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December  2018, 23(10): 4243-4254. doi: 10.3934/dcdsb.2018135

## Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems

 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

* Corresponding author: Juntang Ding

Received  September 2017 Published  December 2018 Early access  April 2018

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 61473180).

In this paper, we consider a quasilinear reaction diffusion equation with Neumann boundary conditions in a bounded domain. Basing on Sobolev inequality and differential inequality technique, we obtain upper and lower bounds for the blow-up time of the solution. An example is also given to illustrate the abstract results obtained of this paper.

Citation: Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4243-4254. doi: 10.3934/dcdsb.2018135
##### References:
 [1] K. Baghaei and M. Hesaaraki, Blow-up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions, Math. Methods Appl. Sci., 38 (2015), 527-536.  doi: 10.1002/mma.3085.  Google Scholar [2] C. Bandle and H. Brunner, Blow-up in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), 3-22.  doi: 10.1016/S0377-0427(98)00100-9.  Google Scholar [3] J. T. Ding, Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions, Comput. Math. Appl., 65 (2013), 1808-1822.  doi: 10.1016/j.camwa.2013.03.013.  Google Scholar [4] J. T. Ding and H. J. Hu, Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions, J. Math. Anal. Appl., 433 (2016), 1718-1735.  doi: 10.1016/j.jmaa.2015.08.046.  Google Scholar [5] J. T. Ding and H. J. Hu, Blow-up solutions for nonlinear reaction diffusion equations under Neumann boundary conditions, Appl. Anal., 96 (2016), 549-562.  doi: 10.1080/00036811.2016.1143933.  Google Scholar [6] J. T. Ding and X. H. Shen, Blow-up in p-Laplacian heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys. 67 (2016), Art. 125, 18 pp.  Google Scholar [7] C. Enache, Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 2864-2874.  doi: 10.1016/j.na.2007.08.063.  Google Scholar [8] C. Enache, Lower bounds for blow-up time in some non-linear parabolic problems under Neumann boundary conditions, Glasgow Math. J., 53 (2011), 569-575.  doi: 10.1017/S0017089511000139.  Google Scholar [9] L. C. Evans, Partial Differential Equations, AMS, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.  Google Scholar [10] Z. B. Fang and L. W. Ma, Blow-up analysis for a reaction-diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux, Nonlinear Anal. RWA, 32 (2016), 338-354.  doi: 10.1016/j.nonrwa.2016.05.005.  Google Scholar [11] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 2018, Springer, Heidelberg, 2011.  Google Scholar [12] F. S. Li and J. L. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary, J. Math. Anal. Appl., 385 (2012), 1005-1014.  doi: 10.1016/j.jmaa.2011.07.018.  Google Scholar [13] F. S. Li and J. L. Li, Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions, Bound. Value. Prob., 2014 (2014), 1-14.   Google Scholar [14] M. Marras and S. Vernier-Piro, On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients, Discrete Contin. Dyn. Syst., (2013), 535-544.   Google Scholar [15] M. Marras, S. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Math. Methods Appl. Sci., 39 (2016), 2787-2798.  doi: 10.1002/mma.3728.  Google Scholar [16] M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulgare Sci., 69 (2016), 687-696.   Google Scholar [17] L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in paraboblic problems under Neumann conditions, Appl. Anal., 85 (2006), 1301-1311.  doi: 10.1080/00036810600915730.  Google Scholar [18] L. E. Payne and P. W. Schaefer, Bounds for the blow-up time for the heat equation under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Section A, 139 (2009), 1289-1296.  doi: 10.1017/S0308210508000802.  Google Scholar [19] X. F. Song and X. S. Lv, Bounds for the blowup time and blowup rate estimates for a type of parabolic equations with weighted source, Appl. Math. Comput., 236 (2014), 78-92.  doi: 10.1016/j.amc.2014.03.023.  Google Scholar [20] R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York, 1981.  Google Scholar

show all references

##### References:
 [1] K. Baghaei and M. Hesaaraki, Blow-up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions, Math. Methods Appl. Sci., 38 (2015), 527-536.  doi: 10.1002/mma.3085.  Google Scholar [2] C. Bandle and H. Brunner, Blow-up in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), 3-22.  doi: 10.1016/S0377-0427(98)00100-9.  Google Scholar [3] J. T. Ding, Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions, Comput. Math. Appl., 65 (2013), 1808-1822.  doi: 10.1016/j.camwa.2013.03.013.  Google Scholar [4] J. T. Ding and H. J. Hu, Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions, J. Math. Anal. Appl., 433 (2016), 1718-1735.  doi: 10.1016/j.jmaa.2015.08.046.  Google Scholar [5] J. T. Ding and H. J. Hu, Blow-up solutions for nonlinear reaction diffusion equations under Neumann boundary conditions, Appl. Anal., 96 (2016), 549-562.  doi: 10.1080/00036811.2016.1143933.  Google Scholar [6] J. T. Ding and X. H. Shen, Blow-up in p-Laplacian heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys. 67 (2016), Art. 125, 18 pp.  Google Scholar [7] C. Enache, Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 2864-2874.  doi: 10.1016/j.na.2007.08.063.  Google Scholar [8] C. Enache, Lower bounds for blow-up time in some non-linear parabolic problems under Neumann boundary conditions, Glasgow Math. J., 53 (2011), 569-575.  doi: 10.1017/S0017089511000139.  Google Scholar [9] L. C. Evans, Partial Differential Equations, AMS, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.  Google Scholar [10] Z. B. Fang and L. W. Ma, Blow-up analysis for a reaction-diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux, Nonlinear Anal. RWA, 32 (2016), 338-354.  doi: 10.1016/j.nonrwa.2016.05.005.  Google Scholar [11] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 2018, Springer, Heidelberg, 2011.  Google Scholar [12] F. S. Li and J. L. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary, J. Math. Anal. Appl., 385 (2012), 1005-1014.  doi: 10.1016/j.jmaa.2011.07.018.  Google Scholar [13] F. S. Li and J. L. Li, Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions, Bound. Value. Prob., 2014 (2014), 1-14.   Google Scholar [14] M. Marras and S. Vernier-Piro, On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients, Discrete Contin. Dyn. Syst., (2013), 535-544.   Google Scholar [15] M. Marras, S. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Math. Methods Appl. Sci., 39 (2016), 2787-2798.  doi: 10.1002/mma.3728.  Google Scholar [16] M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulgare Sci., 69 (2016), 687-696.   Google Scholar [17] L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in paraboblic problems under Neumann conditions, Appl. Anal., 85 (2006), 1301-1311.  doi: 10.1080/00036810600915730.  Google Scholar [18] L. E. Payne and P. W. Schaefer, Bounds for the blow-up time for the heat equation under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Section A, 139 (2009), 1289-1296.  doi: 10.1017/S0308210508000802.  Google Scholar [19] X. F. Song and X. S. Lv, Bounds for the blowup time and blowup rate estimates for a type of parabolic equations with weighted source, Appl. Math. Comput., 236 (2014), 78-92.  doi: 10.1016/j.amc.2014.03.023.  Google Scholar [20] R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York, 1981.  Google Scholar
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