设 $\left\{ \begin{array}{l} x = \mathrm{e}^{-t}, \\ y = \int_{0}^{t} \ln (1 + u^{2}) \, \mathrm{d}u, \end{array} \right.$ 则 $\left. \frac{\mathrm{d}^2y}{\mathrm{d}x^2} \right|_{t=0} = \_$ .