论文标题
带有内存的热方程：大型行为
A heat equation with memory: large-time behavior
论文作者
论文摘要
我们研究了所有$ l^p $规范中的大型行为，以及在$ \ mathbb {r}^n $中提出的带有caputo $α$α$α$ time派生的热量方程的不同时空尺度。假定初始数据是可以集成的，并且在需要时也可以在$ l^p $中。分析的主要困难来自于〜$ n> 1 $的基本解决方案的原点的空间奇点。 $ l^p $规范在特征刻度中的衰减率，$ | x | \ asymp t^{α/2} $，由方程式的缩放不变性为$ t^{ - \ frac {αn} {2} {2}（1- \ frac1p）} $。在紧凑的集合中，对于$ n = 1 $，$ n = 1 $，$ t^{ - α} $的$ t^{ - α/2} $，对于$ n \ ge 3 $，和$ t^{ - t^{ - α} \ log t $在关键尺寸$ n = 2 $中。在中间尺度上，进入无穷大，但比$ t^{α/2} $更慢，我们的中间衰减率。在快速尺度上，无限速度比$ t^{α/2} $快，没有通用速率，对所有解决方案有效，因为我们将通过某些示例显示。无论如何，在这种尺度上，解决方案的衰减速度比特征性的衰减速度快。 当除以衰减率时，解决方案在特征量表中的大规模行为，例如$ m $ $倍的基本解决方案，其中$ m $是初始基准的组成部分。如果$ n \ ge 3 $，本文的主要新颖性之一，如果$ n = 1,2 $，则情况在紧凑型集合的情况大不相同，它们会收敛到初始基准的牛顿潜力。在中间尺度上，如果$ n \ ge 3 $，则接近拉普拉斯式的基本解决方案的倍数，并且在低维度中的常数。比特征更快地进入无穷大的尺度上的渐近行为在很大程度上取决于无穷大的初始基准的行为。我们为特定衰减提供了某些初始数据的结果。
We study the large-time behavior in all $L^p$ norms and in different space-time scales of solutions to a heat equation with a Caputo $α$-time derivative posed in $\mathbb{R}^N$. The initial data are assumed to be integrable, and, when required, to be also in $L^p$. A main difficulty in the analysis comes from the singularity in space at the origin of the fundamental solution of the equation when~$N>1$. The rate of decay in $L^p$ norm in the characteristic scale, $|x|\asymp t^{α/2}$, dictated by the scaling invariance of the equation, is $t^{-\frac{αN}{2}(1-\frac1p)}$. In compact sets it is $t^{-α/2}$ for $N=1$, $t^{-α}$ for $N\ge 3$, and $t^{-α}\log t$ in the critical dimension $N=2$. In intermediate scales, going to infinity but more slowly than $t^{α/2}$, we have an intermediate decay rate. In fast scales, going to infinity faster than $t^{α/2}$, there is no universal rate, valid for all solutions, as we will show by means of some examples. Anyway, in such scales solutions decay faster than in the characteristic one. When divided by the decay rate, solutions behave for large times in the characteristic scale like $M$ times the fundamental solution, where $M$ is the integral of the initial datum. The situation is very different in compact sets, where they converge to the Newtonian potential of the initial datum if $N\ge 3$, one of the main novelties of the paper, and to a constant if $N=1,2$. In intermediate scales they approach a multiple of the fundamental solution of the Laplacian if $N\ge 3$, and a constant in low dimensions. The asymptotic behavior in scales that go to infinity faster than the characteristic one depends strongly on the behavior of the initial datum at infinity. We give results for certain initial data with specific decays.
