On the growth of mass for a viscous Hamilton-Jacobi equation

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equation u(t) = Deltau + \delu\(p), t > 0, x is an element of R-N, where p greater than or equal to 1 and u(0,.) = u(0) greater than or equal to 0, u(0) not equivalent to 0, u(0) is an element of L-1. Denoting I-infinity = lim(t-->infinity) parallel tou(t)parallel to(1) less than or equal to infinity, we show that the asymptotic behavior of the mass can be classified along three cases as follows: if p less than or equal to (N + 2)/(N + 1), then I-infinity = infinity for all u(0); if (N + 2)/(N + 1) < p < 2, then both I-infinity = infinity and I-infinity < infinity occur; if p greater than or equal to 2, then I-infinity < infinity for all u(0). We also consider a similar question for the equation u(t) = Deltau + u(p).