Heat equation Dirichlet Boundary Conditions Let $\Omega \subset R^n$ be a bounded domain with piecewise 
$C^1$ boundary. Suppose that
$u(t, x)$ is real-valued and satisfies the heat equation:
$$\frac{∂u}{∂t} − Δu = 0 $$ 
on $(0,∞) \times \Omega$.
Define
$$\eta(t) :=∫_Ωu^2(t, x) d^n x $$
(a) Assume that $u$ satisfies the Dirichlet boundary conditions:
$$u(t, x)|_{x∈∂Ω} = 0$$ 
for $t ≥ 0$. Show that $\eta$ decreases as a function of $t$.
(b) Use (a) to show that a solution $u$ satisfying boundary and initial conditions
$$u|_{t=0} = g $$ $$ u|_{x∈∂Ω} = h$$
for some continuous functions $g$ on $\Omega$ and $h$ on $\partial \Omega$, is uniquely determined by $g$ and $h$.
Need help answering this. I understand for (a) to show that $\eta$ is decreasing we need to show the derivative is strictly less than zero. Unsure how to prove this I have seen people use Gauss's theorem to show. 
(b) I am unsure how to show it is unique. Also have had the hint to let $w = u-v$, where $u$ and $v$ are solutions to the heat equation. Then to show that $w$ is always zero, thus $u = v$ but I am not sure where to begin.
 A: With
$u_t = \dfrac{\partial u}{\partial t} = \nabla^2 u = \nabla \cdot \nabla u \tag 1$
and
$\eta(t) = \displaystyle \int_\Omega u^2(x, t) \; d^nx, \tag 2$
we have
$\eta_t(t) = \dfrac{d\eta(t)}{dt} = 2\displaystyle \int_\Omega u(x, t) u_t(x, t) \; d^n x; \tag 3$
consider the integral on the right of (3); by virtue of (1),
$\displaystyle \int_\Omega u(x, t) u_t(x, t) \; d^n x = \int_\Omega u(x, t) \nabla \cdot \nabla u(x, t) \; d^n x; \tag 4$
now,
$\nabla \cdot (u \nabla u) = \nabla u \cdot \nabla u + u \nabla \cdot \nabla u, \tag 5$
from which
$u \nabla \cdot \nabla u  = \nabla \cdot (u\nabla u) - \nabla u \cdot \nabla u, \tag 6$
and hence
$\displaystyle\int_\Omega u\nabla \cdot \nabla u \; d^n x = \int _\Omega  \nabla \cdot (u\nabla u) \;  d^nx - \int_\Omega \nabla u \cdot \nabla u,     \tag 7$
where we temporarily suppressed the $x, t$ arguments of $u$ for the sake of notational convenience; now the first integral on the right may be written
$ \displaystyle \int _\Omega  \nabla \cdot (u\nabla u) \;  d^nx = \int_{\partial \Omega} u \nabla u \cdot \mathbf n \; dS,  \tag 8$
where $\mathbf n$ is the outward-pointing unit normal field to, and $dS$ is the volume element on, $\partial \Omega$;  we are given that
$u(x, t) \vert_{x \in \partial \Omega} = 0, \tag 9$
whence
$\displaystyle  \int_{\partial \Omega} u(x, t) \nabla u(x, t) \cdot \mathbf n \; dS = 0; \tag{10}$
continuing in this direction we apply (8)-(10) to (7) and thus find
$\displaystyle \int_\Omega  u\nabla \cdot \nabla u \; d^n x =  - \int_\Omega \nabla u \cdot \nabla u;     \tag{11}$
working our way back to (3):
$\eta_t(t) = \dfrac{d\eta(t)}{dt} = -2\displaystyle \int_\Omega \nabla u(x, t) \cdot \nabla u(x, t)\; d^n x < 0; \tag{12}$
that is, $\eta(t)$ decreases with increasing $t$, provided $u(x, t)$ is not constant on $\Omega$.  Since $u(x, t)$ obeys the Dirichlet boundary condition (9), it is constant on $\Omega$ precisely when it vanishes there; thus we conclude $\eta(t)$ will decrease as long as $u(x, t) \ne 0$ on $\Omega$.  
We have, I believe, adequately addressed part (a); as for (b), 
we assume with our OP Total-fury the existence of distinct solutions $u$ and $v$ which share the same boundary and initial conditions:
$u \mid_{t = 0} = v \mid_{t = 0} = g, \tag{13}$
$u \mid_{x \in \Omega} = v \mid_{x \in \Omega} = h; \tag{14}$
we set
$w = u - v; \tag{15}$
then in accord with (13)-(15),
$w \mid_{t = 0} = 0, \tag{16}$
$w \mid_{x \in \Omega} = 0.  \tag{17}$
Now $w$ satisfies the same boundary condition (9) as was stipulated for $u(x, t)$; thus the results of part (a) apply to $w$ and allow us to write
$\eta_t = \dfrac{d\eta}{dt} = \dfrac{d}{dt} \displaystyle \int_{\Omega} w^2(x, t) \; d^n x \le 0, \tag{18}$
with equality holding precisely when $w \equiv 0$ on $\Omega$; it follows then that $w(x, t)$ cannot take any 
non-zero values for $t \ge 0$, since in such event
$\eta(t) = \displaystyle \int_\Omega w^2(x, t) \; d^nx > 0; \tag{19}$
but this is precluded by virtue of (18).  We thus affirm that
$w(x, t) = 0, \tag{20}$
and hence
$u(x, t) = v(x, t); \tag{21}$
the solutions of (1) with the given boundary/initial conditions are thus unique.
