Let us consider the equation $$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$ subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((0,T)\times \mathbb{R}^{n})$ is a compactly supported function and the coefficients $\{a_{jk}\}_{j,k=0}^n$ are strictly hyperbolic with signature $(-,+,\ldots,+)$ and additionally $a^{jk} \in L^2((0,T)\times\mathbb{R}^n)$ for all $j,k=0,\ldots,n$.
Is it true that there exists a unique solution $u \in H^1((0,T)\times \mathbb{R}^n)$ to this equation? Could you point me to a reference. My primary concern in this question is obviously due to the lower regularity of the coefficients $a_{jk}$ otherwise this is classical.
