A question on the space of solution of the wave equation due to initial data

Question

I would like to ask a question that: Given the 1D wave equation $u_{tt}-u_{xx} = 0$ where the Dirichlet condition $u(0,t) = u(L,t) = 0$ and initial data $u(x,0) \in H^2$, $u_t(x,0) \in H^1$. How can we prove that $u(.,t) \in H^2$ and $u_t(.,t) \in H^1$ for every $t \geq 0$ ?

Answer 1

The separated solutions are $u_n(t,x)=\sin(n\pi x/L)\{ A_n\cos(n\pi t/L)+B_n\sin(n\pi t/L) \}$, leading to the solution $u = \sum_{n=1}^{\infty}u_n$, where $A_n$, $B_n$ are determined by $$ u(0,x) = \sum_{n=1}^{\infty}A_n\sin(n\pi x/L) \\ u_{t}(0,x) = \sum_{n=1}^{\infty}B_n\frac{n\pi}{L}\cos(n\pi x/L). $$ A function $g=\sum_{n=1}^{\infty}C_n\cos(n\pi x/L)+D_n\sin(n\pi x/L)$ is in $H^1$ iff $$ \sum_{n=1}^{\infty}C_n^2(n\pi /L)^2+D_n^2(n\pi /L)^2 < \infty, $$ and is in $H^2$ iff $$ \sum_{n=1}^{\infty}C_n^2(n\pi/L)^4+D_n^2(n\pi/L)^4 < \infty. $$