SOME ASYMPTOTIC BOUNDARY BEHAVIOR OF SOLUTIONS OF NONLINEAR PARABOLIC INITIAL-BOUNDARY VALUE-PROBLEMS

For parabolic initial boundary value problems various results such as limt ↓ 0{ ( ∂u t6x)(0, t) ( ∂uα ∂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, uαsatisfies ( ∂uα ∂t) = α( ∂2uα ∂x2), 0 < x < 1, 0 < t ≤ T, uα(x, 0) = 0, uα(0, t) = |1(t), 0 < t ≤ T, uα(1, t) = |2(t), 0 < t ≤ T, and α = a(0), are demonstrated via the maximum principle and potential theoretic estimates. © 1979.