An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities
An asymptotic of the following ordinary differential operator (ODO) with operator coefficients and with nonlinear operators \(N_0,N_1,\dots,N_n\) \[ A(D_t)u(t)=\sum^n_{j=0}{\mathcal D}^{n-j}_tN_j(t;u(t),\dots,D^{\ell-n}_tu(t)),\quad t\in\mathbb R,\tag{1} \] \(\ell>n\), is developed. The conditions on \(A\) and \(N_j\) include elliptic and parabolic PDO. The operators \(N_j\) are defined in appropriate Sobolev spaces \(W^{\ell-n}_{p,\text{loc}}(\mathbb{R};\{B_k\}^{\ell-n}_{k=0})\) and \(B_k\) are Banach spaces. It is shown that for a certain class of equations (1) the problem of the asymptotic behaviour of \(u\) as \(t\to \infty\) can be reduced to that for a finite dimensional dynamical system perturbed by a nonlocal operator, namely \[ \frac d{dt}h(t)+\mathbb{N}(t;h(t))+\mathbb K[h](t)=0,\tag{2} \] where \(\mathbb K\) is a nonlocal nonlinear operator and the components \(\mathbb N_k\) of \(\mathbb N\) are written explicitly. Under some natural assumptions on the nonlinearities \(N_j\) a beautiful representation formula for \((u,\dots,D^{\ell-n-1}_tu)\) is given. It can be viewed as a very interesting generalization of the variation of constants formula in the classical theory of ODO.
scientific article; zbMATH DE number 2131178