Discontinuous Parabolic Problems with a Nonlocal Initial Condition

Let Ω be a an open bounded domain in R , N ≥ 2, with a smooth boundary ∂Ω. Let QT Ω × 0, T and ΓT ∂Ω × 0, T where T is a positive real number. Then ΓT is smooth and any point on ΓT satisfies the inside and outside strong sphere property see 1 . For u : QT → R we denote its partial derivatives in the distributional sense when they exist byDtu ∂u/∂t, Diu ∂u/∂xi,DiDju ∂u/∂xi∂xj , i, j 1, . . . ,N. In this paper, we study the following parabolic differential equation with a nonlocal initial condition