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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). 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. 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. doi: 10.1016/j.crma.2013.09.024. Google Scholar [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer-Verlag, (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). 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). 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. doi: 10.1007/s00033-015-0537-7. Google Scholar [8] J. Filo, Diffusivity versus absorption through the boundary,, J. Differ. Eq., 99 (1992), 281. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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, (1995). doi: 10.1515/9783110889864.535. Google Scholar [21] B. Straughan, Explosive Instabilities in Mechanics,, Springer-Verlag, (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). 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). 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. 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. doi: 10.1016/j.crma.2013.09.024. Google Scholar [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer-Verlag, (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). 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). 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. doi: 10.1007/s00033-015-0537-7. Google Scholar [8] J. Filo, Diffusivity versus absorption through the boundary,, J. Differ. Eq., 99 (1992), 281. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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, (1995). doi: 10.1515/9783110889864.535. Google Scholar [21] B. Straughan, Explosive Instabilities in Mechanics,, Springer-Verlag, (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). doi: 10.1186/s13661-014-0265-5. Google Scholar
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