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December  2016, 21(10): 3619-3635. doi: 10.3934/dcdsb.2016113

## Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux

 1 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China, China

Received  January 2016 Revised  April 2016 Published  November 2016

This paper deals with blow-up phenomena for an initial boundary value problem of a nonlocal quasilinear parabolic equation with time-dependent coefficients in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficients and nonlinearities to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, upper and lower bounds of $t^{\ast}$ are obtained under suitable measure in high-dimensional spaces. Finally, some application examples are presented.
Citation: 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
##### References:
 [1] I. Ahmed, C. L. Mu, P. Zheng and F. C. Zhang, Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient, Bound. Value. Probl., 2013 (2013), 6 pages. doi: 10.1186/1687-2770-2013-239.  Google Scholar [2] W. Allegretto, G. Fragnelli, P. Nistri and D. Papin, Coexistence and optimal control problems for a degenerate predator-prey model, J. Math. Anal. Appl., 378 (2011), 528-540. doi: 10.1016/j.jmaa.2010.12.036.  Google Scholar [3] K. Baghaei and M. Hesaaraki, Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations, C. R. Acad. Sci. Paris. Ser. I., 351 (2013), 731-735. doi: 10.1016/j.crma.2013.09.024.  Google Scholar [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar [5] Z. B. Fang and Y. Chai, Blow-up analysis for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition, Abstr. Appl. Anal., 2014 (2014), Art. ID 289245, 8 pp. doi: 10.1155/2014/289245.  Google Scholar [6] Z. B. Fang, R. Yang and Y. Chai, Lower bounds estimate for the blow-up time of a slow diffusion equation with nonlocal source and inner absorption, Math. Probl. Eng., 2014 (2014), Art. ID 764248, 6 pp. doi: 10.1155/2014/764248.  Google Scholar [7] Z. B. Fang and Y. X. Wang, Blow-up analysis for a semilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux, Z. Angew. Math. Phys., 66 (2015), 2525-2541. doi: 10.1007/s00033-015-0537-7.  Google Scholar [8] J. Filo, Diffusivity versus absorption through the boundary, J. Differ. Eq., 99 (1992), 281-305. doi: 10.1016/0022-0396(92)90024-H.  Google Scholar [9] J. Furter and M. Grinfield, Local vs. nonlocal interactions in populations dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081.  Google Scholar [10] V. A. Galaktionov and J. L. Vázquez, The problem of blow up in nonlinear parabolic equations, Discrete Cont. Dyn. Syst., 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.  Google Scholar [11] H. A. Levine, Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: The method of unbounded Fourier coefficients, Math. Ann., 214 (1975), 205-220. doi: 10.1007/BF01352106.  Google Scholar [12] Y. Liu, Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions, Math. Comput. Model., 57 (2013), 926-931. doi: 10.1016/j.mcm.2012.10.002.  Google Scholar [13] M. Marras and S. Vernier Piro, On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients, Discrete Cont. Dyn. Syst., 2013 (2013), 535-544. doi: 10.3934/proc.2013.2013.535.  Google Scholar [14] L. E. Payne, G. A. Philippin and S. Vernier Piro, Blow-up phenomena for semilinear heat equation with nonlinear boundary condition I, Z.Angew Math. Phys., 61 (2010), 999-1007. doi: 10.1007/s00033-010-0071-6.  Google Scholar [15] L. E. Payne, G. A. Philippin and S. Vernier Piro, Blow-up phenonmena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Anal., 73 (2010), 971-978. doi: 10.1016/j.na.2010.04.023.  Google Scholar [16] L. E. Payne and G. A. Philippin, Blow-up phenonmena in parabolic problems with time dependent coefficients under Neumann boundary conditions, Proc. R. Soc. Edinb. A., 142 (2012), 625-631. doi: 10.1017/S0308210511000485.  Google Scholar [17] L. E. Payne and G. A. Philippin, Blow up in a class of non-linear parabolic problems with time dependent coefficients under Robin type boundary conditions, Appl. Anal., 91 (2012), 2245-2256. doi: 10.1080/00036811.2011.598865.  Google Scholar [18] L. E. Payne and G. A. Philippin, Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet Boundary conditions, Proc. Am. Math. Soc., 141 (2013), 2309-2318. doi: 10.1090/S0002-9939-2013-11493-0.  Google Scholar [19] R. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar [20] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-Up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.535.  Google Scholar [21] B. Straughan, Explosive Instabilities in Mechanics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58807-5.  Google Scholar [22] G. S. Tang, Y. F. Li and X. T. Yang, Lower bounds for the blow-up time of the nonlinear non-local reaction diffusion problems in $R^N(N\geq3)$, Bound. Value. Probl., 2014 (2014), 5 pages. doi: 10.1186/s13661-014-0265-5.  Google Scholar

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##### References:
 [1] I. Ahmed, C. L. Mu, P. Zheng and F. C. Zhang, Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient, Bound. Value. Probl., 2013 (2013), 6 pages. doi: 10.1186/1687-2770-2013-239.  Google Scholar [2] W. Allegretto, G. Fragnelli, P. Nistri and D. Papin, Coexistence and optimal control problems for a degenerate predator-prey model, J. Math. Anal. Appl., 378 (2011), 528-540. doi: 10.1016/j.jmaa.2010.12.036.  Google Scholar [3] K. Baghaei and M. Hesaaraki, Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations, C. R. Acad. Sci. Paris. Ser. I., 351 (2013), 731-735. doi: 10.1016/j.crma.2013.09.024.  Google Scholar [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar [5] Z. B. Fang and Y. Chai, Blow-up analysis for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition, Abstr. Appl. Anal., 2014 (2014), Art. ID 289245, 8 pp. doi: 10.1155/2014/289245.  Google Scholar [6] Z. B. Fang, R. Yang and Y. Chai, Lower bounds estimate for the blow-up time of a slow diffusion equation with nonlocal source and inner absorption, Math. Probl. Eng., 2014 (2014), Art. ID 764248, 6 pp. doi: 10.1155/2014/764248.  Google Scholar [7] Z. B. Fang and Y. X. Wang, Blow-up analysis for a semilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux, Z. Angew. Math. Phys., 66 (2015), 2525-2541. doi: 10.1007/s00033-015-0537-7.  Google Scholar [8] J. Filo, Diffusivity versus absorption through the boundary, J. Differ. Eq., 99 (1992), 281-305. doi: 10.1016/0022-0396(92)90024-H.  Google Scholar [9] J. Furter and M. Grinfield, Local vs. nonlocal interactions in populations dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081.  Google Scholar [10] V. A. Galaktionov and J. L. Vázquez, The problem of blow up in nonlinear parabolic equations, Discrete Cont. Dyn. Syst., 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.  Google Scholar [11] H. A. Levine, Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: The method of unbounded Fourier coefficients, Math. Ann., 214 (1975), 205-220. doi: 10.1007/BF01352106.  Google Scholar [12] Y. Liu, Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions, Math. Comput. Model., 57 (2013), 926-931. doi: 10.1016/j.mcm.2012.10.002.  Google Scholar [13] M. Marras and S. Vernier Piro, On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients, Discrete Cont. Dyn. Syst., 2013 (2013), 535-544. doi: 10.3934/proc.2013.2013.535.  Google Scholar [14] L. E. Payne, G. A. Philippin and S. Vernier Piro, Blow-up phenomena for semilinear heat equation with nonlinear boundary condition I, Z.Angew Math. Phys., 61 (2010), 999-1007. doi: 10.1007/s00033-010-0071-6.  Google Scholar [15] L. E. Payne, G. A. Philippin and S. Vernier Piro, Blow-up phenonmena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Anal., 73 (2010), 971-978. doi: 10.1016/j.na.2010.04.023.  Google Scholar [16] L. E. Payne and G. A. Philippin, Blow-up phenonmena in parabolic problems with time dependent coefficients under Neumann boundary conditions, Proc. R. Soc. Edinb. A., 142 (2012), 625-631. doi: 10.1017/S0308210511000485.  Google Scholar [17] L. E. Payne and G. A. Philippin, Blow up in a class of non-linear parabolic problems with time dependent coefficients under Robin type boundary conditions, Appl. Anal., 91 (2012), 2245-2256. doi: 10.1080/00036811.2011.598865.  Google Scholar [18] L. E. Payne and G. A. Philippin, Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet Boundary conditions, Proc. Am. Math. Soc., 141 (2013), 2309-2318. doi: 10.1090/S0002-9939-2013-11493-0.  Google Scholar [19] R. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar [20] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-Up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.535.  Google Scholar [21] B. Straughan, Explosive Instabilities in Mechanics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58807-5.  Google Scholar [22] G. S. Tang, Y. F. Li and X. T. Yang, Lower bounds for the blow-up time of the nonlinear non-local reaction diffusion problems in $R^N(N\geq3)$, Bound. Value. Probl., 2014 (2014), 5 pages. doi: 10.1186/s13661-014-0265-5.  Google Scholar
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