A note on the Fujita exponent in fractional heat equation involving the Hardy potential

Abstract

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the fractional Cauchy problem with the Hardy potential, namely, $ {u_t} + {( - \Delta )^s}u = \lambda \frac{u}{{|x{|^{2s}}}} + {u^p}{\rm{ }}\;{\rm{in}}\;{I\!\!R}^N,u(x,0) = {u_0}(x)\;{\rm{in}}\;{I\!\!R}^N, $ where \lt i \gt N \lt /i \gt \gt 2 \lt i \gt s \lt /i \gt , 0 \lt \lt i \gt s \lt /i \gt \lt 1, (-∆) \lt sup \gt \lt i \gt s \lt /i \gt \lt /sup \gt is the fractional laplacian of order 2 \lt i \gt s \lt /i \gt , \lt i \gt λ \lt /i \gt \gt 0, \lt i \gt u \lt /i \gt \lt sub \gt 0 \lt /sub \gt ≥ 0, and 1 \lt \lt i \gt p \lt /i \gt \lt \lt i \gt p \lt /i \gt \lt sub \gt + \lt /sub \gt ( \lt i \gt s \lt /i \gt , \lt i \gt λ \lt /i \gt ), where \lt i \gt p \lt /i \gt \lt sub \gt + \lt /sub \gt ( \lt i \gt λ \lt /i \gt , \lt i \gt s \lt /i \gt ) is the critical existence power to be given subsequently.