Abstract: We considered  the Cauchy problem of a class of quasilinear parabolic equations with strong absorption terms. Due to the effect of the strong absorption term, the solution to the problem could  possess compact support and extinguish  at a finite time. Firstly, by using  the comparison principle and constructing suitable supersolutions, it was proven that the solution possessed a uniform compact support after a certain time and even after any positive time. Secondly, under some conditions, it was proven that the solution extinguished at a finite time by using the L¹ norm estimates of the solution to the problem at different times.

Key words: quasilinear parabolic equation, strong absorption term, compact support, extinction