Abstract

We study the blow-up phenomenon for the porous-medium equation in R N, 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?T?(T? - t) 1/m-1u(t, x) = ?j = 1kw*(|x - xj|). Here w*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation ?wm + wm - 1/m-1 w = 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\?j = 1kB?(xj, R*). The question becomes reduced to that of proving that the ?-limit set in the problem v t = ?vm + vm - 1/m-1 v consists of a single point when its initial condition is nonnegative and compactly supported. © 2002 Éditions scientifiques et médicales Elsevier SAS.