Abstract

For parabolic initial boundary value problems various results such as lim_{t ↓ 0}{(∂u/t6x)(0, t)/(∂u_α/∂x)(0, t)} = 1, where u satisfies ∂u/∂t = a(u)(∂²u/∂x²), 0 < x < 1, 0 < t T, u(x, 0) = 0, u(0, t) = |₁(t), 0 < t T, u(1, t) = |₂(t), 0 < t T, u_αsatisfies (∂u_α/∂t) = α(∂²u_α/∂x²), 0 < x < 1, 0 < t T, u_α(x, 0) = 0, u_α(0, t) = |₁(t), 0 < t T, u_α(1, t) = |₂(t), 0 < t T, and α = a(0), are demonstrated via the maximum principle and potential theoretic estimates.

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*1 This research was supported in part by the National Science Foundation.