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Multidimensional singular diffusion

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Abstract

We consider the nonlinear diffusion equation u t = ▽(u −n▽u) for dimension N ⩾ 2 in the cases n>1, N = 2 and n ⩾ 1, N ⩾ 3 in which there are no finite mass solutions. We concentrate on the physically motivated case of a small but non-zero background concentration, using asymptotic methods to analyse the limit in which this background concentration goes to zero.

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References

  1. J.L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type. Preprint.

  2. J.R. King, ‘Instantaneous source’ solutions to a singular nonlinear diffusion equation, J. Eng. Math. 27 (1993) 31–72.

    Google Scholar 

  3. J.G. Berryman and C.J. Holland, Asymptotic behaviour of the nonlinear diffusion equation n t = (n −1 n x ) x . J. Math. Phys. 23 (1982) 983–987.

    Google Scholar 

  4. J.R. King, Integral results for nonlinear diffusion equations. J. Eng. Math. 25 (1991) 191–205.

    Google Scholar 

  5. J.R. King, Asymptotic results for nonlinear outdiffusion, submitted.

  6. M. Bertsch and M. Ughi, Positivity properties of viscosity solutions of a degenerate parabolic equation. Nonlinear Analysis 14 (1990) 571–592.

    Google Scholar 

  7. P. Rosenau, P.S. Hagan, R.L. Northcutt and D.S. Cohen, Delayed diffusion due to flux limitation. Phys. Lett. A 142 (1989) 26–30.

    Google Scholar 

  8. M. Bertsch and R. Dal Passo, Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch. Rat. Mech. Anal. 117 (1992) 349–387.

    Google Scholar 

  9. P.S. Hagan and D. Brenner, Sorption limits with increasing diffusion coefficients. SIAM J. Appl. Math. 48 (1988) 917–920.

    Google Scholar 

  10. C.M. Elliott, M.A. Herrero, J.R. King and J.R. Ockendon, The mesa problem: diffusion patterns for u t = ▽ · (u m▽u) as m → + ∞. IMA J. Appl. Math. 37 (1986) 147–154.

    Google Scholar 

  11. J.R. Philip, Theory of infiltration. Adv. Hydrosci. 5 (1969) 215–296.

    Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Mechanics, University of Nottingham, NG7 2RD, Nottingham, U.K.

    J. R. King

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  1. J. R. King
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King, J.R. Multidimensional singular diffusion. J Eng Math 27, 357–387 (1993). https://doi.org/10.1007/BF00128761

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  • DOI: https://doi.org/10.1007/BF00128761

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