= Solution
Take real $\kappa$ and use the <L2 norm> on the spatial interval. Existence, uniqueness and continuous dependence are the three requirements of <Hadamard well-posedness>. An <energy method> supplies the decisive estimate. For a smooth solution with homogeneous <Dirichlet boundary conditions>, <integration by parts> gives
$$
\frac12\frac d{dt}\|u(t)\|_2^2
=\operatorname{Re}\int_0^1\overline u(u_{xx}+\kappa u_x)\,dx
=-\|u_x\|_2^2+\frac{\kappa}{2}[|u|^2]_0^1
=-\|u_x\|_2^2.
$$
The drift contributes only a boundary term, which vanishes. The <Poincare inequality> $\|u_x\|_2^2\geq\pi^2\|u\|_2^2$ further gives
$$
\boxed{\|u(t)\|_2\leq e^{-\pi^2t}\|u(0)\|_2.}
$$
Apply the same argument to the difference of two solutions to obtain uniqueness and continuous dependence on the initial data.
For existence, use the <Dirichlet gauge transform for constant drift>: $w=e^{\kappa x/2}u$ satisfies $w_t=w_{xx}-\kappa^2w/4$ with zero boundary values. Expanding $w_0$ in its <Fourier sine series> gives
$$
w(x,t)=\sum_{j=1}^{\infty}b_j
e^{-[(j\pi)^2+\kappa^2/4]t}\sin(j\pi x).
$$
For $u_0\in L^2(0,1)$ the series defines a solution continuous in $L^2$ down to $t=0$ and smooth for positive time; multiplication by the fixed bounded exponentials preserves this interpretation. Its energy estimate follows by approximation with smooth initial data. For a classical solution at the initial corners, require the usual smoothness and boundary compatibility instead. \b[The problem is well posed in $L^2$, with a contraction estimate independent of the initial data.]
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