Convergence rate of solutions to the generalized telegraph equation with an inhomogeneous force

Abstract

The aim of this note is to study the convergence rate of the solution to the generalized telegraph equationεutt−div(|∇u|p(x)−2∇u)+ut=f. Roughly speaking, we borrow some ideas originated from the study of second order dynamical system to establish the strong convergence relation between u(x,t) and u⁎(x), in which u⁎ is the solution to the steady-state equation. Further, we also construct a new error functional to build up the first order differential inequality associated with the difference u(x,t)−u⁎(x), and then give explicit convergence rate estimates for the difference u(x,t)−u⁎(x). Namely, there exists a positive constant M such that∫Ω|∇u(x,t)−∇u⁎(x)|p(x)dx⩽Mt11−p+ with p+=max⁡p(x).