Ultimate bound of solutions to second-order evolution equation with nonlinear damping term

We estimate the ultimate bound of the energy of the solutions to the equation $$\begin{aligned}u''(t)+ Au(t)+g(u'(t))=f(t),\;\; t\ge 0, \end{aligned}$$ u ′ ′ ( t ) + A u ( t ) + g ( u ′ ( t ) ) = f ( t ) , t ≥ 0 , where A is a positive selfadjoint operator on a Hilbert space H, g is a nonlinear damping operator and f is a bounded forcing term with values in H. The paper concerns the asymptotic behavior of the solution. We prove that the bound of the solution is of the form $$\begin{aligned} C(1+\Vert f\Vert ^{2(a+2)/(a+1)}_{\infty }) \end{aligned}$$ C ( 1 + ‖ f ‖ ∞ 2 ( a + 2 ) / ( a + 1 ) ) where the constant $$a\ge 0$$ a ≥ 0 appears in the coercivity condition on the nonlinear damping operator g. and $$\Vert f\Vert $$ ‖ f ‖ stands for the $$L^{\infty }$$ L ∞ norm of f with values in H. Moreover, we investigate other cases, for instance, in the case of a special class of damping operator and the case of antiperiodic solutions. The last part of this paper is devoted to study the rate of decay for the solution of the equation as $$t\rightarrow +\infty $$ t → + ∞ .