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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 $Omegasubset mathbb{R}^d$ 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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