Finite time blow-up for the fractional critical heat equation in Rn. (April 2020) - Chen, Guoyuan; Wei, Juncheng; Zhou, Yifu
Abstract: We consider the fractional critical heat equation u t + ( − Δ ) s u = u n + 2 s n − 2 s, in R n × ( 0, T ), u ( x, 0 ) = u 0 ( x ), in R n, where 4 s < n < 6 s, 0 < s < 1 . For sufficiently small T > 0 and given k distinct points q 1, …, q k, we show that there exists an initial condition u 0 such that the solution u ( x, t ) blows up at these k distinct points as t → T . More precisely, the blow-up profile around each concentration point takes the form of sharply scaled bubble and ‖ u ( ⋅, t ) ‖ L ∞ ( R n ) ∼ ( T − t ) − n − 2 s 6 s − n as t → T .