Existence and decay of solutions of an abstract second order nonlinear problem

Abstract

In this paper we consider the following nonlinear evolution problem with damping: (∗) | u ″ ( t ) + A u ( t ) + B α u ′ ( t ) = 0 , t > 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 where A is a monotone nonlinear operator, B α a linear operator and α a real number with 0 < α ⩽ 1 . We study the global existence and decay of solutions of problem (∗) .