= Solution
Separate variables as $u=X(x)T(t)$. The zero values at both plates give the <Fourier sine basis>
$$
X_j(x)=\sin\frac{j\pi x}{L},\qquad T_j(t)=e^{-\nu j^2\pi^2t/L^2},\qquad j\geq1.
$$
Expand the initial profile using <orthogonality>. With $s=x/L$, its coefficients are
$$
A_j=\frac2L\int_0^LU_0\left(\frac xL-\frac{x^2}{L^2}\right)\sin\frac{j\pi x}{L}\,dx=2U_0\int_0^1s(1-s)\sin(j\pi s)\,ds=\frac{4U_0[1-(-1)^j]}{j^3\pi^3}.
$$
Consequently
$$
\boxed{u(x,t)=\frac{8U_0}{\pi^3}\sum_{\substack{j\geq1\\j\ {
m odd}}}\frac{\sin(j\pi x/L)}{j^3}e^{-\nu j^2\pi^2t/L^2}.}
$$
Only odd modes occur because the initial profile is symmetric about the midpoint. The coefficients are absolutely summable, so <dominated convergence> gives $\boxed{u(x,t)\to0}$ uniformly as $t\to\infty$. The slowest decay rate is $\nu\pi^2/L^2$, associated with $j=1$.
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