I'm having trouble of proving the uniqueness of the initial boundary value problem
$$u_t-ku_{xx}\ =0, \ 0<x<l, \ 0<t<T$$ $$u(x,0)=f(x), \ 0<x<l$$ $$u_x(0,t)=0, \ u(l,t)=h(t),\ 0<t<T$$
This is what I have so far:
Let $w(x, t)=u_1(x,t)-u_2(x,t)$, where $u_1(x,t) $ and $u_2(x,t)$ are two solutions of the problem, and hence so is $w(x,t)=0$, so the goal is to prove $w(x,t)=0$
Let energy integral $E(t)=\int_0^lw^2 \ dx$
Then $d/dt(E)=\int_0^l2ww_t\ dx$
Integral by parts we have $E'(t)=2k(w(l,t)w_x(l,t)-w(0,t)w_x(0,t))-2k\int_0^lw_x^2 \ dx$
Then $E'(t)=2kh(t)w_x(l,t)-2k\int_0^lw_x^2 \ dx$
Since $ u(l,t)=h(t)$
Then $u_x(l,t)=h_x(t)=0$, so $w_x(l,t)=0$
Thus $E'(t)=-2k\int_0^lw_x^2 \ dx$
Not so sure how to continue here. Supposedly $E'(t)<0$ suggests that $E(t)$ is non increasing, and then $E(0)=f(x)$
How do we conclude $E=\int_0^lw^2 \ dx =0$?
Please help, thanks!