Qualitative theory for strongly damped wave equations

Abstract

We investigate the asymptotic periodicity, L p -boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications. KEYWORDSasymptotic behavior, boundedness, classical solutions, damped wave equations, strong solutions, topological structure of solutions set where X 1 2 is the fractional power space associated with A as in the work of Henry. 19 Equations like (1.1) have a lot of nontrivial and interesting features and appear in the literature under the name of strongly damped wave equations. An example of mathematical model represented in the form (1.1) is the wave equation with structural damping (see Carvalho and Cholewa 9 and Chen and Triggiani [20][21][22] ). This manuscript is a natural continuation of the work of Cuevas et al, 18 which investigates the existence and uniqueness of asymptotically almost-periodic mild solutions for strongly damped wave equations of type 1.1. Here we are concerned with the asymptotic periodicity, L p -boundedness properties, classical (resp., strong) solutions, and the topologicalProof. The proof requires straightforward modifications in the proof of Lemma 4.1 by Andrede et al. 27 We omit the details.Throughout this paper we always assume that ( , A) is an admissible pair. Pseudo S-asymptotically -periodic functionsLet Y be an arbitrary Banach space. In this work C b ([0, ∞); Y) denotes the space consisting of the continuous and bounded functions from [0, ∞) into Y, endowed with the norm of the uniform convergence. Definition 2.3. (Pierri and Rolnik[23]) A function f ∈ C b ([0, ∞); Y) is called pseudo S-asymptotically periodic if there is > 0 such that lim h→∞

Qualitative theory for Volterra difference equations

“…To state the next result, we need to introduce the following notations: The following theorem corresponds to an Azevedo et al‐type theorem for strongly damped wave equations (see Azevedo et al, Theorem 1.8).…”

“…There are only few results in the literature dealing with the asymptotic periodicity of Equation and by cause of the rapid evolution of the pseudo‐S asymptotically ω ‐periodic ( P S A P ω ) notion, we study the existence of this class of solutions to Equation . Among other things, interesting applications of this new type of functions are discussed in several branches of the evolution equations, like fractional systems, flexible structures, and strongly damped wave equations (see Azevedo et al, Cuevas et al, and de Andrade et al).…”

Strongly damped wave equations with Stepanov type nonlinear forcing term

“…as small as needed, and therefore (see [2], Example 4.3) (η, A) will be an admissible pair for any η > 0. From [21,Sect.…”

“…Author details 1 Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, Colombia. 2 Departamento de Matemáticas, Universidad del Atlántico, Barranquilla, Colombia. 3 Departamento de Ciencias Básicas, Universidad de la Costa, Barranquilla, Colombia.…”

“…Yang and Zhu investigated the properties of the p-mean Stepanov-like doubly weighted pseudo almost automorphic processes and their application to Sobolev-type stochastic differential equations driven by G-Brownian motion, see [24]. Cuevas et al in [2] studied the asymptotic periodicity, L pboundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. Recently, Xia et al in [22] introduced and investigated the (μ, ν)-pseudo S-asymptotically ω-periodic functions of class r and gave some applications.…”