Large time behaviour of solutions of a singular diffusion equation in [formula omitted

Abstract

Let A > 0 , n ⩾ 2 > q , n / 2 < p < n / q , u 0 = u ¯ 0 + φ ⩾ 0 , where 0 ⩽ u ¯ 0 ∈ L loc p ( R n ) and φ ∈ L 1 ( R n ) ∩ L p ( R n ) satisfy A ( 1 - 1 / k ) | x | - q ⩽ u ¯ 0 ( x ) ⩽ A ( 1 + 1 / k ) | x | - q ∀ | x | ⩾ R 0 , k and A ( 1 - 1 / k ) | x | - q + φ ( x ) ⩾ 0 ∀ R 0 , k ⩽ | x | < R 0 , k + 1 , k = 2 , 3 , … for some constants 0 < R 0 , k < R 0 , k + 1 ∀ k = 2 , 3 , … . Suppose u is the maximal solution of u t = Δ log u , u > 0 , in R n × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) in R n and ψ is the unique self-similar solution of the equation with initial value A | x | - q . Then the rescaled function v ( x , t ) = t q / ( 2 - q ) u ( t 1 / ( 2 - q ) x , t ) will converge uniformly on every compact subset of R n to ψ ( x , 1 ) as t → ∞ .