Prove uniqueness of solution of the following problem using energy method:
$$u_{tt}-k^2u_{xx}=F(x,t), \hspace{0.5cm}0<x<L, \hspace{0.5cm}t>0\\ u_x(0,t)=t^2,\hspace{0.5cm}u(L,t)=-t,\hspace{0.5cm}t\geq0\\ u(x,0)=x^2-L^2,\hspace{0.5cm}0\leq x\leq L\\ u_t(x,0)=\text{sin}^2\left(\dfrac{\pi x}{L}\right),\hspace{0.5cm}0\leq x\leq L$$
What I did:
First I need an expression for the energy, so multiplied by $u_t$ and then integrated: $$\int\limits_0^Lu_{tt}u_tdx-k^2\int\limits_0^Lu_{xx}u_tdx=\int\limits_0^LF(x,t)u_tdx$$
First integral is $\int\limits_0^L\dfrac{1}{2}\dfrac{d}{dt}u_t^2dx$
The second one, integrating by parts: $$\int\limits_0^Lu_{xx}u_tdx=u_tu_x|_0^L-\int\limits_0^Lu_xu_{tx}dx=\underbrace{u_t(L,t)}_0u_x(L,t)-u_t(0,t)\underbrace{u_x(0,t)}_{t^2}-\int\limits_0^L\dfrac{1}{2}\dfrac{d}{dt}u_x^2dx$$ and I'm stuck at this point.
In other problems that I've solved, $u_tu_x|_0^L=0$ which gave me easy expressions for the energy $E(t)$, but this is not the case and I don't know how to proceed. Any hint will be appreciated.