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Following the Perron–Ta Li line of results, we give a characterization of the uniform exponential dichotomy property for the abstract continuous-time evolution families using a discrete-time approach.

We construct topological conjugacies between linear and nonlinear evolution operators that admit either a nonuniform exponential contraction or a nonuniform exponential dichotomy. We consider evolution operators defined by nonautonomous differential equations x′=A(t)x+f(t,x) in a Banach space. The conjugacies are obtained by first considering sufficiently small linear and nonlinear perturbations of linear equations x′=A(t)x. In the case of linear perturbations, we construct in a more or less explicit manner topological conjugacies between the two linear flows. In the case of nonlinear perturbations, we obtain a version of the Grobman–Hartman theorem for nonuniformly hyperbolic dynamics. Furthermore, all the conjugacies that we construct are locally Hölder continuous provided that the vectors fields are of class C1. As a byproduct of our approach, we give conditions for the robustness of strong nonuniform exponential behavior, in the sense that under sufficiently small perturbations the structure determined by the stable and unstable bundles persists up to small variations. We also show that the constants determining the nonuniform exponential contraction or nonuniform exponential dichotomy vary continuously with the perturbation. All the results are obtained in Banach spaces.

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the Lp−Lq type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the Lp−Lq estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|αu. Our result covers the whole super critical case α >1 , where the α=1 is well known as the Fujita exponent when n=2.

In this paper, we study the mixed initial–boundary value problem of Neumann type for the nonlinear elastic wave equation outside a domain. The local existence of solutions to this problem is proved by iteration. To get this result, we prove the existence of solutions for the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.