Singular limit of solutions of the very fast diffusion equation

Abstract

We prove that the distribution solutions of the very fast diffusion equation ∂ u / ∂ t = Δ ( 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 u 0 ⁄ ∈ L 1 ( R n ) , 0 ≤ u 0 ∈ L loc p ( R n ) for some constant p > n / 2 , and u 0 ( x ) ≥ ε / | x | 2 α for any | x | ≥ R 1 where ε > 0 , R 1 > 0 , m 0 < 0 , α < min ( 1 / ( 1 − m 0 ) , 1 / | m 0 | ) , are constants satisfying p > ( 1 − m 0 ) n / 2 , 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 v t = Δ log v , v ( x , 0 ) = u 0 ( x ) , as m ↗ 0 − . For any smooth bounded domain Ω ⊂ R n , m 0 < 0 , m ∈ [ m 0 , 0 ) ∪ ( 0 , 1 ) , 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 = Δ ( 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 .