Viscoelasticity equation with nonlinear boundary conditions (Q1850170)

The paper deals with the global existence, uniqueness and exponential decay of the energy of solutions to the following problem \[ Lu=u_{tt}-\frac{\partial}{\partial x}F(u_x)- \frac{\partial^2}{\partial x\partial t}F_0(u_x)=f(x,t) Q\in ]0,1[\times]0,T[,\tag{1} \] \[ u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in ]0,1[,\tag{2} \] \[ u| _{x=0}=0,\;[u_x+\phi(u_t)]| _{x=1}=0,\;t\in ]0,T[,\tag{3} \] where \(F(s)\) and \(F_0(s)\) are smooth functions such that \(F_0^\prime\geq\delta_0>0\) and \(F^\prime(s)\) can change its sign. Introducing the potential \(u:u_x=v\) and \(u_t=z,\) where \(z\) is the velocity of a flow and \(v\) is the specific volume of a gas, equation (1) can be deduced from the system describing one-dimensional nonstationary motions of a barotropic gas in Lagrange's coordinates. \(\frac{\partial}{\partial x}F(u_x)\) simulates elasticity or the pressure stress and -- \(\frac{\partial^2}{\partial x\partial t}F_0(u_x)\) stands for the viscosity stress. In particular, the existence and uniqueness of solutions to problem (1)-(3) in some Sobolev type spaces is proved.