  We study the asymptotics of solutions of logistic type equations with
fractional Laplacian as time goes to infinity and as the exponent in nonlinear
part goes to infinity. We prove strong convergence of solutions in the energy
space and uniform convergence to the solution of an obstacle problem. As a
by-product, we also prove the cut-off property for eigenvalues of the Dirichlet
fractional Laplace operator perturbed by exploding potentials.
