Show that the initial-boundary value problem

\begin{align} & {{u}_{tt}}={{u}_{xx}}\text{ }(x,t)\in \left( 0,l \right)\times \left( 0,T \right),\text{ }T,l>0 \\ & u\left( x,0 \right)=0,\text{ }x\in \left[ 0,l \right] \\ & {{u}_{x}}\left( 0,t \right)-u\left( 0,t \right)=0,\text{ }{{u}_{x}}\left( l,t \right)+u\left( l,t \right)=0,\text{ }t\in \left[ 0,T \right]\\ \end{align}

has zero solution only.

My attempt 2:

Previously I tried separation by variables but got stuck at the end. Inspired by BCLC, I try energy method this time.


$$E\left( t \right)=\frac{1}{2}\int_{0}^{L}{\left( u_{x}^{2}\left( x,t \right)+u_{t}^{2}\left( x,t \right) \right)dx}.$$

The equation ${{u}_{tt}}={{u}_{xx}}$ and the Robin b.c. gives

$$\begin{align} & \frac{dE}{dt}=\int_{0}^{L}{\left( {{u}_{x}}{{u}_{xt}}+{{u}_{t}}{{u}_{tt}} \right)dx} \\ & \text{ }=\int_{0}^{L}{\left( -{{u}_{t}}{{u}_{xx}}+{{u}_{t}}{{u}_{tt}} \right)dx}+\left. {{u}_{t}}{{u}_{x}} \right|_{0}^{L} \\ & \text{ }={{u}_{t}}\left( l,t \right){{u}_{x}}\left( l,t \right)-{{u}_{t}}\left( 0,t \right){{u}_{x}}\left( 0,t \right) \\ & \text{ }=-{{u}_{t}}\left( l,t \right)u\left( l,t \right)-{{u}_{t}}\left( 0,t \right)u\left( 0,t \right)\le 0\text{ }\left( ? \right) \\ \end{align}$$

Therefore, $E\left( t \right)\le E\left( 0 \right)$ for all $t\ge 0$. Since $E\left( t \right)\ge 0$ and , we obtain $E\left( 0 \right)=0 (?)$ for all $t\ge 0$, thus $E\equiv 0$ and hence $u\equiv 0$ .

Is the proof correct?


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