A parabolic trace type functional-differential equation

The authors consider as a paradigm for parabolic trace type functional differential equations the problem ut = a(q→(t)) uxx, -∞ < x < ∞, 0 < t ≤ T, u(x, 0) = f(hook)(x) -∞ < x < ∞, where q →(t) = (u(x0, t), ∂u ∂x(x1, t), …, ∂nu ∂xn(xn, t), …). Under various conditions on a(·) and f, existence and unicity of the classical solution are demonstrated.