A class of dissipative wave equations with time-dependent speed and damping (Q1931560)

The initial value problem to the equation \[ u_{tt}-\lambda (t)^2\Delta u+b(t)u_t+\lambda (t)\sum_{j=1}^nb_j(t)u_{x_j}+e(t)u=0\;\text{ in } (0+\infty)\times \mathbb{R}^n \] is considered. Under some assumptions on the coefficients of the equation the authors prove the energy estimate \[ \| u_t(t,\cdot)\| _{L_2}^2+\lambda (t)^2\| \nabla u_t(t,\cdot )\| _{L_2}^2\leq C\lambda (t)\exp(-\int_0^tb(s)\text{d}s)\left(\| u(0,\cdot )\|_{H^1}^2+\| u_t(0,\cdot )\| _{L_2}^2\right). \]