Abstract
Abstract In this paper, in a cylindrical domain $D=\Omega \times (0,T)$ D = Ω × ( 0 , T ) with $\Omega \subset {R}^{n}$ Ω ⊂ R n , we consider a mixed Cauchy problem with a potential lateral boundary condition for the following noncharacteristic degenerated equation $$\begin{aligned} Lu=u_{tt}-k(t)\Delta _{x}u(x,t)=f(x,t), \end{aligned}$$ L u = u t t − k ( t ) Δ x u ( x , t ) = f ( x , t ) , where $k(t)\geq 0$ k ( t ) ≥ 0 . As in the case for strictly hyperbolic equations, we first establish that $u\in W_{2}^{1}(D)$ u ∈ W 2 1 ( D ) and $u\in W_{2}^{2}(D)$ u ∈ W 2 2 ( D ) under the assumptions $\Vert \frac{f}{k} \Vert _{L_{2}(\Omega )}(t)<\infty $ ∥ f k ∥ L 2 ( Ω ) ( t ) < ∞ and $\Vert \frac{\mathrm{grad}_{x} f}{k} \Vert _{L_{2}(\Omega )}(t)<\infty $ ∥ grad x f k ∥ L 2 ( Ω ) ( t ) < ∞ for every $t\in [0,T]$ t ∈ [ 0 , T ] , respectively.