Boundary Effect on Asymptotic Behavior of Solutions to the p-System with Time-Dependent Damping

Abstract EN

In this paper, we consider the asymptotic behavior of solutions to the p-system with time-dependent damping on the half-line R + = 0,+∞ , v t − u x =0, u t +p v x =− α/ 1+t λ 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=+∞.

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 R + × H 2 R + .

Our proof is based on the time-weighted energy method.