Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains

We consider the Dirichlet problem for the energy-critical heat equation {u(t )= Delta u + u(5 )in Omega x R+, u = 0 on partial derivative Omega x R+, u(x, 0) = u(0)(x) in Omega, where Omega is a bounded smooth domain in R-3. Let H-gamma(x, y) be the regular part of the Green function of - Delta - gamma in Omega, where gamma is an element of (0, lambda(1)) and lambda(1 )is the first Dirichleteigen value of -Delta. Then, given a point q is an element of Omega such that 3 gamma(q) < lambda(1), where gamma(q) := sup{gamma > 0 : H-gamma(q, q) > 0}, we prove the existence of a non-radial global positive and smooth solution u(x, t) which blows up in infinite time with spike in q. The solution has the asymptotic profile u(x, t) similar to 3(1/4 )(mu(t)/mu(t)(2 )+ |x - xi(t)|(2))(1/2 )as t -> infinity, (0.1) where -ln(mu(t)) = 2 gamma(q)t(1 + o(1)), xi(t) = q + O(mu(t)) as t -> infinity.