Localization and blow-up of thermal waves in nonlinear heat-conduction with peaking

Abstract

The authors consider the initial-boundary value problem for the porous medium equation ut =(um)xx in (0,∞)×(0,T), where m>1, 0<T<∞, with initial and boundary conditions u(x,0)= u0(x)≥0 in (0,∞), sup u0<∞, u0 has compact support, u(0,t)=ψ(t) for t  (0,T), u0 and ψ are given nonnegative continuous functions and ψ(t)is monotonic increasing. The behaviour of the solution u(x,t) and the free boundary ζ(t)=sup{x[0,∞) : u(x,t)>0}as t↑T under the hypothesis that ψ(t)↑∞ as t↑T is investigated. The effect of localization of the blowing-up boundary function when lim sup t↑T ζ(t)<∞ is investigated. It is established that localization occurs if and only if lim sup t↑T (∫ t 0 ψ m (s)ds)/ψ(t)<∞, and some estimates concerning the asymptotic behaviour of the solution near the singular point t=T and in the blow-up set Ω={x≥0: lim sup t↑T u(x,t)=∞} are given. Various estimates from above and below on the length ω=supΩ 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. Samarskiĭ, 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.