We consider the Cauchy problem for the porous medium equation with strong absorption ut= (um)xx? upforx ? R, t & 0, with continuous compactly supported initial data u(x, 0) = u0(x) ? 0in the critical case m + p = 2of the range of parameters m & 1, p < 1. We study the regularity of solutions and interfaces and compare the results with the purely diffusive case ut= (um)xx. Important differences are found in the interface behaviour and equations, in the occurrence of turning points and inflection points of the interfaces, and in the fact that bounded solutions extinguish in finite time. All these phenomena are examined and described. The critical case studied here allows for a comparatively simpler and richer analysis of the qualitative behaviour of non-linear parabolic equations involving finite propagation (m & 1) and strong absorption (p < 1).
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Victor A. Galaktionov, Sergey Shmarev, Juan Luis Vázquez, Behaviour of interfaces in a diffusion-absorption equation with critical exponents. Interfaces Free Bound. 2 (2000), no. 4, pp. 425–448DOI 10.4171/IFB/28