Global solutions of the fast diffusion equation with gradient absorption terms

AbstractWe investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−|∇u|p in Rn×(0,∞) with (1−2/n)+<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u0∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u0(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 L1-norm of solutions for 1<p⩽(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)‖L∞ for (2+mn)/(n+1)<p