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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
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show all references

References:
[1]

Bound. Value. Probl., 2013 (2013), 6 pages. doi: 10.1186/1687-2770-2013-239.  Google Scholar

[2]

J. Math. Anal. Appl., 378 (2011), 528-540. doi: 10.1016/j.jmaa.2010.12.036.  Google Scholar

[3]

C. R. Acad. Sci. Paris. Ser. I., 351 (2013), 731-735. doi: 10.1016/j.crma.2013.09.024.  Google Scholar

[4]

Springer-Verlag, New York, 2011.  Google Scholar

[5]

Abstr. Appl. Anal., 2014 (2014), Art. ID 289245, 8 pp. doi: 10.1155/2014/289245.  Google Scholar

[6]

Math. Probl. Eng., 2014 (2014), Art. ID 764248, 6 pp. doi: 10.1155/2014/764248.  Google Scholar

[7]

Z. Angew. Math. Phys., 66 (2015), 2525-2541. doi: 10.1007/s00033-015-0537-7.  Google Scholar

[8]

J. Differ. Eq., 99 (1992), 281-305. doi: 10.1016/0022-0396(92)90024-H.  Google Scholar

[9]

J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081.  Google Scholar

[10]

Discrete Cont. Dyn. Syst., 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.  Google Scholar

[11]

Math. Ann., 214 (1975), 205-220. doi: 10.1007/BF01352106.  Google Scholar

[12]

Math. Comput. Model., 57 (2013), 926-931. doi: 10.1016/j.mcm.2012.10.002.  Google Scholar

[13]

Discrete Cont. Dyn. Syst., 2013 (2013), 535-544. doi: 10.3934/proc.2013.2013.535.  Google Scholar

[14]

Z.Angew Math. Phys., 61 (2010), 999-1007. doi: 10.1007/s00033-010-0071-6.  Google Scholar

[15]

Nonlinear Anal., 73 (2010), 971-978. doi: 10.1016/j.na.2010.04.023.  Google Scholar

[16]

Proc. R. Soc. Edinb. A., 142 (2012), 625-631. doi: 10.1017/S0308210511000485.  Google Scholar

[17]

Appl. Anal., 91 (2012), 2245-2256. doi: 10.1080/00036811.2011.598865.  Google Scholar

[18]

Proc. Am. Math. Soc., 141 (2013), 2309-2318. doi: 10.1090/S0002-9939-2013-11493-0.  Google Scholar

[19]

Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar

[20]

Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.535.  Google Scholar

[21]

Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58807-5.  Google Scholar

[22]

Bound. Value. Probl., 2014 (2014), 5 pages. doi: 10.1186/s13661-014-0265-5.  Google Scholar

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