While solving the PDF of 2-D heat equation $U_t= C (U_{xx}+U_{yy})$, via classic separation method to set $U(x,y,t)=X(x)Y(y)T(t)$, then by plugging into the PDE, we would have the following expression: $$\frac{T'}{CT}=\frac{X''}{X}+\frac{Y''}{Y}= \lambda\tag{1}$$
where $\lambda$ is a constant. To further analyze steady state where $t\rightarrow \infty$ and $ T'= 0$, it seems right to have:
$$\frac{X''}{X}+\frac{Y''}{Y}=0\tag{2}$$
My question is, according to $(1)$, the constant $\lambda$ should be a fixed number that satisfies B.C. & I.C. for all $t,x$ and $y$. But $(2)$ seems to say equation $(1)$ has an exception as $t \rightarrow\infty$, and the fix number shrinks to zero instead. This seems paradox to me because I feel $(1)$ shall be hold for any $t$ without exception. Hope my misconception can be pointed out.