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Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in N . (English) Zbl 0805.35065

From the introduction: We consider the Cauchy problem

t β(u)=Δu+f(u)in(x,t) N ×(0,T),u(x,0)=u 0 (x)inx N ,(1)

where t =/t, Δ is the N- dimensional Laplacian and β(v), f(v) with v0 and u 0 (x) are nonnegative functions. Equation (1) describes the combustion process in a stationary medium, in which the thermal conductivity β ' (u) -1 and the volume heat source f(u) are depending in a nonlinear way on the temperature β(u)=β(u(x,t)) of the medium. The main purpose of the present paper is the study of blow-up solutions near the blow-up time. Especially, we are interested in the shape of the blow-up set which locates the “hot-spots” at the blow-up time. In addition, since our quasilinear equation (1) has a property of finite propagation, there are some interesting subjects such as the regularity of the interface and its asymptotic behavior near the blow-up time. These problems have been studied by one of the authors [R. Suzuki, Publ. Res. Inst. Math. Sci. 27, No. 3, 375-398 (1991; Zbl 0789.35024)], in the case N=1. This paper extends some of his results to higher dimensional problems.

35K65Parabolic equations of degenerate type
35B40Asymptotic behavior of solutions of PDE
80A25Combustion, interior ballistics