Boundary effect on asymptotic behavior of solutions to the \(p\)-system with time-dependent damping (Q2308018)

Summary: In this paper, we consider the asymptotic behavior of solutions to the \(p\)-system with time-dependent damping on the half-line \(\mathbb{R}^+ =(0,+\infty)\), \(v_t - u_x = 0\), \(u_t + p(v)_x =- (\alpha /(1+t)^\lambda)u\) with the Dirichlet boundary condition \(u|_{x = 0} = 0\), in particular, including the constant and nonconstant coefficient damping. The initial data \((v_0, u_0)(x)\) have the constant state \((v_+, u_+)\) at \(x = + \infty\). We prove that the solutions time-asymptotically converge to \((v_+, 0)\) as \(t\) tends to infinity. Compared with previous results about the \(p\)-system with constant coefficient damping, we obtain a general result when the initial perturbation belongs to \(H^3 (\mathbb{R}^+) \times H^2 (\mathbb{R}^+)\). Our proof is based on the time-weighted energy method.