# Strong stability of the wave equation with time depending potential

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.