Global solutions of the fast diffusion equation with gradient absorption terms

摘要

We investigate the existence of nonnegative weak solutions to the problem u(t) = Delta(u(m)) - vertical bar del u vertical bar(P) in R-n x (0, infinity) with (2 - 2)(+) < m < 1. It will be proved that: (i) When 1 < p < 2, if the initial datum u(0) is an element of D(R-n) then there exists a solution; (ii) When 1 < p < (2 + mn)/(n + 1), if the initial datum u(0)(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2 + mn)/(n + 1) <= p < 2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L-1-norm of solutions for 1 < p <= (2 + mn)/(n + 1), and the large time behavior of t(1/beta)parallel to u((.), t) - E-c((.), t)parallel to(L)infinity for (2 + mn)/(n + 1) < p < 2. (c)@ 2006 Elsevier Inc. All rights reserved.