An inverse problem of finding a parameter in a semi-linear heat equation

We consider the following inverse problem of finding the pair (u, p) which satisfies the following: u1 = uxx + p(t)u + F(x, t, u, ux, p(t), 0 < x < 1, 0 < t ≤ T; u(x,0) = u0(x), 0 < x < 1, ux(0, t) = f(t), ux(1, t) = g(t), 0 < t ≤ T; and ∝01ψ(x,t) u(x, t)dx = E(t), 0 < t ≤ T; where u0, f, g, F, ψ, and E are known functions. The existence, uniqueness, regularity, and the continuous dependence of the solution upon the data are demonstrated.