Solution
= Solution
The slowest surviving mode has $n=1,m=0$. Hence
$$
\boxed{T(x,y,t)\sim\frac{4a^2}{\pi^3}\sin\frac{\pi x}{a}\,e^{-\lambda\pi^2t/a^2}\qquad(t\to\infty).}
$$
It is independent of $y$: insulation allows the constant transverse mode, while every nonconstant transverse mode decays faster. The next possible relative decay rates are $8\lambda\pi^2/a^2$ from $n=3,m=0$ and $16\lambda\pi^2/b^2$ from $n=1,m=4$.