Sign-changing blowing-up solutions for thecritical nonlinear heat equation

Abstract

Let Omega be a smooth bounded domain in R-n and denote the regular part of the Green function on Omega with Dirichlet boundary condition by H(x, y). Assume the integer k0 is sufficiently large, q is an element of Omega and n >= 5. For k >= k(0) we prove that there exist initial data u0 and smooth parameter functions xi(t) -> q and 0 < mu(t) -> 0 for t ->+infinity such that the solution uq of the critical nonlinear heat equation

{u(t) = Delta u + vertical bar u vertical bar(4/n-2)u in Omega x (0, infinity) u = 0 on partial derivative Omega x (0, infinity) u(., 0) = u(0) in Omega

has the form

u(q)(x, t) approximate to mu(t)(-n-2/2) (Q(k)(x-xi(t)/mu(t))-H(x, q)),

where the profile Q(k) is the non-radial sign-changing solution of the Yamabe equation

Delta Q + vertical bar Q vertical bar(4/n-2) Q=0 in R-n,

constructed in [9]. In dimension 5 and 6 we also investigate the stability of u(q) (x, t).