  We study a semilinear PDE generalizing the Fujita equation whose evolution
operator is the sum of a fractional power of the Laplacian and a convex
non-linearity. Using the Feynman-Kac representation we prove criteria for
asymptotic extinction versus finite time blow up of positive solutions based on
comparison with global solutions. For a critical power non-linearity we obtain
a two-parameter family of radially symmetric stationary solutions.
  By extending the method of moving planes to fractional powers of the
Laplacian we prove that all positive steady states of the corresponding
equation in a finite ball are radially symmetric.
