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 $$u^{\prime \prime }+c{u}^{\prime }+(b+a(t))u=0$$ where $b,c>0$ 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$$u^{\prime \prime }+c{u}^{\prime }+g(u)=f(t)$$where $c>0$, $f\in L^{\infty }(R)$ and $g\in C^1(R)$ satisfies some sign hypotheses. The typical case is $g(u)=bu+a|u{|}^{p}u$ with $a\ge 0,b>0.$ Similar properties are valid for evolution equations of the form $$u^{\prime \prime }+c{u}^{\prime }+(B+A(t))u=0$$ where $A(t)$ 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 $L^{\infty }(R+,L(H))$ .