Unicity of wave equation pde problem we consider the problem $$ \begin{cases} \dfrac{\partial^2 u}{\partial t^2}= a^2 \dfrac{\partial^2 u}{\partial x^2}\\\ u(0,t)=u(l,t)=0\\\ u(x,0)=f(x)\\\ \dfrac{\partial u}{\partial t}(x,0)=g(x) \end{cases} $$ I want to prove the unicity of the solution of this problem. For this, we put $f=g=0$, we multiply the equation by $\dfrac{\partial u}{\partial t}$ and integrate in $[0,l]$. Using that $$ \dfrac{\partial w}{\partial t} w= \dfrac{1}{2} \dfrac{\partial}{\partial t} w^2, $$ we obtain that $$ \dfrac{1}{2} \displaystyle\int_0^l \dfrac{\partial}{\partial t} \\{(\dfrac{\partial u}{\partial t})^2+a^2 (\dfrac{\partial u}{\partial x})^2\\} dx =0. $$ We put $$ I(t)= \dfrac{1}{2} \displaystyle\int_0^l \dfrac{\partial}{\partial t} \\{(\dfrac{\partial u}{\partial t})^2+a^2 (\dfrac{\partial u}{\partial x})^2\\} dx. $$ My question is: how do we prove that $$ \forall t \geq 0: I(t)=I(0) $$ and how do we deduce the unicity of the solution $u$? Please

You are meant to set $$ I(t)= \frac{1}{2} \int_0^l \left\\{\left(\frac{∂u}{∂t}\right)^2+a^2 \left(\frac{∂u}{∂x}\right)^2\right\\} dx $$ so that $$ \frac{d}{dt}I(t) = \frac{1}{2} \int_0^l \frac{∂}{∂t} \left\\{\left(\frac{∂u}{∂t}\right)^2+a^2 \left(\frac{∂u}{∂x}\right)^2\right\\} dx=0 $$ which implies that $I$ is a constant function.

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As to the uniqueness, any difference between two solutions solves the homogeneous equation with homogeneous boundary conditions. As $I(0)=0$, the difference function has zero gradient everywhere, thus has to be constant zero.