Asymptotic Estimates of Solutions ofut−12Δu=−|∇u| in R+×Rd,d⩾2

We express the solution,uμ, ofut−12Δu=−|∇u| in R+×Rd,d⩾2;uμ(0, ·)=μthrough a functional of ad-dimensional Bessel process, whereμis invariant by rotations and profiled. This allows us to estimate the asymptotic behaviour of ‖uμ(t, ·)‖1and ‖uμ(t, ·)‖∞astgoes to infinity.