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Cauchy problem for the heat equation with Dirichlet boundary condition
Problem
Let $U(t,x):[0,T]\times \Omega \rightarrow \mathbb{R}$, solve the following Cauchy problem for the heat equation:
\begin{cases}
\begin{array}{rl}
\frac{\partial U}{\partial t}(t,x)=\alpha \frac{\partial^2 U}{\partial x^2} (t,x), x \in \Omega, t \in [0,T]\\
U(0,x)=0, x \in \Omega \\
U(t,\omega)=1, t \in [0,T], \omega=\partial \Omega\\
\end{array}
\end{cases}
where $\Omega\subset \mathbb{R}^d$ is a bounded domain with a smooth boundary $\partial \Omega$
Comments
There exists a unique solution $U(t,x)$ for this Cauchy problem.
However, I do not know how to prove this.
My attempt
By direct differentiation:
\begin{align*}
\frac{\partial U}{\partial t}(t,\omega)&=-\alpha \frac{\partial^2 U}{\partial x^2}(t,\omega),
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