Mixed Cauchy problem with lateral boundary condition for noncharacteristic degenerate hyperbolic equations

Abstract

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

Keywords

• Mixed Cauchy boundary value problem
• Hyperbolic equation
• Newton potential