General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

Abstract

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.

Exponential decay rate bound of one-dimensional distributed port-Hamiltonian systems with boundary dissipation

“…for all ε ∈ [0, min(ε 0 , ε 1 )], completing the proof. Theorem 1: Consider the port-Hamiltonian system with boundary dissipation given by (1)- (5). Define…”

“…From Lemma 2, there are definitions for M (ε) and α(ε) for all ε on the interval [0, min{ε 0 , ε 1 }], and Theorem 1 provides explicit expressions of ε 0 and ε 1 for system (1)- (5). Using the parametrization ε = ξ min{ε 0 , ε 1 } with 0 < ξ < 1 leads to…”

“…Furthermore, determining not only exponential stability but also an expression for the exponential decay rate in terms of the system parameters is of theoretical interest, and also in practical performance such as analyzing control system performance. An important strategy to obtain an explicit expression for the exponential decay rate of dynamical infinite dimensional systems is the multiplier method [5]- [10]. Several approaches of the multiplier method can be found in literature.…”

General energy decay estimates for a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions

“…$$ By imposing specific conditions on the memory kernel, they derived general decay rates that are connected to an ordinary differential equation (ODE) through the utilization of the logarithmic Sobolev inequality and Lyapunov method. For more information, refer to [23–25] and their references. Here are also other logarithmic nonlinearities such as $$$ {\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\mathit{\log}\kern2pt \mid u\mid $$$.…”