Stability of global bounded solutions to a nonautonomous nonlinear second order integro-differential equation (Q1708547)

The author studies the asymptotic behavior as \(t\to \infty\) of global bounded solutions \(u(t)\) of the nonlinear second order evolutionary equation \[ u''+k\ast u' + \triangledown E(u) =g \] in finite dimensions. Here \(\triangledown E(u)\) is the gradient of the scalar function \(E\in C^2(\mathbb R)\), \(k\in L^1(\mathbb R^+)\) is a nonnegative convex kernel, \(k\ast u= \int ^t_0 k(t-s)u(s)ds\) and the forcing term \(g\) tends to \(0\) exponentially or polynomially. Define the \(\omega\)-limit set of a global solution \(u\) by \(\{\phi | \lim _{n\to \infty}u(t_n)=\phi \}\) for some \(t_n\to \infty\). Assume that the function \(E\) satisfies the Lojasiewicz inequality near \(\phi\), i.e., there are constants \(\theta \in (0,\frac{1}{2}]\), \(\sigma , \beta >0\) such that \[ | E(x)-E(\phi )| ^{1-\theta }\leq \beta \| \triangledown E(x)\| , \] for all \(x\in \mathbb R^n\) such that \(\| x-\phi\| \leq \sigma\). Assume that the kernel \(k\) satisfies for some constant \(c>0\), \(dk'(s)+ck'(s)ds \geq 0\), where \(dk'\) is the distributional derivative of \(k'\). Under certain asymptotic conditions on \(g\) it is shown that as \(t\to \infty\), \(\| u'(t)\| +\| u(t)-\phi \| \;to 0.\) Certain assertions on the rate of convergence of \(u\) to \(\phi\) are also obtained.