Behaviour of solutions of a singular diffusion equation near the extinction time

✍ Scribed by Shu-Yu Hsu

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 521 KB
• Volume: 56
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2003.07.018

✦ Synopsis

We prove that if ¿ 2 and 0 6 u0 ∈ L 1 (R 2 ) ∩ L p (R 2 ) for some constant p ¿ 1 is a radially symmetric function, u0 ≡ 0, and u is the unique solution of the equation ut

and rur(x; t)=u(x; t) → -uniformly on [a; b] as r = |x| → ∞ for any 0 ¡ a ¡ b ¡ T where T = R 2 u0 d x=2 , then there exist unique constants ¿ 0; ÿ ¿ -1=2; = 2ÿ + 1, such that the rescaled function v(y; s) = u(y=(T -t) ÿ ; t)=(T -t) with s = -log(T -t) will converge uniformly on every compact subset of R 2 to the solution ; ÿ (|y|) of the ODE (r = ) =r + + ÿr = 0 in [0; ∞] with r (0) = 0; (0) = 1= for some constant ¿ 0 as s → ∞.