A sharp stability criterion for single well Duffing and Duffing-like equations

Abstract

We refine some previous sufficient conditions for exponential stability of the linear ODE u′′+cu′+(b+a(t))u=0where b,c>0 and a is a bounded nonnegative time dependent coefficient. This allows to improve some results on uniqueness and asymptotic stability of periodic or almost periodic solutions of the equation u′′+cu′+g(u)=f(t)where c>0, f∈L∞(R) and g∈C1(R) satisfies some sign hypotheses. The typical case is g(u)=bu+a|u|pu with a≥0,b>0. Similar properties are valid for evolution equations of the form u′′+cu′+(B+A(t))u=0where A(t) and B are self-adjoint operators on a real Hilbert space H with B coercive and A(t) bounded in L(H) with a sufficiently small bound of its norm in L∞(R+,L(H)).