Regular solutions and energy decay for the equation of viscoelasticity with nonlinear damping on the boundary (Q1270908)

The authors study the mixed problem \[ u_{tt}- u_{xxt}- (F(u_x))_x= f\quad\text{in }(0,1)\times (0,T) \] with initial and boundary conditions \[ u(0,t)= (u_x+| u_t|^\rho u_t)|_{x= 1}= 0\quad (\rho\geq 0),\quad t\in(0,T), \] \[ u(x,0)= u^0(x),\quad u_t(x,0)= u^1(x),\quad x\in(0, 1),\quad u^0(0)= u^1(0)= 0; \] \(F\) is a twice continuously differentiable function. This equation models longitudinal oscillations of a bar under the Kelvin state equation. To establish the existence of local solutions, the authors use a variant of the contraction mapping theorem combined with weak convergence arguments. The existence of global solutions is proved on the basis of a priori estimates for local solutions independent of \(t\). Moreover, it is proved the exponential decay of solutions for \(F(s)= s+| s|^{\alpha- 2}s\) and also the algebraic decay for \(F(s)=| s|^{\alpha- 2}s\).