I have two questions about the classical wave equation. We consider the following wave equation: \begin{equation}\label{s6} \left\{ \begin{array}{lll} \frac{\partial^{2}\Phi(x,t)}{\partial{t}^{2}}-\frac{\partial^{2}\Phi(x,t)}{\partial{x}^{2}} = 0, &x \in (0,1),\ t \in \mathbb{R}^{+},& \\ \Phi(0,t) = \Phi(1,t)= 0, &\forall t >0,&\\ \Phi(x,0) = \Phi^{0}(x) , \frac{\partial \Phi(x,0)}{\partial{t}}=\Phi^{1}(x), &\forall x \in (0,1),& \\ \end{array} \right. \end{equation} With $(\Phi^{0},\Phi^{1}) \in L^{2}(0,1)\times H^{-1}(0,1),$ by the Fourier formula, the solution my be expressed in the form $$ \Phi(t, x)=\sum_{n=1}^{\infty}\Big(a_{n} \cos n \pi t+\frac{b_{n}}{n \pi} \sin n \pi t\Big) \sin n \pi x , $$ where $(a_{n})$ and $(b_{n})$ are the sequences of Fourier coefficients in the orthogonal basis of $L^{2}(0, 1)$ : $$ \theta_{n}(x) = \sin(n\pi x),\quad n=1,2,\ldots $$ 1) I know that this result of Fourier formula is true if $(\Phi^{0},\Phi^{1}) \in H^{1}_{0}(0,1) \times L^{2}(0,1) ,$ but I'm not sure in this case, is it true or not ? (can someone confirm it to me ?)
2) For the second question. I want to show the following lemma:
For all $(\Phi^{0},\Phi^{1}) \in L^{2}(0,1)\times H^{-1}(0,1),$ there is a constant $c>0$ such that $$ \int_{0}^{1}\mid \Phi(x,t)\mid^{2}dx \geq c \| \lbrace\Phi^{0},\Phi^{1} \rbrace\|_{L^{2}\times H^{-1}} .$$ It is equivalent to show the next lemma:
For all $(\Phi^{0},\Phi^{1}) \in H^{1}_{0}(0.1)\times L^{2}(0.1),$ there is a constant $c>0$ such that $$ \int_{0}^{1}\Big\vert \frac{\partial\Phi(x,t)}{\partial t}\Big\vert^{2}dx \geq c \| \lbrace\Phi^{0},\Phi^{1} \rbrace\|_{H^{1}_{0}\times L^{2}}.$$
NB: I think they are equivalent, because if we put $L(x,t) = \partial_t \Phi(x,t)$, So $L$ is a solution of the system, and, $ L(x,0) =\partial_x \Phi(x,0) = \Phi^{1}$ and $\partial_x L(x,0) =\partial^{2}_{t} \Phi(x,0) =\partial^{2}_{x} \Phi(x,0) = L^{1}$ !!