论文标题
在不均匀的热方程和平方势势上
On inhomogeneous heat equation with inverse square potential
论文作者
论文摘要
我们研究具有反平势的不均匀热方程，即，\ [\ partial_tu + \ \ \ \ \ \ \ \ \ \ \ a} _a u = \ pm | \ cdot | \ cdot | \ cdot |^{ - b} | u |^au，\ \ y $ \ mathcal {l} l} _a = - _a = -guirition soldimient $ e^{ - t \ Mathcal {l} _a} $与Lebesgue Spaces中的不均匀非固定性$ | \ cdot | \ cdot |^{ - b} $关联。然后，我们在$ l^q- $缩放关键和超临界制度以及小数据全球范围内的小数据中发展本地理论。我们通过使用自相似解决方案进一步研究了全球解决方案的渐近行为，只要初始数据满足某些界限。我们的证明方法灵感来自Slimene-Tayachi-Weissler（2017）的工作，他们考虑了经典案例，即$ a = 0 $。
We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^αu,\] where $\mathcal{L}_a=-Δ+ a |x|^{-2}.$ We establish some fixed-time decay estimate for $e^{-t\mathcal{L}_a}$ associated with inhomogeneous nonlinearity $|\cdot|^{-b}$ in Lebesgue spaces. We then develop local theory in $L^q-$ scaling critical and super-critical regime and small data global well-posedness in critical Lebegue spaces. We further study asymptotic behaviour of global solutions by using self-similar solutions, provided the initial data satisfies certain bounds. Our method of proof is inspired from the work of Slimene-Tayachi-Weissler (2017) where they considered the classical case, i.e. $a=0$.
