On the stability of the zero solution of a one-dimensional mathematical model of viscoelasticity (Q808758)

An initial boundary value problem with homogeneous Dirichlet boundary data is studied describing in Lagrangian coordinates the motion of a one- dimensional, physical linear viscoelastic medium. The partial differential equation is \(u_{tt}-\mu_ 1u_{xx}-\mu_ 2[(1+u_ x)^{-1}u_{tx}]_ x=f(t,x).\) The nonlinearity arises because of the difference between Lagrangian and Eulerian coordinates. It is shown that a global in time solution of this problem exists for small right hand side f and for small initial data. The solution is contained in \(L_ q([0,\infty)\times [0,1])\) for \(q>3\). To prove this result a corresponding linear differential equation in Banach spaces is considered. The estimates derived for the linear equation are used to prove existence for the nonlinear problem.