In this paper, we address the decay of solutions to the four-dimensional energy-critical nonlinear heat equation in the critical space ${\dot{H}}^1$. Recently, it was proven that the ${\dot{H}}^1$ norm of solutions goes to zero when time goes to infinity, but no decay rates were established. By means of the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions.

中文翻译：

---

4D 能量关键非线性热方程的衰减率

在本文中，我们解决了临界空间中四维能量临界非线性热方程解的衰减问题${\dot{H}}^1$。近日，事实证明，${\dot{H}}^1$当时间趋于无穷大时，解的范数为零，但没有建立衰减率。通过傅里叶分裂方法并利用尺度不变性产生的性质，我们获得了解衰减率的代数上限。