Global periodic attractor for strongly damped wave equations with time-periodic driving force

Abstract

In this paper, we consider the existence of a global periodic attractor for a strongly damped nonlinear wave equation with time-periodic driving force under homogeneous Dirichlet boundary condition. It is proved that in certain parameter region, for arbitrary time-periodic driving force, the system has a unique periodic solution attracting any bounded set exponentially. This implies that the system behaves exactly as a one-dimensional system. We mention, in particular, that the obtained result can be used to prove the existence of global periodic attractor of the usual damped and driven wave equations.

Qualitative theory for strongly damped wave equations

“…The strongly damped wave equations are of a great current interest, and they have been investigated in several contexts by many mathematicians under different assumptions. The literature offers topics such as existence, long time asymptotic behavior, attractors, well‐posedness, decay estimates, blowup, controllability, bootstrapping and regularity, and asymptotic periodicity . These equations are also considered in physical areas, such as heat conduction and solid mechanics.…”

Asymptotic Periodicity for Strongly Damped Wave Equations

“…Corollary 14. Let : R + × 1/2 × → be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15)…”

“…Remark 15. Let : R + × 1/2 × → be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15). We can avoid the condition (32) by using the fixed-point iteration method.…”

“…An example of mathematical model represented in form (1) is the wave equation with structural damping (see [2][3][4][5]). The strongly damped wave equations has been investigated in several contexts by many authors in the last years, for example, existence [6,7], global classical solution [6,8], long-time asymptotic behavior [9][10][11], attractor [2,[12][13][14][15][16], well-posedness [17], decay estimates [18], blowup [8,19,20], controllability [19], bootstrapping, and regularity [21]. Another important aspect of the qualitative study of the solutions of strongly damped wave equations is their asymptotic periodicity.…”

Conservation laws for a strongly damped wave equation

“…[6, 8-10, 13-15, 18, 20, 24, 25] and references therein. In [21] an initial-boundary value problem for a nonlinear wave equation in one space dimension with a nonlinear damping term It is well known [10,18,20,21] that when the damping term |u| m−1 u t is absent from the equation, then the source term |u| p−1 u yields the solution to blow up in finite time.…”