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

抄録

We consider the initial-boundary value problem
(P) {$¥frac{∂}{∂t}$u = Δu-V(|x|)u   in Ω_L×(0,∞),
μu+(1-μ)$¥frac{∂}{∂ν}$u = 0   on ∂Ω_L×(0,∞),
u(·,0) = φ(·)∈L^p(Ω_L),   p≥1,
where Ω_L={x∈R^N:|x|>L}, N≥2, L>0, 0≤μ≤1, &\ u; is the outer unit normal vector to ∂Ω_L, and V is a nonnegative smooth function such that V(r)=O(r^-2) as r→∞. In this paper, we study the decay rates of the derivatives ∇_x^ju of the solution u to (P) as t→∞.