This paper deals with the following initial value problem: (1) ?u/?t??u+u p=0 on RN, N?1, 0 0 and ??0. For any x?RN, the extinction time of x is defined as follows: TE(x)=sup{t>0,u(x,t)>0}, u(x,t) being the solution of (1). The following results are established. (i) If u0(x)?A(|x|)+B|x?a|2/(1?p) for some a?RN where A(|x|)=o(|x| 2/(1?p)) as |x|?+? and
0?BTE(y). (ii) It is assumed that N=1. Let u(x,t) be the solution of (1). For a convenient behaviour of u0(x) as |x|?+?, then u(x,t) tends to a limit as t?+?, uniformly in compact sets on R. (iii) Let u0(r) be a nonnegative radial function satisfying (2). Then the solution of (1) with initial value u0(r) is radially symmetric and for a convenient behaviour of u0(r) as r?+?, u(r,t) tends to a limit as r?+?, uniformly on compact sets in R. For N?2, the authors restrict themselves to the radial case.
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