Large-time geometrical properties of solutions of the Barenblatt equation of elasto-plastic filtration

We study the geometric behavior for large times of the solutions of the following equationut+γ|ut|=Δu,0<|γ|<1, posed in the whole space RNRN, for N⩾1N⩾1 when the initial data are nonnegative, continuous and compactly supported. We prove that, after a finite time, log(u)log(u) becomes a concave function in the space variable and converges to all orders of differentiability to a certain parabolic shape, so-called Barenblatt-type profile, which was described in Kamin et al. (1991) [20]. Extensions to more general fully nonlinear equations are considered.