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In this review, we extend the boundary control method – an approach to inverse problems based on control theory for dynamical systems – to inverse problems for discrete dynamical systems. We apply our results to classical moment problems, Toda lattices, Weyl functions, de Branges spaces, Krein–Stieltjes strings, and also to problems of numerical simulations. [ABSTRACT FROM AUTHOR]

Let M α be the spherical maximal operators of complex order α on R n . In this article we show that when n ≥ 2 , suppose ‖ M α f ‖ L p (R n) ≤ C ‖ f ‖ L p (R n) holds for some α and p ≥ 2 , then we must have that Re α ≥ max { 1 / p - (n - 1) / 2 , - (n - 1) / p }. In particular, when n = 2 , we prove that ‖ M α f ‖ L p (R 2) ≤ C ‖ f ‖ L p (R 2) if Re α > max { 1 / p - 1 / 2 , - 1 / p } , and consequently the range of α is sharp in the sense that the estimate fails for Re α < max { 1 / p - 1 / 2 , - 1 / p }. [ABSTRACT FROM AUTHOR]

A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions. [ABSTRACT FROM AUTHOR]

This paper proves the qualitative properties like existence, uniqueness and Ulam-Hyers-Rassias stability for a solution of a nonlinear impulsive Hilfer fractional stochastic differential equation with non-instantaneous impulses. The Banach fixed point theorem and Cauchy inequality are utilized for obtaining our results. The Ulam-Hyers-Rassias stability is presented under specific assumptions. To verify the theoretical results an example is presented at the end. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for semilinear classical wave equations u tt - Δ u = | u | p S (n) μ (| u |) with the Strauss exponent p S (n) and a modulus of continuity μ = μ (τ) , which provides an additional regularity of nonlinearities in u = 0 comparing with the power nonlinearity | u | p S (n) . We obtain a sharp condition on μ as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted L t ∞ L r ∞ estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply a conjecture for the critical regularity of source nonlinearities for semilinear classical wave equations. We verify this conjecture in the 3d case. [ABSTRACT FROM AUTHOR]

We prove the nonexistence of global solutions for the following wave equations with structural damping and nonlinear memory source term utt+(−Δ)α2u+(−Δ)β2ut=∫0t(t−s)δ−1∣u(s)∣pds{u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| u\left(s)| }^{p}{\rm{d}}s and utt+(−Δ)α2u+(−Δ)β2ut=∫0t(t−s)δ−1∣us(s)∣pds,{u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| {u}_{s}\left(s)| }^{p}{\rm{d}}s, posed in (x,t)∈RN×[0,∞)\left(x,t)\in {{\mathbb{R}}}^{N}\times \left[0,\infty ), where u=u(x,t)u=u\left(x,t) is the real-valued unknown function, p>1p\gt 1, α,β∈(0,2)\alpha ,\beta \in \left(0,2), δ∈(0,1)\delta \in \left(0,1), by using the test function method under suitable sign assumptions on the initial data. Furthermore, we give an upper bound estimate of the life span of solutions.

We consider a network of coupled strings that contains a cycle. We show that boundary feedback stabilization that uses only data of the state at boundary nodes that are not contained in the cycle is in general impossible. We demonstrate that also with control actuators that act at the interior nodes contained in the cycle with certain standard control laws that use only Neumann control action the situation does not improve and the system is still not stable. We prove that with an actuator that uses also Dirichlet control action and is located within the cycle, exponential stabilization is possible. [ABSTRACT FROM AUTHOR]

By the Klein-Gordon potential, we call a convolution-type integral with a kernel, which is the fundamental solution of the Klein-Gordon equation and also a solution of the Cauchy problem to the same equation. An interesting question having several important applications (in general) is what boundary condition can be imposed on the Klein-Gordon potential on the boundary of a given domain so that the Klein-Gordon equation with initial conditions complemented by this “transparent” boundary condition would have a unique solution within that domain still given by the Klein-Gordon potential. It amounts to finding the trace of the Klein-Gordon potential to the boundary of the given domain. In this article, we analyze this question and construct a novel initial boundary-value problem for the Klein-Gordon equation in characteristic coordinates.

In this paper, we pursue our series of papers aiming to show the applicability of the concept of very weak solutions. We consider a wave model with irregular position-dependent mass and dissipation terms, in particular, allowing for δ -like coefficients and prove that the problem has a very weak solution. Furthermore, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory. A special case of the model considered here is the so-called telegraph equation. [ABSTRACT FROM AUTHOR]

This note addresses the well-posedness of weak solutions for a general linear evolution problem on a separable Hilbert space. For this classical problem there is a well known challenge of obtaining a priori estimates, as a constructed weak solution may not be regular enough to be utilized as a test function. This issue presents an obstacle for obtaining uniqueness and continuous dependence of solutions. Utilizing a generic weak formulation (involving the adjoint of the system’s evolution operator), the classical reference (Ball in Proceedings of the American Mathematical Society 63:370-373, 1977) provides a characterization which makes equivalent well-posedness of weak solutions and generation of a C 0 -semigroup. On the other hand, the approach in (Ball in Proceedings of the American Mathematical Society 63:370-373, 1977) does not take into account any underlying energy estimate, and requires a characterization of the adjoint operator, the latter often posing a non-trivial task. We propose an alternative approach, when the problem is posed on a Hilbert space and admits an underlying “formal" energy estimate. For such a Cauchy problem, we provide a general notion of weak solution and through a straightforward observation, obtain that arbitrary weak solutions have additional time regularity and obey an a priori estimate. This yields weak well-posedness. Our result rests upon a central hypothesis asserting the existence of a “good" Galerkin basis for the construction of a weak solution. A posteriori, a C 0 -semigroup may be obtained for weak solutions, and by uniqueness, weak and semigroup solutions are equivalent. [ABSTRACT FROM AUTHOR]

In this essay, we investigate the blow-up scenario, global solution and propagation speed for a modified Camassa–Holm (MCH) equation both dissipation and dispersion in Sobolev space H s , p (R) , s ≥ 1 , p ∈ (1 , ∞) . First of all, by the mathematical induction of index s, we establish the precise blow-up criteria, which extends the result obtained by Gui et al. in article (Comm Math Phys 319: 731–759, 2013). Secondly, we derive the global existence of the strong solution of MCH equation both dissipation and dispersion. Eventually, the propagation speed of the equation is studied when the initial data are compactly supported. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b (a , b are positive integers) and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf ηϱ(x) > 0 is not required. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a system of coupled wave equations featuring boundary fractional damping applied to a portion of the domain. We first establish the well-posedness of the system, proving the existence and uniqueness of solutions through semi-group theory. While the system does not exhibit exponential stability, we demonstrate its strong stability. Furthermore, leveraging Arendt and Batty's general criteria and certain geometric conditions, we prove a polynomial rate of energy decay for the solutions. [ABSTRACT FROM AUTHOR]

Existence of global attractors for a structural-acoustic system, subject to restricted boundary dissipation, is considered. Dynamics of the acoustic environment is given by a linear 3-D wave equation subject to locally distributed boundary dissipation, while the dynamics on the (flat) structural wall is given by a 2D-Kirchhoff-Boussinesq plate equation, subject to linear dissipation and supercritical nonlinear restoring forces. It is shown that the trajectories of the dynamical system defined on finite energy phase space are attracted asymptotically to a global attractor. The main challenges of the problem are related to: (i) superlinearity of the elastic energy of the structural component, (ii) Boussinesq effects of internal forces potentially leading to a finite time blowing up solutions, (iii) partially-restricted boundary dissipation placed on the interface only. The resulting system lacks dissipativity along with the suitable compactness properties, both corner stones of PDE dynamical system theories [1]. To contend with the difficulties, a new hybrid approach based on a suitable adaptation of the so called "energy methods" [2-3] and compensated compactness [4] has been developed. The geometry of the acoustic chamber plays a critical role. [ABSTRACT FROM AUTHOR]

In this article we study the solution u(t, x) of the Cauchy problem of linear damped wave equation. We obtain the sufficient and necessary condition of the boundedness of the solution u(t, x) on the Triebel–Lizorkin space at a fixed time t. [ABSTRACT FROM AUTHOR]

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation. [ABSTRACT FROM AUTHOR]

Traveling wave solutions are a class of invariant solutions which are critical for shallow water wave equations. In this paper, traveling wave solutions for two perturbed KP-MEW equations with a local delay convolution kernel are examined. The model equation is reduced to a planar near-Hamiltonian system via geometric singular perturbation theorem, and the qualitative properties of the corresponding unperturbed system are analyzed by using dynamical system approach. The persistence of the bounded traveling wave solutions for the perturbed KP-MEW equations with delay is investigated. By using a criterion for the monotonicity of ratio of two Abelian integrals and Melnikov’s method, the existence of kink (anti-kink) wave solutions and periodic wave solutions of the model equation are established. The result shows that the delayed KP-MEW equations with positive perturbation and the one with negative perturbation exhibit completely diverse dynamical properties. These new findings greatly enrich the understanding of dynamical properties of the traveling wave solutions of perturbed nonlinear wave equations with local delay convolution kernel. Numerical experiments further confirm and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of a coupled wave system in one-dimensional space with thermal effects, which is coupled by velocities. We prove the well-posedness of the system by using Lumer-Phillips theorem, and then turn out the exponential stability of the system via the semigroup method. The main novelty of this paper is that the heat-affected system achieves exponential stability compared with the previous non-thermal system. [ABSTRACT FROM AUTHOR]

In this paper we highlight a type of hyperbolic equation relating the logarithmic source term with distributed delay and dynamic boundary condition. We get, under comfortable primary data is the weak solution to local existence. The results of the solutions were found using the Faydo–Galerkin method and Schoder’s fixed point theorem. Then, the minimum blow-up result was studied. Our work is an extension of some previous work. [ABSTRACT FROM AUTHOR]

In this paper, we study an analogue of Tricomi problem for a parabolichyperbolic equation of the first type. To solve the problem for the heat equation, we use properties of the spectral problem with a spectral parameter in the boundary conditions. We show the basicity of the spectral problem with the spectral parameter in the boundary conditions. The conditions for the correctness of the formulated Tricomi problem are found. [ABSTRACT FROM AUTHOR]

In this article, we investigate some new asymptotic behaviors of the solution for a two-component b-family equations. We first prove the persistence properties of the solution for Eq. (1.1) when the initial data decay logarithmically, algebraically at infinity with the power β ∈ (0 , ∞) . Subsequently, we obtain the infinite propagation of the solution to Eq. (1.1). If the initial data satisfy certain compact condition, then the nontrivial solution u of Eq. (1.1) immediately loses compactly supported. Meanwhile, the solution u decays exponentially as | x | → ∞ . [ABSTRACT FROM AUTHOR]

This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.

In this article, we study the ill-posedness of the viscous nonlinear wave equation for any polynomial nonlinearity in negative Sobolev spaces. In particular, we prove a norm inflation result above the scaling critical regularity in some cases. We also show failure of C k -continuity, for k the power of the nonlinearity, up to some regularity threshold. [ABSTRACT FROM AUTHOR]

In the previous paper (Ding et al. in J Differ Equ 385:183–253, 2024), for the 3D quasilinear wave equation - (1 + (∂ t ϕ) p) ∂ t 2 ϕ + Δ ϕ = 0 with short pulse initial data (ϕ , ∂ t ϕ) (1 , x) = (δ 2 - ε 0 ϕ 0 (r - 1 δ , ω) , δ 1 - ε 0 ϕ 1 (r - 1 δ , ω)) , where p ∈ N , 0 < ε 0 < 1 , under the outgoing constraint condition (∂ t + ∂ r) k ϕ (1 , x) = O (δ 2 - ε 0 - k max { 0 , 1 - (1 - ε 0) p } ) for k = 1 , 2 , the authors establish the global existence of smooth large solution ϕ when p > p c with p c = 1 1 - ε 0 . In the present paper, under the same outgoing constraint condition, when 1 ≤ p ≤ p c , we will show that the smooth solution ϕ may blow up and further the outgoing shock is formed in finite time. [ABSTRACT FROM AUTHOR]

The ability to deal with complex geometries and to go to higher orders is the main advantage of space-time finite element methods. Therefore, we want to develop a solid background from which we can construct appropriate space-time methods. In this paper, we will treat time as another space direction, which is the main idea of space-time methods. First, we will briefly discuss how exactly the vectorial wave equation is derived from Maxwell's equations in a space-time structure, taking into account Ohm's law. Then we will derive a space-time variational formulation for the vectorial wave equation using different trial and test spaces. This paper has two main goals. First, we prove unique solvability for the resulting Galerkin–Petrov variational formulation. Second, we analyze the discrete equivalent of the equation in a tensor product and show conditional stability, i.e., under a CFL condition. Understanding the vectorial wave equation and the corresponding space-time finite element methods is crucial for improving the existing theory of Maxwell's equations and paves the way to computations of more complicated electromagnetic problems. [ABSTRACT FROM AUTHOR]

In this paper we study Strichartz estimates for the half wave, the half Klein–Gordon and the Dirac Equations on compact manifolds without boundary, proving in particular for each of these flows local in time estimates both for the wave and Schrödinger admissible couples (in this latter case with an additional loss of regularity). The strategy for the proof is based on a refined version of the WKB approximation. [ABSTRACT FROM AUTHOR]

The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method's distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM's approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems. [ABSTRACT FROM AUTHOR]

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with a quasilinear wave equation with nonlinear damping and source terms of variable exponents-type acting in a part of the boundary. Under suitable conditions on the exponents and the initial data, we study the blow-up properties. Firstly, by using Faedo-Galerkin method and Banach-Fixed-Point Theorem, we establish the existence of a weak solution, under suitable assumptions on the variable exponents and the initial data. Secondly, we show a finite time blow-up with lower and upper bound as well. Next, an infinite time blow-up is proved under some conditions in the exponents and the initial data. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of quasi-periodic response solutions (i.e., solutions that are quasi-periodic with the same frequencies as forcing term) for a class of forced reversible wave equations with derivative nonlinearity. The forcing frequency ω ∈ R 2 would be Liouvillean which is weaker than the usual Diophantine and Brjuno conditions. The derivative nonlinearity in the equation also leads to some difficulty in measure estimate. To overcome it, we also use the Töplitz–Lipschitz property of vector field. The proof is based on an infinite dimensional Kolmogorov–Arnold–Moser theorem for reversible systems. [ABSTRACT FROM AUTHOR]

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations. [ABSTRACT FROM AUTHOR]

The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces. [ABSTRACT FROM AUTHOR]

In this article, we consider the Timoshenko beam system with a delay term in the internal fractional feedback in a bounded domain: ρ 1 φ tt - K φ x + ψ x = 0 , ρ 2 ψ tt - b ψ xx + K φ x + ψ + μ 1 ψ t + μ 2 ∂ t α , β ψ (x , t - τ) = 0. Under a hypothesis between the weight of the delay term in the fractional feedback and the weight of the term without delay, using the energy method combined with the Faedo–Galerkin procedure, we prove the global existence of the solutions. Under the imposed constraints on the weights of the feedbacks and the wave speeds, general decay results of the energy are proved via suitable Lyapunov functionals. [ABSTRACT FROM AUTHOR]

We consider wave equations with a nonlocal polynomial type of damping depending on a small parameter θ ∈ (0 , 1) . This research is a trial to consider a new type of dissipation mechanisms produced by a bounded linear operator for wave equations. These researches were initiated in a series of our previous works with various dissipations modeled by a logarithmic function published in (Charão et al. in Math Methods Appl Sci 44:14003-14024, 2021; Charão and Ikehata in Angew Math Phys 71:26, 2020; Piske et al. in J Diff Eqns 311:188-228, 2022). The model of dissipation considered in this work is probably the first defined by more than one sentence and it opens field to consider other more general. We obtain an asymptotic profile and optimal estimates in time of solutions as t → ∞ in L 2 -sense, particularly, to the case 0 < θ < 1 / 2 . [ABSTRACT FROM AUTHOR]

We study the motion of closed hypersurfaces with friction in central force fields. This kind of closed hypersurfaces can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The parabolic flow case was first established by Schnürer & Smoczyk (J. Reine. Angew. Math. 500 (2002) 77–95). The hyperbolic flow without central force fields and friction was studied by LeFloch & Smoczyk (J. Math. Pures Appl. 90 (2008) 591–614), and a general case by Notz (Comm. Pure. Appl. Math. 66 (2013) 790–819), and a dissipative model by Shao (Arch. Ration. Mech. Anal. 243 (2022) 501–557). In this paper, we consider a general case of the motion of closed hypersurfaces with friction. It also models the motion of elastic membrane with external friction and internal friction. A quasi-linear degenerate dissipative hyperbolic system including the mean curvature operator is introduced to describe it. Our main result shows that closed hypersurfaces converges smoothly to a uniquely determined smooth immersion (e.g. sphere) if the initial shape of elastic membrane being a small perturbation of this immersion. [ABSTRACT FROM AUTHOR]

In this paper, we provide high order hybrid asymptotic augmented finite volume schemes on a uniform grid for nonlinear weakly degenerate and strongly degenerate wave equations. The whole domain is divided into singular and regular subdomains by introducing an intermediate point. Puiseux series asymptotic technique is used in singular subdomain, and augmented numerical method is used in regular subdomain. The key to the method are the recovery of Puiseux series for the nonlinear degenerate wave equation in singular subdomain and the organic combination between the singular and regular subdomains by means of augmented variables related to singularity. In particular, through imposing a condition at the intermediate point, we can not only improve the accuracy of the augmented variables, but also avoid the restriction conditions when the mesh is divided in regular subdomain. The advantage of this method is that the global convergence order of the degenerate wave equation is determined by the augmented numerical scheme in regular subdomain. A rigorous error estimate is conducted for the solution of the degenerate wave equation. Numerical examples on weakly degenerate and strongly degenerate problems are provided to illustrate the effectiveness of the proposed method. Especially, we use the method to solve an interesting example of a degenerate wave equation with coefficient blow-up. [ABSTRACT FROM AUTHOR]

In this paper, we consider one-dimensional nonlinear piezoelectric beams with thermal and magnetic effects in the presence of a distributed delay term acting on the heat equation. First, we show that the system is well-posed in the sense of a semigroup. Through the construction of an appropriate Lyapunov functional, we establish a general decay result for the solutions of the system, for which the exponential and polynomial decays are only special cases, under a suitable assumption on the weight of the delay that the damping effect through heat conduction is strong enough to stabilize the system even in the presence of a time delay. Furthermore, our result does not depend on any relationship between system parameters. [ABSTRACT FROM AUTHOR]

The aim of this article is to give an alternative proof of Tataru's dispersive estimates for wave equations posed on the hyperbolic space. Based on the formula for the wave kernel on ℍ n , we give the proof from the perspective of Bessel potentials, by exploiting various facts about Gamma functions, modified Bessel functions, and Bessel potentials. This leads to our proof being more self-contained than that in [D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 2001, 2, 795–807]. [ABSTRACT FROM AUTHOR]