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  • Cited by 2
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Abdulla, Ugur G. 2002. Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption. Nonlinear Analysis: Theory, Methods & Applications, Vol. 50, Issue. 4, p. 541.


    Abdulla, Ugur G. and King, John R. 2000. Interface Development and Local Solutions to Reaction-Diffusion Equations. SIAM Journal on Mathematical Analysis, Vol. 32, Issue. 2, p. 235.


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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 123, Issue 5
  • January 1993, pp. 803-817

On the initial growth of interfaces in reaction–diffusion equations with strong absorption

  • Luis Alvarez (a1) and Jesus Ildefonso Diaz (a2)
  • DOI: http://dx.doi.org/10.1017/S0308210500029504
  • Published online: 14 November 2011
Abstract

We study the initial growth of the interfaces of non-negative local solutions of the equation ut = (um)xx−λuq when m ≧ 1 and 0<q <1. We show that if with C < C0, for some explicit C0 = C0(λ, m, q), then the free boundary Ϛ(t) = sup {x:u(x, t) > 0} is a ‘heating front’. More precisely Ϛ(t) ≧at(m−q)/2(1−q) for any t small enough and for some a>0. If on the contrary, with C<C0, then Ϛ(t) is a ‘cooling front’ and in fact Ϛ(t) ≧ −atm−q)/2(1−q) for any t small enough and for some a > 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.

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1L. Alvarez . On the behavior of the free boundary of some nonhomogeneous elliptic problems. Applicable Anal. 36 (1990), 131144.

3L. Alvarez and J. I. Díaz . Sufficient and necessary initial mass conditions for the existence of a waiting time in nonlinear-convection processes. J. Math. Anal. Appl. 155 (1991), 378392.

4L. Alvarez , J. I. Díaz and R. Kersner . On the initial growth of the interfaces in nonlinear diffusion-convection processes. In Nonlinear Diffusion Equations and Their Equilibrium States, eds W.-M. Ni , L. A. Peletier and J. Serrin , pp. 120 (Berlin: Springer, 1988).

6C. Bandle and I. Stakgold . The formation of the dead core in parabolic reaction-diffusion problems. Trans. Amer. Math. Soc. 286 (1984), 275293.

7Ph. Benilan , M. G. Crandall and M. Pierre . Solutions of the porous medium equation in RN under optimal conditions on initial value. Indiana Univ. Math. J. 33 (1984), 5187.

9M. Bertsch . A class of degenerate diffusion equations with a singular nonlinear term. Nonlinear Anal. 7(1983), 117–127.

10J. Crank and R. S. Gupta . A moving boundary problem arising from the diffusion of oxygen in absorbing tissue. J. Inst. Math. Appl. 10 (1972), 1933.

16A. Friedman and M. A. Herrero . Extinction properties of semilinear heat equations with strong absorption. J. Math. Anal. Appl. 124 (1987), 530546.

17R. E. Grundy . Asymptotic solutions of a model diffusion-reaction equation. IMA J. Appl. Math. 40 (1988), 5372.

20S. Gutman and R. H. Martin Jr The porous medium equation with nonlinear absorption and moving boundaries. Israel J. Math. 54 (1986), 81109.

21M. A. Herrero and J. L. Vazquez . The one-dimensional nonlinear heat equation with absorption: Regularity of solutions and interfaces. SIAM J. Math. Anal. 18 (1987), 149167.

22M. A. Herrero and J. L. Vazquez . Thermal waves in absorbing media. J. Differential Equations 74 (1988), 218233.

23M. A. Herrero and J. J. L. Velazquez . On the dynamics of a semilinear heat equation with strong absorption. Comm. Partial Differential Equations 14 (1989), 16531715.

24A. S. Kalashnikov . The propagation of disturbances in problems of non-linear heat conduction with absorption. USSR. Comput. Math, and Math. Phys. 14 (1974), 7085.

25A. S. Kalashnikov . The effect of absorption on heat propagation in a medium in which the thermal conductivity depends on temperature. U.S.S.R. Comput. Math, and Math. Phys. 16 (1976), 141149.

26S. Kamin , L. A. Peletier and J. L. Vazquez . A nonlinear diffusion-absorption equation with unbounded data. In Nonlinear diffusion equations and their equilibrium states. III, eds N. G. Lloyd et al., pp. 243263 (Boston: Birkhauser, 1992).

28R. Kersner .Degenerate parabolic equations with general nonlinearities. Nonlinear Anal. 4 (1984), 10431062.

29B. F. Knerr . The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension. Trans. Amer. Math. Soc. 249 (1979), 409424.

30M. Langlais and D. Phillips . Stabilization of solutions of nonlinear and degenerate evolution problems. Nonlinear Anal. 9 (1985), 321333.

32P. Rosenau and S. Kamin . Thermal waves in an absorbing and convecting medium. Phys. D 8 (1983), 273283.

33J. L. Vazquez . The interfaces of one-dimensional flows in porous media. Trans. Amer. Math. Soc. 285 (1984), 717737.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
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  • EISSN: 1473-7124
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