Resumen:
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The authors consider the initial-boundary value problem for the porous medium equation ut =(um)xx in (0,?)×(0,T), where m>1, 00}as t?T under the hypothesis that ?(t)?? as t?T is investigated. The effect of localization of the blowing-up boundary function when lim?sup t?T ?(t) is investigated. It is established that localization occurs if and only if lim?sup t?T (? t 0 ? m (s)ds)/?(t), and some estimates concerning the asymptotic behaviour of the solution near the singular point t=T and in the blow-up set ?={x?0: lim?sup t?T u(x,t)=?} are given. Various estimates from above and below on the length ?=sup? of the blow-up set are obtained. These theorems make more precise some previous results concerning the localization of the boundary blowing-up function which were given in the book by A. A. Samarski?, the reviewer et al. [Peaking modes in problems for quasilinear parabolic equations(Russian), "Nauka'', Moscow, 1987].
Proofs of the theorems are based on comparison with some explicit solutions and on construction of different kinds of weak sub- and supersolutions. The authors use some special integral identities and estimates of the solution and its derivatives by means of the maximum principle. A special comparison theorem above blow-up sets for different boundary functions is proved.
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