25 research outputs found
On the limit of solutions of ut=Îum as mââ
Let fâL1(RN), Nâ„1, fâ„0, and consider the Cauchy problem ut=Îum on ]0,â[ĂRN, u(0,â
)=f on RN. The authors prove that as mââ, the corresponding solutions um(t)âu_=f+Îw in L1(RN), uniformly for t in a compact set in ]0,â[, where 0â€w_âL1(Rn) is the solution of the variational inequality Îw_âL1(RN), 0â€f+Îw_â€1, w_(f+Îw_ â1)=0 a.e. The authors also show similar results for the same equation on a bounded open set Ω in RN with Dirichlet or Neumann boundary condition
Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions
The nonnegative viscosity solutions to the infinite heat equation with
homogeneous Dirichlet boundary conditions are shown to converge as time
increases to infinity to a uniquely determined limit after a suitable time
rescaling. The proof relies on the half-relaxed limits technique as well as
interior positivity estimates and boundary estimates. The expansion of the
support is also studied