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Nonlocal parabolic equation with conserved spatial integral

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Abstract

In this short note, we investigate the behavior of the solution for a scalar nonlocal semi-linear parabolic equation, in which the nonlocal term acts to conserve the spatial integral of the solution as time evolves. For the solution blowing up in finite time, the blow-up rate is estimated. For the global solution, the global convergence is studied.

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References

  1. C. Budd, B. Dold, and A. Stuart, Blowup in a partial differential equation with conserved first integral. SIAM J. Appl. Math. 53 (1993), 718–742.

  2. A. El Soufi, M. Jazar, and R. Monneau, A gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 17–39.

  3. Freitas P.: Stability of stationary solutions for a scalar non-local reaction-diffusion equation. Quart. J. Mech. Appl. Math. 48, 557–582 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fila M., Levine H.A.: On the boundedness of global solutions of abstract semilinear parabolic equations.J. Math. Anal. Appl. 216, 654–666 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Freitas and M. P. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space. Differential Integral Equations 13 (2000), 265–288.

  6. Gao W.J., Han Y.Z.: Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl. Math. Lett. 24, 784–788 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Hu and H.M. Yin, Semilinear parabolic equations with prescribed energy. Rend. Circ. Mat. Palermo (2) 44 (1995), 479–505.

  8. O. A. Ladyženskaja, V. A Solonnikov and N. N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967.

  9. G.M. Lieberman, Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

  10. Jazar M., Kiwan R.: Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. H. Poincar Anal. Non Linaire 25, 215–218 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162.

  12. P. Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 2007.

  13. J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48 (1992), 249–264.

  14. Wang M., Wang Y.: Properties of positive solutions for non-local reaction-diffusion problems, Math. Methods Appl. Sci. 19, 1141–1156 (1996)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Xiao-Liu Wang, Fang-Zheng Tian & Gen Li

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  1. Xiao-Liu Wang
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  2. Fang-Zheng Tian
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  3. Gen Li
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Correspondence to Xiao-Liu Wang.

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Wang, XL., Tian, FZ. & Li, G. Nonlocal parabolic equation with conserved spatial integral. Arch. Math. 105, 93–100 (2015). https://doi.org/10.1007/s00013-015-0782-1

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  • DOI: https://doi.org/10.1007/s00013-015-0782-1

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