11.5.1.3 d'Alembert method

condition: Let \( u(x, t)- \) solution \( \quad \) in \( \quad[0,1] \times \bar{R}_{+} \) of the mixed problem \[ \begin{array}{l} u_{t t}(x, t)=u_{x x}(x, t), \quad u(0, t)=u(1, t)=0, \\ u(x, 0)=0, u_{t}(x, 0)=x^{2}(1-x) . \\ \text { Find } \lim _{t \rightarrow+\infty} \int_{0}^{1}\left[u_{t}^{2}(x, t)+u_{x}^{2}(x, t)\right] d x \\ \text { and } \lim _{t \rightarrow 0} \int_{0}^{1}\left[u_{t}^{2}(x, t)+u_{x}^{2}(x, t)\right] d x . \end{array}\]