  We show the existence and the uniqueness of initial traces of nonnegative
solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$
under the zero Dirichlet boundary condition. Furthermore, we obtain necessary
conditions and sufficient conditions on the initial data for the solvability of
the corresponding Cauchy--Dirichlet problem. Our necessary conditions and
sufficient conditions are sharp and enable us to find optimal singularities of
initial data for the solvability of the Cauchy--Dirichlet problem.
