For the following Cauchy problem: \begin{cases} \partial^2_tu-a^2(x,t)\partial^2_xu=f(x,t), x\in\mathbb{R},t>0\\u(x,0)=\varphi(x), \partial_t u(x,0)=\psi(x), x\in\mathbb{R},\end{cases} where \begin{cases} \|a\|_{L^\infty}+\|\partial a\|_{L^\infty}\leq M_0\\ \varphi(x)=\psi(x)=0, |x|\geq M_1\\f(x,t)=0, |x|>M_1+M_0t.\end{cases} Prove that when $|x|\geq M_1+M_0t, u(x,t)=0$.
I just learned the energy esimate of $a$ being a constant. For the problem, it seems the first equation multiplying $\partial_t u$ cannot be writen as the form of divergence. I don't know how to handle the additional terms by using remaining conditions.
Some calculation I have done: $$ 2\int_{K_\tau}f(x,t)\partial_t u \mathrm{d}x\mathrm{d}t=\int_{K_\tau}[2\partial_t^2u\partial_t u-2a^2(x,t)\partial_x^2u\partial_t u ]\mathrm{d}x\mathrm{d}t $$ $$ 2\partial_t^2u\partial_t u-2a^2(x,t)\partial_x^2u\partial_t u=\partial_t[(\partial_t u)^2+a^2(x,t)(\partial_xu)^2]-\partial_x[2a^2(x,t)\partial_xu\partial_tu]\\+2a(x,t)\partial_ta(x,t) (\partial_xu)^2-4a(x,t)\partial_xa(x,t) \partial_xu\partial_tu $$ I don't kwon what to do with the last two terms. Or multiple other term instead of $2\partial_t u$?
Appreciate any help,hint, or recommendation of relevant textbooks.