5.671.  Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, 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), 363-441.  Google Scholar [3] D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math., 77 (1992), 127-159. 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), 376-406. 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), 1-37. 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), 733-764. 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), 253-293. 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), 449-470.  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), 1323-1330. 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, Regularity and Singularities, Banach Center Publ., 52, Polish Acad. Sci., Warsaw, 2000, 61-71.  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), 169-192.  Google Scholar [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.  Google Scholar [13] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 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), 215-220.  Google Scholar [15] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comm. Math. Univ. Carol. 30 (1989), 479-484.  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), 197-205. doi: 10.1002/mma.1670140304.  Google Scholar [17] J. Filo and J. Kačur, Local existence of general nonlinear parabolic systems, Nonlinear Anal., 24 (1995), 1597-1618. 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), 125-146. 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., 79, Birkhäuser Boston, Inc., Boston, MA, 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), 28-49. 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), 271-295. 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), 301-313.  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), 251-276.  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), 891-901.  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), 117-135. 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), 1235-1260. 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), 429-457. 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), 320-329. 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, Vol. 23, American Mathematical Society, Providence, RI, 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), 365-378. 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), 369-400.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 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), 671-681. doi: 10.3934/dcdss.2012.5.671.  Google Scholar
 [1] Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 [2] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [3] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [4] 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 [5] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [6] Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935 [7] Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032 [8] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [9] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [10] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [11] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 [12] Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 [13] Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025 [14] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [15] Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 [16] Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 [17] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [18] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [19] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [20] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

2019 Impact Factor: 1.338