In this paper, we investigated the stabilization of the damped wave equation through the use of an internal feedback mechanism incorporating time delay. This work builds upon the partial pole placement paradigm, a recent theoretical framework originally developed for functional differential equations, which enables the selective assignment of eigenvalues within a prescribed region of the complex plane. Using this approach, we designed an internal delayed feedback law that guarantees the exponential stabilization of the resulting closed-loop system. A distinctive feature of our control strategy lies in its ability to prescribe the optimal exponential decay rate for each modal cluster, thereby achieving the fastest possible stabilization consistent with the system’s spectral limitations. This can be achieved regardless of the stabilizability domain being delay-independent or delay-dependent. This allows for a highly efficient control mechanism tailored to the specific dynamical behavior of the wave equation. To illustrate the practical relevance of our theoretical findings, we applied the proposed method to the control of transverse vibrations in a taut string. Numerical simulations confirmed the robustness and effectiveness of the feedback design, underscoring its potential for broader applications in the control of distributed parameter systems with delay effects.

In the framework of Timoshenko beam, the material parameters were inherently prescribed on the material moving frame. In this regard, we derived the strong and weak formulations of the dynamics under finite transformation in Lagrangian coordinates. Accordingly, analytical mechanics tools were used to deduce a new Hamiltonian formulation of the model which proved to be remarkably simple and synthetic.

In this article, a heat equation with an interior logarithmic source and a nonlinear dynamic boundary condition is considered. After establishing the local well-posedness, the global existence and finite time blow-up results of the solutions with different initial energy levels are given. Moreover, the lower bound of the maximal existence time of the weak solution is deduced for the special case m = 2.

This study focuses on a class of fractional sublinear operators, denoted as S_Ω,α, and their commutators S_Ω,α,b with rough kernels. We establish the boundedness of these operators on newly emerging vanishing generalized Morrey spaces that are characterized by (V_∞) or (V^∗). The primary innovation of this paper lies in the novel approach of controlling S_Ω,α via the Riesz potential I_α and the management of S_Ω,α,b through fractional maximal commutators with rough kernels M_Ω,α−ε,b and M_Ω,α+ε,b for some ε > 0.

In this paper, we investigate the stability and instability of standing waves for the Schrödinger- Choquard equation with mixed fractional Laplacians. We first establish the existence and stability of normalized standing waves in theL²-subcritical case. Subsequently, we prove the existence and strong instability of normalized ground state standing waves in theL²-supercritical case.

This paper considers the application of the orthogonality sampling method (OSM) with single and multiple sources for a fast identification of small objects in the limited-aperture inverse scattering problem. First, we apply the OSM with a single source and demonstrate that the indicator function of the OSM with a single source can be expressed by the Bessel function of order zero of the first kind, an infinite series of Bessel functions of nonzero integer order of the first kind, the range of the signal receiver, and the emitter location. We then explain that the objects can be identified using the OSM with a single source; however, the identification is strongly influenced by the location of the source and the applied frequency. To realize effective improvement, we consider the OSM with multiple sources. Based on the identified structure of the OSM with a single source, we propose an indicator function for the OSM with multiple sources and demonstrate that it can be expressed by the square of the Bessel function of order zero of the first kind and an infinite series of the square of the Bessel function of nonzero integer order of the first kind. This result shows that the locations of objects can be uniquely identified using the designed OSM. Simulation results with experimental data provided by the Institute Fresnel demonstrate the advantages and disadvantages of the OSM with a single source and how the proposed OSM with multiple sources behaves.

This paper investigates the existence, energy decay, and finite-time blow-up of solutions for a class of fourth-order viscoelastic evolution equations involving variable exponent nonlinearities, logarithmic terms, and strong damping effects. Such models arise naturally in the mathematical description of heterogeneous viscoelastic media and nonlinear plate equations with memory. By applying the Nehari manifold method and suitable energy estimates, we establish the global existence of weak solutions under appropriate assumptions on the initial data and the variable exponents. Moreover, using a perturbed energy method combined with a carefully constructed Lyapunov functional, we prove that the solutions exhibit general energy decay rates depending on the properties of the relaxation kernel and the spatially variable nonlinearities. Finally, we derive sufficient conditions ensuring finite-time blow-up for solutions with negative initial energy, thereby highlighting the competing effects of strong damping, viscoelastic memory, and logarithmic nonlinearities.

In recent years, fractional-order dynamical systems have attracted significant attention because of their ability to describe memory effects and hereditary properties found in many physical and engineering processes. Fractional calculus extends the concept of differentiation and integration to non-integer orders, providing a more flexible mathematical framework for modeling and analyzing complex system behaviors. Within this framework, the study of chaotic dynamics in fractional-order systems has become an active research area since many well-known chaotic models show new and qualitatively different behaviors when generalized to fractional derivatives. This study revisits a classical chaotic system originally defined in the integer-order setting and analyzes its behavior using fractional-order calculus. The main objective is to examine how fractional differentiation influences the system’s chaotic dynamics and sensitivity to initial conditions. To achieve this, numerical simulations are performed to demonstrate the effects of different fractional orders on the development and evolution of chaotic behavior.

In this study, we consider a topological derivative-based imaging technique for the fast identification of short, linear perfectly conducting cracks completely embedded in a two-dimensional homogeneous domain with smooth boundary. Unlike conventional approaches, we assume that the background permittivity and permeability are unknown due to their dependence on frequency and temperature, and we propose a normalized imaging function to localize cracks. Despite inaccuracies in background parameters, application of the proposed imaging function enables recognition of the existence of cracks, but it is still impossible to identify accurate crack locations. Furthermore, the shift in crack localization of imaging results is significantly influenced by the applied background parameters. In order to theoretically explain this phenomenon, we show that the imaging function can be expressed in terms of the zero-order Bessel function of the first kind, the crack lengths, and the applied inaccurate background wavenumber corresponding to the applied inaccurate background permittivity and permeability. Various numerical simulation results with synthetic data polluted by random noise validate the theoretical results.

This manuscript was dedicated to exploring the existence and uniqueness of global solutions pertaining to a specific class of stochastic pseudo-parabolic equations. These equations have logarithmic nonlinearity and are driven by Brownian motion. By utilizing the Galerkin method, Prokhorov’s theorem, and Skorohod’s embedding theorem, we proved the existence of a global solution in the weak probabilistic sense. Subsequently, by leveraging the uniqueness of solutions and applying the Yamada-Watanabe theorem, the global existence and uniqueness of a probability strong solution were established. A notable finding, in contrast to the stochastic heat equation, was that as the pseudo- parabolic coefficient µ increased, the growth condition imposed on the noise coefficient within the stochastic pseudo-parabolic equation could be significantly relaxed.