It is well known that the wave equation with frictional damping $$\eqalign{ & {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr & y(t,0) = y(t,1) = 0 \cr} $$ is strongly stable in the energy space $H^1_0 \times L^2$ norm in the case $a(t,x)=a(x)$ if it is non zero everywhere ($a$ is in $L^\infty$).

What can we say about the case $a(t,x)$ ?. Clearly, we can not use the spectral theory here because of the time dpending of $a$. Is there any references or suggestions ? Thank you.