= Solution
For fixed real $\kappa$, consider initial data $u_0\in L^2(0,1)$ and a solution continuous into $L^2$ at time zero, with homogeneous <Dirichlet boundary conditions> for positive times. Existence, uniqueness and continuous dependence in this <norm> give <Hadamard well-posedness>.
To construct a solution, put $v(x,t)=e^{\kappa x}u(x,t)$. Substitution gives
$$
v_t=v_{xx}-\kappa^2v,\qquad v(0,t)=v(1,t)=0.
$$
Let $b_j=2\int_0^1e^{\kappa x}u_0(x)\sin(j\pi x)\,dx$. The <Fourier sine series> solution is
$$
\boxed{u(x,t)=e^{-\kappa x}\sum_{j=1}^\infty b_j\sin(j\pi x)
e^{-[(j\pi)^2+\kappa^2]t}.}
$$
Multiplication by $e^{\pm\kappa x}$ is bounded on $L^2(0,1)$. <Completeness> of the sine basis gives convergence to $u_0$ in $L^2$ at time zero, while the exponential decay gives smoothness for positive times and the required equation and boundary values. For a classical solution continuous up to the endpoints at time zero, the usual initial-boundary compatibility conditions are additionally required.
For the <energy method>, <integration by parts> gives
$$
\begin{aligned}
\frac12\frac d{dt}\|u\|_{L^2}^2
&=\operatorname{Re}\int_0^1\overline u(u_{xx}+2\kappa u_x)\,dx\\
&=-\int_0^1|u_x|^2dx+\kappa[|u|^2]_0^1
=-\|u_x\|_{L^2}^2.
\end{aligned}
$$
Both boundary terms vanish. The same identity applies to the difference of two solutions, proving uniqueness and continuous dependence. The <Poincaré inequality> further gives
$$
\boxed{\|u(t)\|_{L^2}\leq e^{-\pi^2t}\|u_0\|_{L^2},}
$$
with the identical estimate for differences of initial data. This <Dirichlet convection-diffusion contraction> holds for \b[every real $\kappa$], and its energy constant is independent of the drift strength.
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