Bounded Solutions of Second Order Semicoercive Evolution Equations in a Hilbert Space and of Nonlinear Telegraph Equations

Motivated by the problem of the existence of a solution of the nonlinear telegraph equation ut t + cut − ux x + h(u) = f (t, x) , such that u(t, ·) satisfies suitable boundary conditions over (0, π) and ‖u(t, ·)‖ is bounded over for some function space norm ‖ · ‖, we prove the dissipativeness and the existence of bounded solutions over of semilinear evolution equations in a Hilbert space of the form ü + cu̇ + Au + g(t, u) = 0 , where c > 0, A : D(A) ⊂ H → H is self-adjoint, semi-positive definite, has compact resolvant and g : × H → H , bounded and sufficiently regular, satisfies some semicoercivity condition.