Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

We consider the initial-boundary value problem (P) {partial derivative/partial derivative t mu = Delta u -V(vertical bar x vertical bar) u in Omega(L) x (0, infinity), mu u + (1 -mu) partial derivative/partial derivative nu u = 0 on partial derivative Omega(L) x (0, infinity), u(.,0) = phi(.) epsilon L-P(Omega(L))(,) p >= 1, where Omega(L) = {x epsilon R-N : vertical bar x vertical bar > L}, N >= 2, L > 0, 0 <= mu <= 1, v is the outer unit normal vector to Omega partial derivative(L), and V is a nonnegative smooth function such that V(r) = O(r(-2)) as r -> infinity. In this paper, we study the decay rates of the derivatives V(x)(j)u of the solution u to (P) as t -> infinity.