Nonlocal Conditions for Lower Semicontinuous Parabolic Inclusions

Let Ω be an open bounded domain in N , N ≥ 2, with a smooth boundary ∂Ω. We denote the norm usually the Euclidean norm of x ∈ Ω by ‖x‖. Let T be a positive real number. Set QT Ω × 0, T and ΓT ∂Ω × 0, T . For u : D → we denote its partial derivatives when they exist by ut ∂u/∂t, uxi ∂u/∂xi, uxixj ∂ u/∂xi∂xj , i, j 1, . . . ,N. Let X C QT denote the Banach space of continuous functions u : QT → , endowed with the norm