S0294-1449(02)00107-5/FLA UNIQUENESS AND STABILITY OF REGIONAL BLOW-UP IN A POROUS-MEDIUM EQUATION

ABSTRACT. – We study the blow-up phenomenon for the porous-medium equation in RN, N 1, ut =um + um, m> 1, for nonnegative, compactly supported initial data. A solution u(x, t) to this problem blows-up at a finite time T ̄> 0. Our main result asserts that there is a finite number of points x1,..., xk ∈RN, with |xi − xj | 2R ∗ for i = j, such that lim t→T̄ (T ̄ − t) 1m−1 u(t, x)= k∑ j=1 w∗(|x − xj |). Here w∗(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation wm + wm − 1 m−1w = 0 in RN and R ∗ is the radius of its support. Moreover u(x, t) remains uniformly bounded up to its blow-up time on compact subsets of RN \ ⋃kj=1 B̄(xj,R∗). The question becomes reduced to that of proving that the ω-limit set in the problem vt = vm + vm − 1m−1v consists of a single point when its initial condition is nonnegative and compactly supported