We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ \begin{ equation*} \begin{ cases} \pp_t u=\Delta u+|u|^{ \frac{ 4}{ 3}} u ~&\mbox{ in }~ R^5 \times (t_0,\infty), u(\cdot,t_0)=u_0~&\mbox{ in }~ R^5. \end{ cases} \end{ equation*} For $t_0$ sufficiently large, we show the existence of positive solutions for a class of initial value $u_0(x)\sim |x|^{ -\gamma}$ as $|x|\rightarrow \infty$ with $\gamma>\frac32$ such that the global solutions behave asymptotically \begin{ equation*} \| u(\cdot,t) \|_{ L^\infty (\R^5)} \sim \begin{ cases} t^{ -\frac{ 3(2-\gamma)}{ 2}} ~&\mbox{ if }~ \frac32<\gamma<2 (\ln t)^{ -3} ~&\mbox{ if }~ \gamma=2 1 ~&\mbox{ if }~ \gamma>2 \end{ cases} \mbox{ \ for \ } t >t_0, \end{ equation*} which is slower than the self-similar time decay $t^{ -\frac{ 3}{ 4}}$. These rates are inspired by Fila-King \cite[Conjecture 1.1]{ FilaKing12}.
Categories: math.AP