Leibniz rule; Partial Differential Equations

Question

I'm stuck on a question :|

So far I have,

i) $$I'(t)=\int^L_0 2 \frac{\partial v(x,t)}{\partial x} [v(x,t)] dt$$

ii) $$I(0)=\int^L_0 [v(x,0)]^2 dx= 0$$

Answer 1

HINT: $$ I'(t)=\int_0^L 2vv_t\,dx=\int_0^L 2vv_{xx}\,dx=\big[2vv_x\big]_0^L-\int_0^L 2v_x^2\,dx=-2\int_0^L v_x^2\,dx $$