Tokyo Journal of Mathematics

Uniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Source

Jiashan ZHENG

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.

Article information

Source
Tokyo J. Math., Volume 39, Number 1 (2016), 199-214.

Dates
First available in Project Euclid: 30 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1459367265

Digital Object Identifier
doi:10.3836/tjm/1459367265

Mathematical Reviews number (MathSciNet)
MR3543139

Zentralblatt MATH identifier
1359.35100

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B33: Critical exponents 35K10: Second-order parabolic equations

Citation

ZHENG, Jiashan. Uniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Source. Tokyo J. Math. 39 (2016), no. 1, 199--214. doi:10.3836/tjm/1459367265. https://projecteuclid.org/euclid.tjm/1459367265


Export citation

References

  • A. Z. Akcasu and E. Daniels, Fluctuation analysis in simple fluids, Phys. Rev. A 2 (1970), 962.
  • W. E. Alley and B. J. Alder, Generalized transport coefficients for hard spheres, Phys. Rev. A 27 (1983), 3158.
  • F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Non-Local Diffusion Problems, Mathematical Surveys and Monographs, vol. 165, 2010.
  • J. W. Barrett and W. B. Liu, Finite element approximation of the parabolic $p$-Laplacian, SIAM J. Numer. Anal. 31 (1994), 413–428.
  • P. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher dimensions, J. Statistical Phys. 95 (1999), 1119–1139.
  • P. Bates, P. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal. 138 (1997), 105–136.
  • J. Boon and S. Yip, Molecular Hydrodynamics, McGraw-Hill, New York, 1980.
  • P. J. Cadusch, B. D. Todd, J. Zhang and P. J. Daivis, A non-local hydrodynamic model for the shear viscosity of confined fluids: analysis of a homogeneous kernel, J. Phys. A 41(3) (2008), 035501. 23 pp.
  • C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol. 50(2) (2005), 161–188.
  • E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl. 86 (2006), 271–291.
  • X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Diff. Eqns. 2 (1997), 125–160.
  • J. Coville, J. Dáila and S. Martinez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Eqns. 244 (2008), 3080–3118.
  • R. Ferreira and M. Pérez-Llanos, Blow-up for the non-local $p$-Laplacian equation with a reaction term, Nonlinear Anal. TMA. 75 (2012), 5499–5522.
  • P. Fife, Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions, Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003.
  • J. García-Mellán and F. Quirós, Fujita exponents for evolution problems with nonlocal diffusion, J. Evol. Eqns. 10 (2010), 147–161.
  • J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Diff. Eqns. 246 (2009), 21–38.
  • L. Hopf, Introduction to Differential Equations of Physics, Dover, New York, 8 (1948), 55–100.
  • Q. Liu, Y. Li and H. Gao, Uniform blow-up rate for a nonlocal degenerate parabolic equations, Nonlinear Anal. TMA. 66(4) (2007), 881–889.
  • Q. Liu, Y. Li and H. Gao, Uniform blow-up rate for diffusion equations with nonlocal nonlinear source, Nonlinear Anal. TMA. 67(6) (2007), 1947–1957.
  • Q. Liu, Y. Li and H. Gao, Uniform blow-up rate for diffusion equations with localized nonlinear source, J. Math. Anal. Appl. 320(2) (2006), 771–778.
  • M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal. TMA. 70 (2009), 1629–1640.
  • P. Morse and H. Feshback, Methods of Theoretical Physics, McGraw Hill, New York, 1 (1953).
  • J. D. Murray, Mathematical Biology, Springer, New York, 1993.
  • S. Pan, W. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction–diffusion systems and applications, Z. Angew. Math. Phys. 60 (2009), 377–392.
  • C. Qu, X. Bai and S. Zheng, Blow-up versus extinction in a nonlocal $p$-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl. 412(1) (2014), 326–333.
  • P. Souplet, Uniform blow-up profle and boundary behaviour for a non-local reaction–diffusion equation with critical damping, Math. Meth. Appl. Sci. 27 (2004), 1819–1829.
  • P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Diff. Eqns. 153 (1999), 374–406.
  • P. Souplet, Blow-up in nonlocal reaction–diffusion equations, SIAM J. Math. Anal. 29 (1998), 1301–1334.
  • M. Wang and Y. Wang, Properties of positive solutions for non-local reaction–diffusion problems, Math. Method. Appl. Sci. 19 (1996), 1141–1156.
  • C. Wang, S. Zheng and Z. Wang, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity 20(6) (2007), 1343–1359.