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Applicable Analysis
An International Journal
Volume 90, 2011 - Issue 9
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Original Articles

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

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Pages 1373-1389 | Received 29 May 2010, Accepted 01 Sep 2010, Published online: 11 Mar 2011

References

  • Budd , C , Galaktionov , VA and Chen , J . 1998 . Focusing blow-up for quasilinear parabolic equations . Proc. Roy. Soc. Edinb. Sect. A , 128 : 965 – 992 .
  • Kamin , S and Rosenau , P . 1982 . Nonlinear diffusion in a finite mass medium . Commun. Pure Appl. Math. , 35 : 113 – 127 .
  • Kamin , S and Rosenau , P . 1981 . Propagation of thermal waves in an inhomogeneous medium . Commun. Pure Appl. Math. , 34 : 831 – 852 .
  • Anderson , JR . 1991 . Local existence and uniqueness of solution of degenerate parabolic equations . Commun. Partial Differ. Equ. , 16 : 105 – 143 .
  • Deng , K . 1992 . Comparison principle for some nonlocal problems . Quart. Appl. Math. , 50 : 517 – 522 .
  • Liu , QL , Chen , YP and Xie , CH . 2003 . Blow-up for a degenerate parabolic equation with a nonlocal source . J. Math. Anal. Appl. , 285 : 487 – 505 .
  • Chan , CY and Liu , HT . 1998 . Global existence of solutions for degenerate semilinear parabolic problems . Nonlinear Anal. , 34 : 617 – 628 .
  • Chen , YP , Liu , QL and Xie , CH . 2003 . Blow-up for degenerate parabolic equation with nonlocal source . Proc. Amer. Math. Soc. , 132 : 135 – 145 .
  • Du , LL . 2007 . Blow-up for a degenerate reaction-diffusion system with a nonlinear nonlocal source . J. Comput. Appl. Math. , 202 : 237 – 247 .
  • Floater , MS . 1991 . Blow-up at the boundary for degenerate semilinear parabolic equations . Arch. Ration. Mech. Anal. , 114 : 57 – 77 .
  • Li , FC and Xie , CH . 2003 . Global existence and blow-up for a nonlinear porous medium equation . Appl. Math. Lett. , 16 : 185 – 192 .
  • Li , J , Cui , ZJ and Mu , CL . 2008 . Global existence and blow-up for degenerate and singular parabolic system with localized sources . Appl. Math. Comput. , 199 : 292 – 300 .
  • Ockendon , H . 1979 . Channel flow with temperature-dependent viscosity and internal viscous dissipation . J. Fluid Mech. , 93 : 737 – 746 .
  • J. Zhou, C. L. Mu and Z. P. Li, Blow-up for degenerate and singular parabolic system with nonlocal source, Bound. Value Probl. (2006), Article ID 21830, pp. 1–19.
  • Carlson , DE . 1972 . Linear Thermoelasticity, Encyclopedia of Physics , Berlin : Springer .
  • Pao , CV . 1995 . Dynamics of reaction-diffusion equations with nonlocal boundary conditions . Quart. Appl. Math. , 53 : 173 – 186 .
  • Seo , S . 1996 . Blowup of solutions to heat equations with nonlocal boundary conditions . Kobe J. Math. , 13 : 123 – 132 .
  • Day , WA . 1983 . A decreasing property of solutions of parabolic equations with applications to thermoelasticity . Quart. Appl. Math. , 40 : 468 – 475 .
  • Day , WA . 1985 . Heat Conduction within Linear Thermoelasticity , New York : Springer-Verlag .
  • Friedman , A . 1986 . Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions . Quart. Appl. Math. , 44 : 401 – 407 .
  • Lin , ZG and Liu , YR . 2004 . Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary . Acta Math. Sci. Ser. B Engl. Ed. , 24 : 443 – 450 .
  • Cui , ZJ and Yang , ZD . 2008 . Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition . J. Math. Anal. Appl. , 342 : 559 – 570 .
  • Liu , DM and Mu , CL . 2010 . Blow-up analysis for a semilinear parabolic equation with nonlinear memory and nonlocal nonlinear boundary condition . Electron. J. Qual. Theory Differ. Equ. , 51 : 1 – 17 .
  • Liu , DM and Mu , CL . 2010 . Blowup properties for a semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions . Abstr. Appl. Anal. , : 17 Article ID 148035
  • Mu , CL , Liu , DM and Zhou , SM . 2010 . Properties of positive solutions for a nonlocal reaction diffusion equation with nonlocal nonlinear boundary condition . J. Korean Math. Soc. , 47 : 1317 – 1328 .
  • Pao , CV . 1998 . Asymptotic behaviour of solutions of reaction-diffusion equations with nonlocal boundary conditions . J. Comput. Appl. Math. , 88 : 225 – 238 .
  • Seo , S . 2000 . Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions . Pac. J. Math. , 193 : 219 – 226 .
  • Wang , YL , Mu , CL and Xiang , ZY . 2007 . Blow up of solutions to a porous medium equation with nonlocal boundary condition . Appl. Math. Comput. , 192 : 579 – 585 .
  • Wang , YL , Mu , CL and Xiang , ZY . 2007 . Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary . Bound. Value Probl. , : 12 Article ID 64579
  • Souplet , P . 1999 . Uniform blow-up profiles and boundary behaviour for diffusion equations with nonlocal nonlinear source . J. Differ. Equ. , 153 : 374 – 406 .
  • Li , ZP and Mu , CL . 2007 . Blow-up for doubly degenerate parabolic system with nonlocal sources . Dyn. Syst. , 22 : 485 – 509 .
  • Wang , MX and Wang , YM . 1996 . Properties of positive solutions for nonlocal reaction-diffusion problems . Math. Methods Appl. Sci. , 19 : 1141 – 1156 .
  • Zheng , SN and Su , H . 2006 . A quasilinear reaction-diffusion system coupled via nonlocal sources . Appl. Math. Comput. , 180 : 295 – 308 .
  • Friedman , A and Mcleod , B . 1985 . Blowup of positive solutions of semilinear heat equations . Indiana Univ. Math. J. , 34 : 425 – 447 .
  • Mclachlan , NW . 1955 . Bessel Functions For Engineers, , 2nd ed , Oxford, London : Clarendon Press .

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