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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

show all references

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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