From the introduction: We consider the Cauchy problem
where , is the - dimensional Laplacian and , with and are nonnegative functions. Equation (1) describes the combustion process in a stationary medium, in which the thermal conductivity and the volume heat source are depending in a nonlinear way on the temperature 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 . This paper extends some of his results to higher dimensional problems.