Consider the following second-order wave equation $$ u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega $$ with boundary conditions $$ u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0. $$ Here, $\Omega\subset \mathbb R^d$ is bounded, $a: (0,T)\times \Omega \to \mathbb R^{d,d}$ is smooth (but dependent on time). Under the assumption $f\in L^2(0,T;L^2(\Omega))$, $g\in H^1_0(\Omega)$, $h\in L^2(\Omega)$, there exists a uniquely determined weak solution $u\in L^\infty(0,T;H^1_0(\Omega))$ with $u_t\in L^\infty(0,T;L^2(\Omega))$ and $u_{tt}\in L^2(0,T;H^{-1}(\Omega))$. This is proven, e.g., in Evans' book.
My question: Can you point me to a reference that includes a proof that we can improve the regularity of the weak solution from $L^\infty$ to $C$ (i.e., continuous in time)? Due to Corona-virus related restrictions I have no access to our university's library...
If $a$ does not depend on time then such a proof can be found in John Hunters PDE-notes online. The idea is to show continuity of the energy, where one step is to mollify the wave equation in time. This does not work if $a$ depends on $t$, as then $\rho_\epsilon \ast(a\cdot \nabla u) \ne a\cdot \nabla (\rho_\epsilon \ast u)$, and I cannot figure out how to estimate the difference of these terms.
Crossposted from https://math.stackexchange.com/questions/3746894/continuity-of-weak-solutions-to-wave-equation-with-time-dependent-coefficients , which did not attract much attention over there.
