Long-time dynamics for the energy critical heat equation in R5

Abstract

We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ $\left(\begin{array}{cc}{\partial }_{t}u=\Delta u+{\left|u\right|}^{\frac{4}{3}}u& \text{in}{R}^5\times (t_0,\infty ),\\ u(\cdot ,t_0)=u_0& \text{in}{R}^5.\end{array}\right)$For $t_0$ sufficiently large, we show the existence of positive solutions for a class of initial value $u_0\left(x\right)\sim {\left|x\right|}^{-\gamma }$ as $\left|x\right|\to \infty$ with $\gamma >\frac{3}{2}$ such that the global solutions behave asymptotically ${\Vert u(\cdot ,t)\Vert }_{L^{\infty }\left(R^5\right)}\sim \left(\begin{array}{cc}{t}^{-\frac{3(2-\gamma )}{2}}& \text{if}\frac{3}{2}<\gamma <2\\ {(lnt)}^{-3}& \text{if}\gamma =2\\ 1& \text{if}\gamma >2\end{array}\right)\text{for}t>t_0,$which is slower than the self-similar time decay $t^{-\frac{3}{4}}$. These rates are inspired by Fila and King (2012, Conjecture 1.1).