Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

BLOWUP PROPERTIES FOR PARABOLIC EQUATIONS COUPLED VIA NON-STANDARD GROWTH SOURCES

  • Liu, Bingchen (College of Science, China University of Petroleum) ;
  • Hong, Zhenzhen (College of Science, China University of Petroleum)
  • 투고 : 2012.05.10
  • 심사 : 2012.09.23
  • 발행 : 2013.01.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

This paper deals with parabolic equations coupled via nonstandard growth sources, subject to homogeneous Dirichlet boundary conditions. Three kinds of necessary and sufficient conditions are obtained, which determine the complete classifications for non-simultaneous and simultaneous blowup phenomena. Moreover, blowup rates are given.

키워드

참고문헌

  1. M. Escobedo and M.A. Herrero, A semilinear parabolic system in a bounded domain, Annali di Matematica pura ed applicata CLXV (1993), 315-336.
  2. Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999), 374-406. https://doi.org/10.1006/jdeq.1998.3535
  3. E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213-259. https://doi.org/10.1007/s00205-002-0208-7
  4. S.N. Antontsev and J.F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara, Sez. VII Sci. Mat. 52 (2006), 19-36. https://doi.org/10.1007/s11565-006-0002-9
  5. K. Rajagopal and M. Ruzicka, Mathematical modelling of electro-rheological fluids, Contin. Mech. Thermodyn. 13 (2001), 59-78. https://doi.org/10.1007/s001610100034
  6. R. Aboulaicha, D. Meskinea, and A. Souissia, New diffusion models in image processing, Comput. Math. Appl. 56 (2008), 874-882. https://doi.org/10.1016/j.camwa.2008.01.017
  7. Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406. https://doi.org/10.1137/050624522
  8. S. Levine, Y. Chen, and J. Stanich, Image restoration via nonstandard diffusion. Technical Report# 04-01, Dept. of Mathematics and Computer Science, Duquesne University, 2004.
  9. C.V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.
  10. A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov, Blow-up in quasilinear parabolic equations, Walter de Gruyter, Berlin, New York, 1995.
  11. F.C. Li, S.X. Huang, and C.H. Xie, Global existence and blow-up of solutions to a nonlocal reaction-diffsion system, Discrete Contin. Dyn. Syst. 9 (2003), 1519-1532. https://doi.org/10.3934/dcds.2003.9.1519
  12. J.R. Cannon and H.M. Yin, A class of non-linear non-classical parabolic equations, J. Differential Equations 79 (1989), 266-288. https://doi.org/10.1016/0022-0396(89)90103-4
  13. H.L. Li and M.X. Wang, Properties of blow-up solutions to a parabolic system with nonlinear localized terms, Discrete and Continuous Dynamical Systems 13 (2005), 683-700. https://doi.org/10.3934/dcds.2005.13.683
  14. C.V. Pao, Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory, J. Math. Anal. Appl. 166 (1992) 591-600. https://doi.org/10.1016/0022-247X(92)90318-8
  15. Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), 1301-1334. https://doi.org/10.1137/S0036141097318900
  16. J.P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Anal. 71 (2009), 1094-1099. https://doi.org/10.1016/j.na.2008.11.030
  17. S.N. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math. 234 (2010), 2633-2645. https://doi.org/10.1016/j.cam.2010.01.026
  18. R. Ferreira, A.de Pablo, M. Perez-Llanos, and J.D. Rossi, Critical exponents for a semilinear parabolic equation with variable reaction, To appear in The Royal Society of Edinburgh Proceedings A (Mathematics). (http://mate.dm.uba.ar/-jrossi/complete.html).
  19. X.L. Bai and S.N. Zheng, A semilinear parabolic system with coupling variable exponents, Annali di Matematica Pura ed Applicata 190 (2011), 525-537. https://doi.org/10.1007/s10231-010-0161-2