  We prove the time analyticity for weak solutions of inhomogeneous parabolic
equations with measurable coefficients in the half space with either the
Dirichlet boundary condition or the conormal boundary condition under the
assumption that the solution and the source term have the exponential growth of
order $2$ with respect to the space variables. We also obtain the time
analyticity for bounded mild solutions of the incompressible Navier-Stokes
equations in the half space with the Dirichlet boundary condition. Our work is
an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (2020)]
and Dong-Zhang [J. Funct. Anal. (2020)], where the authors proved the time
analyticity of solutions to the homogeneous heat equation and the Navier-Stokes
equations in the whole space.
