Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

TL;DR

This paper investigates the existence, uniqueness, and exponential decay of solutions for a nonlinear viscoelastic wave equation with mixed boundary conditions, expanding understanding of such complex nonlinear PDEs.

Contribution

It provides a global existence and uniqueness theorem under Lipschitz conditions and demonstrates exponential decay of solutions under more restrictive assumptions.

Findings

Proved global existence and uniqueness of solutions.

Established exponential decay of solutions and their derivatives.

Extended analysis to a nonlinear viscoelastic wave equation with mixed boundary conditions.

Abstract

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t), u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.