I'm a little lost and need a little help figuring out where to start.

ut = uxx, 0 < x < 1, t > 0

u(x,0) = f(x)

ux(1,t) = u(1,t)

ux(0,t) = 0

The fact that u(x,0) is equal to f(x) is throwing me off and I'm not sure How I go about solving for lambda = 0, lambda > 0 and lambda < 0.

Thanks!