2 research outputs found
Convergence in relative error for the porous medium equation in a tube
Given a bounded domain D â RN and m > 1, we study the long-time behaviour of solutions
to the porous medium equation (PME) posed in a tube
âtu = um in D Ă R, t > 0,
with homogeneous Dirichlet boundary conditions on the boundary â DĂR and suitable initial
datum at t = 0. In two previous works, VaÌzquez and Gilding & Goncerzewicz proved that a
wide class of solutions exhibit a traveling wave behaviour, when computed at a logarithmic
time-scale and suitably renormalized. In this paper, we show that, for large times, solutions
converge in relative error to the Friendly Giant, i.e., the unique nonnegative solution to the
PME posed in the section D of the tube (with homogeneous Dirichlet boundary conditions)
having a special self-similar form. In addition,sharp rates of convergence and uniform bounds
for the location of the free boundary of solutions are given