Asymptotic behaviour of solutions for some semilinear dissipative wave equations (Q1860735)

The author deals with the asymptotic behaviour of solutions to the following semilinear wave equation in one space dimension: \[ \begin{cases} u_{tt}-u_{xx}+ Q(t,u,u_t)= f(t,x)\quad & \text{in }(0,\infty) \times(a,b),\\ u(x,0)= u_0(x),\;u_t(0,x) =u_1(x)\quad & \text{for }x\in(a,b),\\ u(t,a)=u(t,b)= 0\quad & \text{for }t>0, \end{cases} \] where \(Q=[u_t-V(t)u] |u|^{m-1}\), \(m>1\), \(-\infty <a<b< +\infty\), \(V(t)\) is a positive time-dependent potential function and \(f(t,\cdot)\) is a source function, which is uniformly bounded as time tends to infinity.