On the semilinear wave equations. I (Q1272709)

We work with the triviality of the solutions of the initial-boundary value problems for the semilinear wave equation \[ \square u+f(u)= 0\quad\text{in }[0, T)\times\Omega,\quad u= 0\quad\text{on }[0,T]\times \partial\Omega, \] \[ u(0,\cdot)= u_0\in H^1_0(\Omega),\quad \dot u(0,\cdot)= u_1\in L^2(\Omega), \] where \(0< T\leq\infty\) and \(\Omega\subset \mathbb{R}^n\) is a bounded domain on which the divergent theorem is applicable.