Kin Ming Hui

Abstract: We prove that the distribution solutions of the very fast diffusion equation ∂u/∂t=Δ(um/m)∂u/∂t=Δ(um/m), u>0u>0, in Rn×(0,∞)Rn×(0,∞), u(x,0)=u0(x)u(x,0)=u0(x) in RnRn, where m n/2p>n/2, and u0(x)≥e/|x|2αu0(x)≥e/|x|2α for any |x|≥R1|x|≥R1 where e>0e>0, R1>0R1>0, m0 (1−m0)n/2p>(1−m0)n/2, we prove that the solution of the above problem will converge uniformly on every compact subset of Rn×(0,∞)Rn×(0,∞) to the maximal solution of the equation vt=Δlogvvt=Δlogv, v(x,0)=u0(x)v(x,0)=u0(x), as m↗0−m↗0−. For any smooth bounded domain Ω⊂RnΩ⊂Rn, m0 (1−m0)max(1,n/2)p>(1−m0)max(1,n/2), we prove the existence and uniqueness of solutions of the Dirichlet problem ∂u/∂t=Δ(um/m)∂u/∂t=Δ(um/m), u>0u>0, in Ω×(0,∞)Ω×(0,∞), u=u0u=u0 in ΩΩ, u=gu=g on ∂Ω×(0,∞)∂Ω×(0,∞) with either finite or infinite positive boundary value gg. We also prove a similar convergence result for the solutions of the above Dirichlet problem as m→0m→0.

TL;DR: In this article, the authors proved the existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain D × (0, ∞) where D is a bounded smooth domain in R n with zero lateral values.

TL;DR: In this article, the authors studied the asymptotic large time behavior of singular solutions of the fast diffusion equation in the subcritical case and proved the existence of the singular solution that is trapped in between self-similar solutions.