A Priori Estimates and Continuous Dependence of Solutions of Mixed Problems for Parabolic Equations As Nonlocal Boundary Conditions Pass into Local Ones

摘要

In the rectangle G = (0, 1) × (0, T ), we consider the family of problems 1 a(x, t) auα at -a2uα ax2 = f(x, t), uα(x, 0) = φα(x), uα(0, t) = 0, 0 ≤ α ≤ 1, u0(1, t) = h(t), au1(1, t) ax = h(t), uα(1, t) - uα(α, t) 1 - α = h(t), 0 < α < 1, a1 ≥ a(x, t) ≥ a0 > 0, h∈ W1 2 (0, T ), φα ∈ W1 2 (0, T ), φα(0) = 0, 0 ≤ α ≤ 1, φ0(1) = h(0), φ 1(1) = h(0),φα(1) - φα (0) 1 - α = h(0), 0 < α < 1, f ∈ L2(G). It is well known that, for α = 0 and α = 1, the corresponding problems with local conditions are solvable, and the solutions are unique and belong to W2,1 2 (G).