AbstractWe consider the Dirichlet problem for the energy-critical heat equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+u^5 &{} \text {in} \quad \Omega \times \mathbb {R}^+,\\ u=0 &{}\text {on} \quad \partial \Omega \times \mathbb {R}^+,\\ u(x,0)=u_0(x) &{} \text {in} \quad \Omega , \end{array}\right. } \end{aligned}$$ u t = Δ u + u 5 in Ω × R + , u = 0 on ∂ Ω × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^3$$ R 3 . Let $$H_\gamma (x,y)$$ H γ ( x , y ) be the regular part of the Green function of $$-\Delta -\gamma $$ - Δ - γ in $$\Omega $$ Ω , where $$\gamma \in (0,\lambda _1)$$ γ ∈ ( 0 , λ 1 ) and $$\lambda _1$$ λ 1 is the first Dirichlet eigenvalue of $$-\Delta $$ - Δ . Then, given a point $$q\in \Omega $$ q ∈ Ω such that $$3\gamma (q)<\lambda _1$$ 3 γ ( q ) < λ 1 , where $$\begin{aligned} \gamma (q) :=\sup \{ \gamma>0: H_\gamma (q,q)>0 \}, \end{aligned}$$ γ ( q ) : = sup { γ > 0 : H γ ( 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 $$\begin{aligned} u(x,t)\sim 3^{\frac{1}{4}} \left( \frac{\mu (t)}{\mu (t)^2+\vert x-\xi (t)\vert ^2}\right) ^{\frac{1}{2}} \quad \text {as}\quad t \rightarrow \infty , \end{aligned}$$ u ( x , t ) ∼ 3 1 4 μ ( t ) μ ( t ) 2 + | x - ξ ( t ) | 2 1 2 as t → ∞ , where $$\begin{aligned} -\ln (\mu (t))= 2\gamma (q) t(1+o(1)),\quad \xi (t)=q+O(\mu (t)) \quad \text {as}\quad t \rightarrow \infty . \end{aligned}$$ - ln ( μ ( t ) ) = 2 γ ( q ) t ( 1 + o ( 1 ) ) , ξ ( t ) = q + O ( μ ( t ) ) as t → ∞ .