ALMOST PERIODICITY AND ASYMPTOTIC BEHAVIOR FOR THE SOLUTIONS OF A NONLINEAR WAWE EQUATION

Abstract

This chapter describes almost periodicity and asymptotic behavior for the solutions of a nonlinear wave equation. It presents few preliminaries on almost periodic functions. A set E is said to be relatively dense if there is l > 0 such that every interval contains some point of E. The interest of almost periodic functions is given by the result that the space of almost periodic functions coincides with the closure with respect to the uniform convergence on R of the trigonometric polynomials. It is observed that the space of almost periodic functions is a Banach space for the norm Sup. It is found that if f satisfies suitable almost periodicity hypotheses and β has a suitable polynomial growth to infinity and is uniformly mono tone, the answer is affirmative and precisely there is a unique energy-almost periodic solution u to the problem. The chapter explores a problem that has at most one solution bounded in energy with ut bounded in energy. If f is periodic of period T, there is a periodic solution of periodic T, but nothing is known on the asymptotic behavior of the solutions.