Uniqueness of variational heat equation

Question

$u_{t} - u_{xx} + cu = f $ on $U_{T}$

$u = g$ on $\Gamma_{T}$

Problem is Show uniqueness of solution. I tried to find it with energy method, but the method doesn't working since $\frac{\partial }{\partial t}e(t) = 2\int_{U}ww_{t} + w^{2} dx = -\int_{U}|Dw|^{2}dx+e(t)$ Could you give me a hint?

Edited; Proof : Just let $v(x,t) = e^{ct}u(x,t)$ then I can use energy method since $v(x,t)$ solves non homogeneous heat equation, which I can deal with!

Answer 1

Your method works fine. Here's another idea:

Suppose that $u_1$ and $u_2$ are two solutions to the variational heat equation as given. It follows that $v := u_1-u_2$ satisfies

$v_{t} - v_{xx} + cv = 0 $ on $U_{T}$

$v = 0$ on $\Gamma_{T}$

(Why is this the case?). It follows that $v(x,t) = 0$ (why is this the case)? That is, $u_1(x,t) - u_2(x,t) = 0$, which is to say that $u_1$ and $u_2$ are identical solutions. We conclude that the solution to this IVP is unique.