Hardy inequalities and dynamic instability of singular Yamabe metrics

We study the Cauchy problem(P) {(ut = Δ u + | u |σ - 1 u, in  RN × (0, + ∞),; u (x, 0) = u0 (x), in  RN,) for nonnegative functions u : RN × (0, + ∞) → R+. Here N ≥ 3, σ + 1 = 2* = frac(2 N, N - 2) is the Sobolev exponent of the embedding H1 (RN) {right arrow, hooked} L2* (RN) and u0 = U is a time independent positive solution with nonempty singular set Σ = Sing (U), e.g. a distributional solution associated to a singular Yamabe metric on SN. We show that, if Σ is a finite set, then problem (P) has a weak solution which is smooth for positive time. Hence, time independent singular solutions may be unstable and the Cauchy problem (P) may have infinitely many weak solutions. A similar weaker result is proved for any nonnegative distributional solution U when Σ is a compact set. © 2005 Elsevier SAS. All rights reserved.