Singular limit of solutions of the very fast diffusion equation

✍ Scribed by Kin Ming Hui

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 555 KB
• Volume: 68
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2006.12.009

✦ Synopsis

We prove that the distribution solutions of the very fast diffusion equation ∂u/∂t = ∆(u m /m), u > 0, in R n × (0, ∞), u(x, 0) = u 0 (x) in R n , where m < 0, n ≥ 2, constructed in [P. Daskalopoulos, M.A. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Ration. Mech. Anal. 137 (1997) 363-380] are actually classical maximal solutions of the problem. Under the additional assumption that

we prove that the solution of the above problem will converge uniformly on every compact subset of R n × (0, ∞) to the maximal solution of the equation

), and 0 ≤ u 0 ∈ L p (Ω ) for some constant p > (1 -m 0 ) max(1, n/2), we prove the existence and uniqueness of solutions of the Dirichlet problem ∂u/∂t = ∆(u m /m), u > 0, in Ω × (0, ∞), u = u 0 in Ω , u = g on ∂Ω × (0, ∞) with either finite or infinite positive boundary value g. We also prove a similar convergence result for the solutions of the above Dirichlet problem as m → 0.