Effect of resonance on the existence of periodic solutions for strongly damped wave equation

Abstract

We are interested in the differential equationü(t) = −Au(t) − cAu(t) + λu(t) + F (t, u(t)), where c > 0 is a damping factor, A is a sectorial operator and F is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that λ is an eigenvalue of A and F is a bounded map. We introduce new geometrical conditions for the nonlinearity F and use topological degree methods to find T -periodic solutions for this equation as fixed points of Poincaré operator.

Periodic solutions and multiharmonic expansions for the Westervelt equation

“…Before doing so, we rewrite (31) as a multiharmonic system, equipped with boundary conditons (29). Analogously to above, formally setting u N −1 N = 0, we get…”

“…Since these can also be tackled by maximal L p regularity techniques (see [38] in the non-periodic setting) possibly the results in [9] extended to absorbing boundary conditions as well. We also refer to [29] and the references therein, for some recent result on time periodic strongly damped nonlinear wave equations in a general but semilinear setting. Some helpful inequalities and notations that will be used in this paper are the following:…”

Recurrent solutions of the linearly coupled complex cubic‐quintic Ginzburg‐Landau equations

“…The qualitative behavior of the partial differential equations is an active research field on which there is a great deal of literature. There are many methods to deal with the qualitative behavior of partial differential equations: *Averaging principle method, *The method of upper and lower solutions, *Fourier expansion method, *Fixed point method and exponential dichotomy, *Energy method, *Variation method, *Nash‐Moser iteration method, *Kolmogorov‐Arnold‐Moser iteration method …”

Nonlinear wave equation with damping: Periodic forcing and non‐resonant solutions to the Kuznetsov equation

“…Related time‐periodic problems have been studied by other authors over the years. In particular, we mention the work of Kokocki in which a class of nonlinear wave equations with Kelvin‐Voigt damping is investigated. The work of Kokocki does not cover the Kuznetsov equation however.…”

“…Our investigation of – is based on estimates of solutions to the corresponding linearizations and The novelty of our approach is rooted in the method we employ to establish the estimates for the linearized Dirichlet and Neumann boundary value problem. Instead of relying on a Poincaré map, which is the standard procedure in the investigation of time‐periodic problems, and also the approach used in , we obtain the estimates directly via a representation formula for the solution. We hereby circumvent completely the theory for the corresponding initial‐value problem, which is needed to construct a Poincaré map, and develop a much more direct approach.…”