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Properties of Positive Solution for Nonlocal Reaction-Diffusion Equation with Nonlocal Boundary

Abstract

This paper considers the properties of positive solutions for a nonlocal equation with nonlocal boundary condition on. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the uniform blow-up estimates for the blow-up solution.

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References

  1. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, NY, USA; 1992:xvi+777.

    MATH  Google Scholar 

  2. Souplet P: Blow-up in nonlocal reaction-diffusion equations. SIAM Journal on Mathematical Analysis 1998,29(6):1301–1334. 10.1137/S0036141097318900

    Article  MATH  MathSciNet  Google Scholar 

  3. Souplet P: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Journal of Differential Equations 1999,153(2):374–406. 10.1006/jdeq.1998.3535

    Article  MATH  MathSciNet  Google Scholar 

  4. Wang M, Wang Y: Properties of positive solutions for non-local reaction-diffusion problems. Mathematical Methods in the Applied Sciences 1996,19(14):1141–1156. 10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9

    Article  MATH  MathSciNet  Google Scholar 

  5. Souplet P: Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping. Mathematical Methods in the Applied Sciences 2004,27(15):1819–1829. 10.1002/mma.567

    Article  MATH  MathSciNet  Google Scholar 

  6. Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983,40(4):468–475.

    MATH  Google Scholar 

  7. Day WA: Heat Conduction within Linear Thermoelasticity, Springer Tracts in Natural Philosophy. Volume 30. Springer, New York, NY, USA; 1985:viii+83.

    Book  Google Scholar 

  8. Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1986,44(3):401–407.

    MATH  MathSciNet  Google Scholar 

  9. Deng K: Comparison principle for some nonlocal problems. Quarterly of Applied Mathematics 1992,50(3):517–522.

    MATH  MathSciNet  Google Scholar 

  10. Seo S: Blowup of solutions to heat equations with nonlocal boundary conditions. Kobe Journal of Mathematics 1996,13(2):123–132.

    MATH  MathSciNet  Google Scholar 

  11. Seo S: Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions. Pacific Journal of Mathematics 2000,193(1):219–226. 10.2140/pjm.2000.193.219

    Article  MATH  MathSciNet  Google Scholar 

  12. Lin Z, Liu Y: Uniform blowup profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Mathematica Scientia. Series B 2004,24(3):443–450.

    MATH  MathSciNet  Google Scholar 

  13. Carlson DE: Linear thermoelasticity. In Encyclopedia of Physics. Volume vIa/2. Springer, Berlin, Germany; 1972:319.

    Google Scholar 

  14. Pao CV: Dynamics of reaction-diffusion equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1995,53(1):173–186.

    MATH  MathSciNet  Google Scholar 

  15. Pao CV: Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Journal of Computational and Applied Mathematics 1998,88(1):225–238. 10.1016/S0377-0427(97)00215-X

    Article  MATH  MathSciNet  Google Scholar 

  16. Pao CV: Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. Journal of Computational and Applied Mathematics 2001,136(1–2):227–243. 10.1016/S0377-0427(00)00614-2

    Article  MATH  MathSciNet  Google Scholar 

  17. Wang Y, Mu C, Xiang Z: Blowup of solutions to a porous medium equation with nonlocal boundary condition. to appear in Applied Mathematics and Computation

  18. Xiang Z, Hu X, Mu C: Neumann problem for reaction-diffusion systems with nonlocal nonlinear sources. Nonlinear Analysis 2005,61(7):1209–1224. 10.1016/j.na.2005.01.098

    Article  MATH  MathSciNet  Google Scholar 

  19. Xiang Z, Chen Q, Mu C: Blowup properties for several diffusion systems with localised sources. The ANZIAM Journal 2006,48(1):37–56. 10.1017/S1446181100003400

    Article  MATH  MathSciNet  Google Scholar 

  20. Yin H-M: On a class of parabolic equations with nonlocal boundary conditions. Journal of Mathematical Analysis and Applications 2004,294(2):712–728. 10.1016/j.jmaa.2004.03.021

    Article  MATH  MathSciNet  Google Scholar 

  21. Zhou J, Mu C, Li Z: Blowup for degenerate and singular parabolic system with nonlocal source. Boundary Value Problems 2006, 2006: 19 pages.

    Article  MathSciNet  Google Scholar 

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Correspondence to Yulan Wang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Wang, Y., Mu, C. & Xiang, Z. Properties of Positive Solution for Nonlocal Reaction-Diffusion Equation with Nonlocal Boundary. Bound Value Probl 2007, 064579 (2007). https://doi.org/10.1155/2007/64579

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