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

  • Luis Alvarez (a1) and Jesus Ildefonso Diaz (a2)

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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4 L. 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).

26 S. 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).

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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