I am starting to study PDEs and the professor gave us the following exercise.
For $(\varphi^0,\varphi^1)\in L^2(\Omega)\times H^{-1}(\Omega)$, consider the following equation
$$ \left\{\begin{array}{ll}\varphi^{\prime \prime}-\Delta \varphi=0 & \text { in }(0, T) \times \Omega \\ \varphi=0 & \text { on }(0, T) \times \partial \Omega \\ \varphi(0, \cdot)=\varphi^{0}, \varphi^{\prime}(0, \cdot)=\varphi^{1} & \text { in } \Omega\end{array}\right. $$ If $\left(\varphi, \varphi^{\prime}\right) \in C\left([0, T], L^{2}(\Omega) \times H^{-1}(\Omega)\right)$ is the unique weak solution, prove that $$ \|\varphi\|_{L^{\infty}\left(0, T ; L^{2}(\Omega)\right)}^{2}+\left\|\varphi^{\prime}\right\|_{L^{\infty}\left(0, T ; H^{-1}(\Omega)\right)}^{2} \leq\left\|\left(\varphi^{0}, \varphi^{1}\right)\right\|_{L^{2}(\Omega) \times H^{-1}(\Omega)}^{2}. $$
How can I prove this?