= Solution
The two homogeneous <Dirichlet boundary conditions> on $x$ force negative separation constants. The initial profile is an <even function>, so only the even sine-family <eigenfunctions> are needed:
$$
X_m(x)=\cos(k_mx),\qquad k_m=\left(m+\frac12\right)\pi,\qquad m=0,1,\ldots.
$$
They obey $X_m(\pm1)=0$ and $\int_{-1}^1X_mX_n\,dx=\delta_{mn}$. Odd <eigenfunctions> have zero projection onto the even initial data. Hence the initial coefficients are
$$
\begin{aligned}
a_m&=\int_{-1}^1(1-x^2)\cos(k_mx)\,dx\\
&=2\left[-\frac{2x\cos(k_mx)}{k_m^2}+\frac{2\sin(k_mx)}{k_m^3}\right]_0^1
=\frac{4(-1)^m}{k_m^3}.
\end{aligned}
$$
The desired <eigenfunction expansion> is therefore
$$
\boxed{u(x,y)=4\sum_{m=0}^{\infty}\frac{(-1)^m}{k_m^3}\cos(k_mx)\exp[-(k_m^2+1)y+\cos y-1],\qquad k_m=(m+\tfrac12)\pi.}
$$
At $y=0$ this is the <Fourier series> of $1-x^2$ on the interval; its coefficients are absolutely summable. Each term satisfies the <partial differential equation> and side conditions. For $y>0$ the additional Gaussian decay in $m$ justifies termwise <differentiation>. Since $k_m^2+1>0$ and $e^{\cos y-1}\leq1$, the whole series decays uniformly as $y\to\infty$.
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