= Solution
Remove the first spatial <derivative> by the <Dirichlet gauge transform for constant drift>:
$$
u(x,y,t)=e^{-\beta x/2}v(x,y,t),\qquad v_t=v_{xx}+v_{yy}-\frac{\beta^2}{4}v.
$$
The nonzero exponential factor preserves the zero side and top values; it also preserves $v_y=0$ on the bottom. A separated product $v=X(x)Y(y)T(t)$ has the spatial <eigenvalue problems>
$$
X''+k_x^2X=0,\quad X(0)=X(1)=0;\qquad Y''+k_y^2Y=0,\quad Y'(0)=Y(1)=0.
$$
The nontrivial solutions are $X_m=\sin(m\pi x)$ for $m\geq1$ and $Y_n=\cos((n-1/2)\pi y)$ for $n\geq1$. There is no zero spatial mode in either direction. The temporal equation gives $T=e^{-\lambda_{mn}t}$, where
$$
\lambda_{mn}=m^2\pi^2+(n-1/2)^2\pi^2+\beta^2/4.
$$
Therefore the <mixed-boundary drift-diffusion eigenmodes on a rectangle> give
$$
\boxed{u(x,y,t)=e^{-\beta x/2}\sum_{m=1}^\infty\sum_{n=1}^\infty A_{mn}\sin(m\pi x)\cos((n-1/2)\pi y)e^{-\lambda_{mn}t}.}
$$
At $t=0$ the temporal factors are one, giving the required expansion. The sine and mixed-end cosine bases are complete and orthogonal for $v$. In particular,
$$
A_{mn}=4\int_0^1\int_0^1 e^{\beta x/2}u(x,y,0)\sin(m\pi x)\cos((n-1/2)\pi y)\,dy\,dx.
$$
The factor four comes from the product of the two squared basis norms, each $1/2$.
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