Fractional parabolic problems have attracted much attention in recent years, which are more intricate than the classical parabolic equations. For the semi-linear equation, Sugitani [1] proved non-existence below the Fujita exponent. In the case of above the Fujita exponent, global existence remains open since all the known techniques fail in this case. As far as blow-up is concerned, a theory in the spirit of the one developed by Giga and Kohn [2], is missing. A crucial step in these approaches is to exhibit a monotone quantity. For the fractional heat operator we are considering, such a quantity is missing.
In this paper, we consider positive solutions for the fractional heat equation with critical Sobolev exponent on a bounded domain, and prove the existence of initial datum such that the solution blows up precisely at prescribed multiple distinct points. The main ingredient of the proofs is a new inner-outer gluing scheme for the fractional parabolic problems.
[1] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12:45-51, 1975.
[2] Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semi-linear heat equations, Comm. Pure Appl. Math., 38:297-319, 1985.
Math. Ann.
https://doi.org/10.1007/s00208-018-1784-7
