# American Institute of Mathematical Sciences

November  2017, 16(6): 2053-2068. doi: 10.3934/cpaa.2017101

## Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition

 Department of Mechanics and Mathematics Belarusian State University, Nezavisimosti avenue 4,220030 Minsk, Belarus

Received  December 2016 Revised  March 2017 Published  July 2017

Fund Project: This work is supported by the state program of fundamental research of Belarus, grant 1.2.03.1.

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence result. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.

Citation: Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
##### References:
 [1] J. M. Arrieta and A. Rodrígues-Bernal, Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions, Comm. Partial Differential Equations, 29 (2004), 1127-1148.  doi: 10.1081/PDE-200033760. [2] S. Carl and V. Lakshmikantham, Generalized quasilinearization method for reaction-diffusion equation under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl., 271 (2002), 182-205.  doi: 10.1016/S0022-247X(02)00114-2. [3] C. Cortazar, M. del Pino and M. Elgueta, On the short-time behaviour of the free boundary of a porous medium equation, Duke J. Math., 7 (1997), 133-149.  doi: 10.1215/S0012-7094-97-08706-8. [4] Z. Cui and Z. Yang, Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition, J. Math. Anal. Appl., 342 (2008), 559-570.  doi: 10.1016/j.jmaa.2007.11.055. [5] Z. Cui, Z. Yang and R. Zhang, Blow-up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition, Appl. Math. Comput., 224 (2013), 1-8.  doi: 10.1016/j.amc.2013.08.044. [6] K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math., 50 (1992), 517-522.  doi: 10.1090/qam/1178431. [7] K. Deng and C. L. Zhao, Blow-up for a parabolic system coupled in an equation and a boundary condition, Proc. Royal Soc. Edinb., 131A (2001), 1345-1355.  doi: 10.1017/S0308210500001426. [8] Z. B. Fang and J. Zhang, Global existence and blow-up properties of solutions for porous medium equation with nonlinear memory and weighted nonlocal boundary condition, Z. Angew. Math. Phys., 66 (2015), 67-81.  doi: 10.1007/s00033-013-0382-5. [9] A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math., 44 (1986), 401-407.  doi: 10.1090/qam/860893. [10] Y. Gao and W. Gao, Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition, Appl. Anal., 90 (2011), 799-809.  doi: 10.1080/00036811.2010.511191. [11] A. Gladkov, Initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal boundary condition, preprint, arXiv: 1602.05018. [12] A. Gladkov and M. Guedda, Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition, Nonlinear Anal., 74 (2011), 4573-4580.  doi: 10.1016/j.na.2011.04.027. [13] A. Gladkov and T. Kavitova, Blow-up problem for semilinear heat equation with nonlinear nonlocal boundary condition, Appl. Anal., 95 (2016), 1974-1988.  doi: 10.1080/00036811.2015.1080353. [14] A. Gladkov and K. I. Kim, Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition, J. Math. Anal. Appl., 338 (2008), 264-273.  doi: 10.1016/j.jmaa.2007.05.028. [15] A. Gladkov and A. Nikitin, On the existence of global solutions of a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions, Differential Equations, 52 (2016), 467-482.  doi: 10.1134/S0012266116040078. [16] B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Diff. Integral Equat., 9 (1996), 891-901. [17] 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. [18] C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations, Czechoslovac Math. J., 33 (1983), 262-285. [19] L. Kong and M. Wang, Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries, Science in China, Series A, 50 (2007), 1251-1266.  doi: 10.1007/s11425-007-0105-5. [20] D. Liu and C. Mu, Blowup properties for a semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions Abstr. Appl. Anal. 2010 (2010), Article ID 148035 17 pp. (electronic). doi: 10.1155/2010/148035. [21] M. Marras and S. Vernier Piro, Reaction-diffusion problems under non-local boundary conditions with blow-up solutions Journal of Inequalities and Applications 167 (2014), 11 pp. (electronic). doi: 10.1186/1029-242X-2014-167. [22] C. V. Pao, Asimptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 88 (1998), 225-238.  doi: 10.1016/S0377-0427(97)00215-X. [23] Y. Wang, C. Mu and Z. Xiang, Blowup of solutions to a porous medium equation with nonlocal boundary condition, Appl. Math. Comput., 192 (2007), 579-585.  doi: 10.1016/j.amc.2007.03.036. [24] L. Yang and C. Fan, Global existence and blow-up of solutions to a degenerate parabolic system with nonlocal sources and nonlocal boundaries, Monatshefte für Mathematik, 174 (2014), 493-510.  doi: 10.1007/s00605-013-0580-4. [25] Z. Ye and X. Xu, Global existence and blow-up for a porous medium system with nonlocal boundary conditions and nonlocal sources, Nonlinear Anal., 82 (2013), 115-126.  doi: 10.1016/j.na.2013.01.004. [26] H. M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl., 294 (2004), 712-728.  doi: 10.1016/j.jmaa.2004.03.021. [27] S. Zheng and L. Kong, Roles of weight functions in a nonlinear nonlocal parabolic system, Nonlinear Anal., 68 (2008), 2406-2416.  doi: 10.1016/j.na.2007.01.067.

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##### References:
 [1] J. M. Arrieta and A. Rodrígues-Bernal, Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions, Comm. Partial Differential Equations, 29 (2004), 1127-1148.  doi: 10.1081/PDE-200033760. [2] S. Carl and V. Lakshmikantham, Generalized quasilinearization method for reaction-diffusion equation under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl., 271 (2002), 182-205.  doi: 10.1016/S0022-247X(02)00114-2. [3] C. Cortazar, M. del Pino and M. Elgueta, On the short-time behaviour of the free boundary of a porous medium equation, Duke J. Math., 7 (1997), 133-149.  doi: 10.1215/S0012-7094-97-08706-8. [4] Z. Cui and Z. Yang, Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition, J. Math. Anal. Appl., 342 (2008), 559-570.  doi: 10.1016/j.jmaa.2007.11.055. [5] Z. Cui, Z. Yang and R. Zhang, Blow-up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition, Appl. Math. Comput., 224 (2013), 1-8.  doi: 10.1016/j.amc.2013.08.044. [6] K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math., 50 (1992), 517-522.  doi: 10.1090/qam/1178431. [7] K. Deng and C. L. Zhao, Blow-up for a parabolic system coupled in an equation and a boundary condition, Proc. Royal Soc. Edinb., 131A (2001), 1345-1355.  doi: 10.1017/S0308210500001426. [8] Z. B. Fang and J. Zhang, Global existence and blow-up properties of solutions for porous medium equation with nonlinear memory and weighted nonlocal boundary condition, Z. Angew. Math. Phys., 66 (2015), 67-81.  doi: 10.1007/s00033-013-0382-5. [9] A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math., 44 (1986), 401-407.  doi: 10.1090/qam/860893. [10] Y. Gao and W. Gao, Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition, Appl. Anal., 90 (2011), 799-809.  doi: 10.1080/00036811.2010.511191. [11] A. Gladkov, Initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal boundary condition, preprint, arXiv: 1602.05018. [12] A. Gladkov and M. Guedda, Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition, Nonlinear Anal., 74 (2011), 4573-4580.  doi: 10.1016/j.na.2011.04.027. [13] A. Gladkov and T. Kavitova, Blow-up problem for semilinear heat equation with nonlinear nonlocal boundary condition, Appl. Anal., 95 (2016), 1974-1988.  doi: 10.1080/00036811.2015.1080353. [14] A. Gladkov and K. I. Kim, Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition, J. Math. Anal. Appl., 338 (2008), 264-273.  doi: 10.1016/j.jmaa.2007.05.028. [15] A. Gladkov and A. Nikitin, On the existence of global solutions of a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions, Differential Equations, 52 (2016), 467-482.  doi: 10.1134/S0012266116040078. [16] B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Diff. Integral Equat., 9 (1996), 891-901. [17] 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. [18] C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations, Czechoslovac Math. J., 33 (1983), 262-285. [19] L. Kong and M. Wang, Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries, Science in China, Series A, 50 (2007), 1251-1266.  doi: 10.1007/s11425-007-0105-5. [20] D. Liu and C. Mu, Blowup properties for a semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions Abstr. Appl. Anal. 2010 (2010), Article ID 148035 17 pp. (electronic). doi: 10.1155/2010/148035. [21] M. Marras and S. Vernier Piro, Reaction-diffusion problems under non-local boundary conditions with blow-up solutions Journal of Inequalities and Applications 167 (2014), 11 pp. (electronic). doi: 10.1186/1029-242X-2014-167. [22] C. V. Pao, Asimptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 88 (1998), 225-238.  doi: 10.1016/S0377-0427(97)00215-X. [23] Y. Wang, C. Mu and Z. Xiang, Blowup of solutions to a porous medium equation with nonlocal boundary condition, Appl. Math. Comput., 192 (2007), 579-585.  doi: 10.1016/j.amc.2007.03.036. [24] L. Yang and C. Fan, Global existence and blow-up of solutions to a degenerate parabolic system with nonlocal sources and nonlocal boundaries, Monatshefte für Mathematik, 174 (2014), 493-510.  doi: 10.1007/s00605-013-0580-4. [25] Z. Ye and X. Xu, Global existence and blow-up for a porous medium system with nonlocal boundary conditions and nonlocal sources, Nonlinear Anal., 82 (2013), 115-126.  doi: 10.1016/j.na.2013.01.004. [26] H. M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl., 294 (2004), 712-728.  doi: 10.1016/j.jmaa.2004.03.021. [27] S. Zheng and L. Kong, Roles of weight functions in a nonlinear nonlocal parabolic system, Nonlinear Anal., 68 (2008), 2406-2416.  doi: 10.1016/j.na.2007.01.067.
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