I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.


Do you know where I can find a complete proof of such a result :
There exists a unique solution $u$ to the problem 


$\left\{
\begin{array}{rcll}
\partial\_t u(t,x) - \eta \Delta u(t,x) &=& f(t,x)&\qquad\text{on } ]0,T]\times \mathbb{R}^n \\
u(0,x) &=& u\_0(x)&\qquad\text{on } \mathbb{R}^n
\end{array}
\right.$


with $f\in L^2(0,T;L^2(\mathbb{R}^n))$, $u\_0 \in H^1(\mathbb{R}^n)$ and $\eta>0$ fixed and furthermore $u\in L^2(0,T;H^2(\mathbb{R}^n))$ and $\partial\_t u \in L^2(0,T;L^2(\mathbb{R}^n))$.


Thank you



Heat Equation on $[0,T] \times \mathbb{R}^n$
Fourier transform in $x$ gets you there:
$
\dot v+\eta\vert\xi\vert^2 v=g(t,\xi),\quad v(0)=v\_0,
$
so that
$$
v(t,\xi)=e^{-t\eta \vert\xi\vert^2} v\_0(\xi)+\int\_0^te^{-(t-s)\eta \vert\xi\vert^2} g(s,\xi) ds.
$$
Since $\iint\_0^T\vert g(t,\xi)\vert^2 dtd\xi<+\infty$ and 
$\int\vert v\_0(\xi)\vert^2 d\xi<+\infty$,
you get
$$
\iint\_0^Te^{-2t\eta \vert\xi\vert^2} \vert \xi\vert^2\vert v\_0(\xi)\vert^2 dt d\xi\le \int\frac{[e^{-2t\eta \vert \xi\vert^2}]^{t=0}\_{t=T}}{2\eta}\vert v\_0(\xi)\vert^2 d\xi\le
(2\eta)^{-1}\int\vert v\_0(\xi)\vert^2 d\xi<+\infty,
$$
and similarly with $H=\mathbf 1\_{\mathbb R\_+}$
$$
\vert \xi\vert^2\int H(t-s)e^{-(t-s)\eta \vert \xi\vert^2} g(s)H(s) ds= \vert \xi\vert^2 H(t)e^{-t\eta \vert \xi\vert^2} \ast g(t)H(t)
$$
with $L^2$ norm (in $t$) bounded above by
$$
\vert \xi\vert^2 (\int\_0^T\vert g(t,\xi)\vert^2 dt)^{1/2}\int\_0^{+\infty}e^{-t\eta \vert \xi\vert^2} dt
=\eta^{-1}(\int\_0^T\vert g(t,\xi)\vert^2 dt)^{1/2}
$$
and the square of the $L^2$ norm (in $t,\xi$) is bounded above by
$$
\eta^{-2}\iint\_0^T\vert g(t,\xi)\vert^2 dtd\xi<+\infty.
$$ 
The last statement on $\dot v$
comes from the equation and the previous results.


