The blow-up profile for a nonlocal nonlinear parabolic equation with a nonlocal boundary condition

Authors:
Youpeng Chen and Lihua Liu

Journal:
Quart. Appl. Math. **70** (2012), 759-772

MSC (2000):
Primary 35A07, 35B40, 35K55, 35K57, 35K60

Published electronically:
August 27, 2012

MathSciNet review:
3052089

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a nonlocal reaction source and a nonlocal boundary condition. Under certain conditions, the blow-up criteria is established. Furthermore, under two additional conditions, the global blow-up behavior is shown, and when , the blow-up rate estimates are also obtained.

**1.**Jeffrey R. Anderson and Keng Deng,*Global existence for degenerate parabolic equations with a non-local forcing*, Math. Methods Appl. Sci.**20**(1997), no. 13, 1069–1087. MR**1465394**, 10.1002/(SICI)1099-1476(19970910)20:13<1069::AID-MMA867>3.0.CO;2-Y**2.**Robert Stephen Cantrell and Chris Cosner,*Diffusive logistic equations with indefinite weights: population models in disrupted environments. II*, SIAM J. Math. Anal.**22**(1991), no. 4, 1043–1064. MR**1112065**, 10.1137/0522068**3.**Yujuan Chen and Hongjun Gao,*Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation*, J. Math. Anal. Appl.**330**(2007), no. 2, 852–863. MR**2308412**, 10.1016/j.jmaa.2006.08.014**4.**Zhoujin Cui and Zuodong Yang,*Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition*, J. Math. Anal. Appl.**342**(2008), no. 1, 559–570. MR**2440821**, 10.1016/j.jmaa.2007.11.055**5.**W. A. Day,*Extensions of a property of the heat equation to linear thermoelasticity and other theories*, Quart. Appl. Math.**40**(1982/83), no. 3, 319–330. MR**678203****6.**W. A. Day,*A decreasing property of solutions of parabolic equations with applications to thermoelasticity*, Quart. Appl. Math.**40**(1982/83), no. 4, 468–475. MR**693879****7.**Keng Deng,*Comparison principle for some nonlocal problems*, Quart. Appl. Math.**50**(1992), no. 3, 517–522. MR**1178431****8.**Jesus Ildefonso Diaz and Robert Kersner,*On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium*, J. Differential Equations**69**(1987), no. 3, 368–403. MR**903393**, 10.1016/0022-0396(87)90125-2**9.**Weibing Deng, Yuxiang Li, and Chunhong Xie,*Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations*, Appl. Math. Lett.**16**(2003), no. 5, 803–808. MR**1986054**, 10.1016/S0893-9659(03)80118-0**10.**Weibing Deng, Li Yuxiang, and Xie Chunhong,*Global existence and nonexistence for a class of degenerate parabolic systems*, Nonlinear Anal.**55**(2003), no. 3, 233–244. MR**2007471**, 10.1016/S0362-546X(03)00226-8**11.**M. Escobedo and M. A. Herrero,*A semilinear parabolic system in a bounded domain*, Ann. Mat. Pura Appl. (4)**165**(1993), 315–336. MR**1271424**, 10.1007/BF01765854**12.**Lawrence C. Evans,*Partial differential equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR**1625845****13.**J. Furter and M. Grinfeld,*Local vs. nonlocal interactions in population dynamics*, J. Math. Biol.**27**(1989), no. 1, 65–80. MR**984226**, 10.1007/BF00276081**14.**Avner Friedman and Bryce McLeod,*Blow-up of positive solutions of semilinear heat equations*, Indiana Univ. Math. J.**34**(1985), no. 2, 425–447. MR**783924**, 10.1512/iumj.1985.34.34025**15.**Avner Friedman,*Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions*, Quart. Appl. Math.**44**(1986), no. 3, 401–407. MR**860893****16.**Yoshikazu Giga and Robert V. Kohn,*Asymptotically self-similar blow-up of semilinear heat equations*, Comm. Pure Appl. Math.**38**(1985), no. 3, 297–319. MR**784476**, 10.1002/cpa.3160380304**17.**Zhigui Lin and Yurong Liu,*Uniform blowup profiles for diffusion equations with nonlocal source and nonlocal boundary*, Acta Math. Sci. Ser. B Engl. Ed.**24**(2004), no. 3, 443–450. MR**2073260****18.**Lin Zhigui and Xie Chunhong,*The blow-up rate for a system of heat equations with nonlinear boundary conditions*, Nonlinear Anal.**34**(1998), no. 5, 767–778. MR**1634742**, 10.1016/S0362-546X(97)00573-7**19.**D. F. Rial and J. D. Rossi,*Blow-up results and localization of blow-up points in an 𝑁-dimensional smooth domain*, Duke Math. J.**88**(1997), no. 2, 391–405. MR**1455526**, 10.1215/S0012-7094-97-08816-5**20.**Sangwon Seo,*Blowup of solutions to heat equations with nonlocal boundary conditions*, Kobe J. Math.**13**(1996), no. 2, 123–132. MR**1442200****21.**Mingxin Wang,*Blow-up rate estimates for semilinear parabolic systems*, J. Differential Equations**170**(2001), no. 2, 317–324. MR**1815186**, 10.1006/jdeq.2000.3823**22.**Fred B. Weissler,*An 𝐿^{∞} blow-up estimate for a nonlinear heat equation*, Comm. Pure Appl. Math.**38**(1985), no. 3, 291–295. MR**784475**, 10.1002/cpa.3160380303**23.**Yulan Wang, Chunlai Mu, and Zhaoyin Xiang,*Blowup of solutions to a porous medium equation with nonlocal boundary condition*, Appl. Math. Comput.**192**(2007), no. 2, 579–585. MR**2385623**, 10.1016/j.amc.2007.03.036**24.**Yunfeng Yin,*On nonlinear parabolic equations with nonlocal boundary condition*, J. Math. Anal. Appl.**185**(1994), no. 1, 161–174. MR**1283048**, 10.1006/jmaa.1994.1239

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Additional Information

**Youpeng Chen**

Affiliation:
School of Mathematics, Yancheng Normal University, Yancheng 224002, Jiangsu, People’s Republic of China

Email:
youpengc@yahoo.com.cn

**Lihua Liu**

Affiliation:
School of Mathematics, Yancheng Normal University, Yancheng 224002, Jiangsu, People’s Republic of China, and School of Science, Hehai University, Nanjing 210098, Jiangsu, People’s Republic of China

Email:
liulihua11111@163.com

DOI:
http://dx.doi.org/10.1090/S0033-569X-2012-01278-5

Keywords:
Nonlinear parabolic equation,
nonlocal reaction source,
nonlocal boundary condition,
global blow-up,
blow-up rate

Received by editor(s):
February 17, 2011

Published electronically:
August 27, 2012

Additional Notes:
This research is supported by the research scheme of the natural science of the universities of Jiangsu province (08KJD110017)

Article copyright:
© Copyright 2012
Brown University

The copyright for this article reverts to public domain 28 years after publication.