The main aim of the present paper is to investigate the conditions under which the nonnegative solutions blow-up for the parabolic problem ∂ u ∂ t = − − Δ α 2 u + c x α u in ℝ d × 0 T , $$ \frac{\partial u}{\partial t}=-{\left(-\Delta \right)}^{\frac{\alpha }{2}}u+\frac{c}{{\left|x\right|}^{\alpha }}u\kern1em \mathrm{in}\kern1em {\mathrm{\mathbb{R}}}^{\mathrm{d}}\times \left(0,T\right), $$ where 0 < α < min(2, d ), − Δ α 2 $$ {\left(-\Delta \right)}^{\frac{\alpha }{2}} $$ is the fractional Laplacian on ℝ d and the initial condition u 0 is in L 2 (ℝ d ).

中文翻译：

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整个空间中分数拉普拉斯算子的抛物线方程：非负解的爆破

本文的主要目的是研究抛物线问题 ∂ u ∂ t = − − Δ α 2 u + cx α u in ℝ d × 0 T , $$ \frac {\partial u}{\partial t}=-{\left(-\Delta \right)}^{\frac{\alpha }{2}}u+\frac{c}{{\left|x\right| }^{\alpha }}u\kern1em \mathrm{in}\kern1em {\mathrm{\mathbb{R}}}^{\mathrm{d}}\times \left(0,T\right), $$其中 0 < α < min(2, d ), − Δ α 2 $$ {\left(-\Delta \right)}^{\frac{\alpha }{2}} $$ 是 ℝ d 上的分数拉普拉斯算子并且初始条件 u 0 在 L 2 (ℝ d ) 中。