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In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function. [ABSTRACT FROM AUTHOR]

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation. [ABSTRACT FROM AUTHOR]

In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation. [ABSTRACT FROM AUTHOR]

Considering the importance of correctly understanding the dynamics of microstructure materials for their applications in related technologies, by eliminating the shortcomings and some overlooked physical concepts in the existing asymmetric elastic theories, we have presented an asymmetric elastodynamic model based on a $ U(1) $ U (1) gauge theory with quantum electrodynamics (QED) structure. Accordingly, we have shown that there is a correspondence between an elastic theory, which can explain the behavior of elastic waves within an asymmetric elastic medium and QED. More specific, we have indicated that the corresponding elastic wave equations are somehow analogous to QED ones. In this regard, by adding vibrational degrees of freedom and introducing a gauge property of the waves of displacement for the waves of rotation, we have generalized and modified the related Cosserat theory (CT) for an elastic environment. Thus on macro scales, the elastic waves can possess the QED treatment. This analogy provides a new paradigm of fermions and bosons. Also, from experimental point of view, we have shown that the behavior of elastic waves in a granular medium is equivalent to behavior of light in dispersive media, which can be explained using QED. Hence, contrary to the Cosserat and discrete models, this amended CT has qualitatively been indicated to be consistent with the corresponding empirical observations. [ABSTRACT FROM AUTHOR]

In this paper, analytic solutions and dynamic properties of a variety of Boussinesq-type equations are established via the complete discrimination system for polynomial method. All the existing single traveling wave solutions to these equations as well as some new solutions are shown, and the Hamiltonian and topological properties to these equations are also presented. Considering the significance of the Boussinesq-type equations, our results would have wide applications in ocean engineering and fluid mechanics, like describing and predicting the solitary and periodic waves in various shallow water models. [ABSTRACT FROM AUTHOR]

In the present work, we study the global existence and uniform decay rates of solutions to the initial-boundary value problem related to the dynamic behavior of evolution equations accounting for rotational inertial forces along with a linear time-nonlocal Kelvin-Voigt damping arising in viscoelastic materials. By constructing appropriate Lyapunov functional, we show that the solution converges to the equilibrium state exponentially in the energy space. [ABSTRACT FROM AUTHOR]

The solutions of the Schrὅdinger equation for a molecular Mobius square potential are obtained using two elegant traditional methods. In each case, the energy equation is obtained. The pure vibrational energies for seven different molecules are studied using their respective spectroscopic constants and the obtained energy equation. The present results are compared with the standard results and the results of other potential models. The present study shows that the molecular Mobius square potential produces better results compared to some potential models. [ABSTRACT FROM AUTHOR]

The paper studies the influence of the initial inhomogeneous thermal stresses on the dispersion of the axisymmetric longitudinal waves propagating in the bi-layered hollow cylinder within the scope of the 3D linearized theory of elastic waves in initially stressed bodies. The initial static thermal stresses are determined within the scope of the uncoupled classical linear theory of thermo-elasticity and it is assumed that the temperature change in the cylinder is caused by the heating or cooling of the inner and outer surfaces of the cylinder. The discrete analytical solution method is employed for the solution to the corresponding equations of the wave propagation. Concrete numerical results are obtained for the cylinder, the materials of the layers of which are steel and aluminum, as well for the cylinder made of steel only. Numerical results on the influence of the initial inhomogeneous thermal stresses on the dispersion curves related to the first, second and third modes are presented and discussed. In particular, it is established that the cooling or the heating of the outer and inner surfaces of the bi-layered hollow cylinder influences the dispersion curves not only in the quantitative sense, but also in the qualitative sense. [ABSTRACT FROM AUTHOR]

In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems. [ABSTRACT FROM AUTHOR]

We prove Strichartz estimates for solutions to Maxwell equations in three dimensions with rough permittivities, which have less than three different eigenvalues. To this end, Maxwell equations are conjugated to half-wave equations in phase space. We use the Strichartz estimates in a known combination with energy estimates to show the new well-posedness results for quasilinear Maxwell equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider an evolution linear control system. First, by using Weiss–Staffans perturbation results we prove that, under suitable conditions, our evolution system is well posed. Next, we give a characterisation of observability of the evolution system via some spectral condition. We also provide sufficient condition to ensure the stability of the system in the periodic case. As applications, we consider concrete problems including wave equation in one dimension. [ABSTRACT FROM AUTHOR]

Solutions of the wave equation are analyzed considering the propagation of electromagnetic fields through a non-homogeneous stratified metamaterial medium. These solutions are obtained by two methods: (1) Spectral Parameter Power Series (SPPS) and (2) SPPS with Operator Transmutation (OT). This last method is considered an innovation for this application. Numerical solutions are obtained for left-handed (LH) and right-handed (RH) media, which allow their reflectance and transmittance coefficients to be calculated. The performance of each method is evaluated by comparing their solutions with the exact ones available and with those obtained by the differential transfer matrix method (DTMM). The advantages and limitations of each method are indicated. [ABSTRACT FROM AUTHOR]

In this paper, the H $ {}_2 $ 2 N $ {}_2 $ 2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel $ t^{1-\gamma } $ t 1 − γ . Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When $ K_{1} \lt \frac {3}{2} $ K 1 < 3 2 , the convergence order is $ O(\tau ^{3-\gamma }+ h^{4}+ \varepsilon) $ O (τ 3 − γ + h 4 + ϵ) , where $ K_{1} $ K 1 is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme. [ABSTRACT FROM AUTHOR]

We study the small vibrations of an axially travelling string with a dashpot damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed $ \mathbf{v} $ v . For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower exponentially decaying estimates for the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case $ \mathbf{v}=0 $ v = 0. [ABSTRACT FROM AUTHOR]

We give an elementary new argument for global existence and exponential decay of solutions of quasilinear wave equations on Schwarzschild–de Sitter black hole backgrounds, for appropriately small initial data. The core of the argument is entirely local, based on time translation invariant energy estimates in spacetime slabs of fixed time length. Global existence then follows simply by iterating this local result in consecutive spacetime slabs. We infer that an appropriate future energy flux decays exponentially with respect to the energy flux of the initial data. [ABSTRACT FROM AUTHOR]

The harmonic acoustic waves in a semi-infinite functionally graded (FG) 1D rod with a longitudinal arbitrary inhomogeneity are analyzed by a combined method based on the modified Cauchy formalism and the exponential matrix method. The closed form dispersion equations for harmonic waves are constructed, yielding the implicit dispersion relations for acoustic waves in FG rods. For longitudinal inhomogeneity of polynomial type the corresponding dispersion relations are constructed explicitly. [ABSTRACT FROM AUTHOR]

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein-Gordon equation characterized by logarithmic nonlinearity, specifically in the form wlog |w|k. Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions w0 and w1 in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein-Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls. [ABSTRACT FROM AUTHOR]

Motivated by the L p − L q smoothing properties of heat semigroups on unbounded domains and the conditional stability of hyperbolic semigroups, we develop a unified approach toward the problems on the existence of periodic solutions to the evolution equation u ˙ = A u (t) + B g (u) (t). Our method is based on the analysis of (X , Y , φ) -stability of the semigroup (T (t)) t ≥ 0 generated by A, i.e. ‖ T (t) x ‖ Y ⩽ φ (t) ‖ x ‖ X , t>0, for certain couple of Banach spaces (X , Y) and real-valued function φ satisfying lim t → ∞ φ (t) = 0. Our theory covers both cases corresponding to L p − L q smoothing properties and the conditional stability of hyperbolic semigroups as well as some other important situations relating to the polynomial or exponential stability of semigroups. As illustrations for our theory, we give applications to the existence and uniqueness of periodic solutions to Navier–Stokes and damped wave equations. [ABSTRACT FROM AUTHOR]

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Bresse system that arises in the modeling of arched beams. It is given by a system of three coupled wave equations that reduces to the well-known Timoshenko model when the arch curvature is zero. In a context of nonlinear elastic foundation, we establish the existence of smooth finite-dimensional global attractors, by adding dissipation mechanism in only one of its equations. In addition, we study the uniform boundedness of longtime dynamics with respect to the curvature parameter. These results have not been considered for partially damped semilinear Bresse or Timoshenko systems. [ABSTRACT FROM AUTHOR]

In this work, we prove a comparison result for a general class of nonlinear dispersive unidirectional wave equations. The dispersive nature of one-dimensional waves occurs because of a convolution integral in space. For two specific choices of the kernel function, the Benjamin–Bona–Mahony equation and the Rosenau equation that are particularly suitable to model water waves and elastic waves, respectively, are two members of the class. We first prove an energy estimate for the Cauchy problem of the non-local unidirectional wave equation. Then, for the same initial data, we consider two distinct solutions corresponding to two different kernel functions. Our main result is that the difference between the solutions remains small in a suitable Sobolev norm if the two kernel functions have similar dispersive characteristics in the long-wave limit. As a sample case of this comparison result, we provide the approximations of the hyperbolic conservation law. [ABSTRACT FROM AUTHOR]

The classical Euler–Bernoulli beam model and Kirchhoff plate model are very useful on the macroscopic scale. In the context of Eringen's nonlocal elasticity theory, this article aims to develop the criteria of the applicability of nanoscale Euler–Bernoulli beam and Kirchhoff plate models based on the Timoshenko beam model and the Mindlin plate model via the wave propagation theory. The corresponding governing differential equations for the nanoscale Timoshenko beam and Mindlin plate are derived by the Hamilton's principle, and the dispersion equations of wave are then obtained. By applying Taylor expansion to the corresponding solutions of the dispersion equations, new criteria are developed, simultaneously taking into account the effects of nonlocal parameters and material properties. When the nonlocal parameter is set to zero, the present criteria may be readily degenerated to their macroscopic counterparts. According to the present criteria, this article systematically evaluates the existing studies in literature. Various works in literature did not consider the effect of the nonlocal parameter, and hence, failed to satisfy the application conditions of the Euler–Bernoulli beam and Kirchhoff plate models on the nanoscale. The work in this article is of scientific significance to various studies on nanostructures. [ABSTRACT FROM AUTHOR]

In this paper, we consider a transmission wave problem with Kelvin–Voigt damping in two embedded domains in R n . We construct a global Carleman estimate for the transmission strongly damped wave equation with discontinuous coefficients. With the new Carleman estimate, we prove the Lipschitz stability and uniqueness of the inverse problem of determining the coefficient of the zeroth-order term in the equation with a single measurement observed in any small domain located in the outer domain. [ABSTRACT FROM AUTHOR]

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument, and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions. [ABSTRACT FROM AUTHOR]

We study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and then Ionescu-Pausader, to model the nonlinear effects from the Einstein-Klein-Gordon equation in harmonic coordinates. We first go over a slightly simplified version of global existence based on LeFloch-Ma, and then derive the asymptotic behavior of the system. The asymptotics of the Klein-Gordon field consist of a modified phase times a homogeneous function, and the asymptotics of the wave equation consist of a radiation field in the wave zone and an interior homogeneous solution coupled to the Klein-Gordon asymptotics. We then consider the inverse problem, the scattering from infinity. We show that given the type of asymptotic behavior at infinity, there exist solutions of the system that present the exact same behavior. [ABSTRACT FROM AUTHOR]

Hyperbolic equations are used in several physical phenomena to describe dynamical processes where information propagates with a finite speed. Recently, the wave equations with damping terms turned out to be fundamental hyperbolic equations in certain branches of physics mainly scattering processes and fractal medium. The aim of the present study is double shooting, first to prove that damped quantum wave equations may be obtained using the notion of non-standard Lagrangians and second to show that linear and nonlinear damping terms may be obtained if the concept of 'two-occurrences of integrals' is used, hence reducing the damped quantum wave equation to the conventional quantum wave equation known as the Klein-Gordon equation. This study supports the idea of non-standard Lagrangians and its usefulness in the theory of partial differential equations. [ABSTRACT FROM AUTHOR]

Lump solutions are empirical rational function solutions found in all directions in space. One of the essential solutions to both linear and nonlinear partial differential equations is lump solutions. The current work studies a class of lump interaction phenomena to the generalized (3 + 1) -dimensional nonlinear-wave equation with time-dependent-coefficient. Variable-coefficient nonlinear partial differential equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counterparts constant-coefficient in many physical cases. The Hirota bilinear form is the fundamental concept that has been used to derive the novel lump-periodic and breather wave solutions. The acquired solutions are constructed using symbolic computations called Maple. The physical characteristics of the acquired solutions are shown with three-dimensional and contour plots in order to shed more light on the acquired novel solutions. [ABSTRACT FROM AUTHOR]

The investigation of an initial-boundary value problem for a fractional wave equation with space-dependent variable-order wherein the coefficients have a dependency on the spatial and time variables is the concern of this work. This type of variable-order fractional differential operator originates in the modelling of viscoelastic materials. The global in time existence of a unique weak solution to the model problem has been proved under appropriate conditions on the data. Rothe's time discretization method is applied to achieve that purpose. [ABSTRACT FROM AUTHOR]

Under Dirichlet boundary conditions, we consider here a new type of viscoelastic Petrovsky wave equation involving variable sources and memory term u t t + Δ 2 u − ∫ 0 t g (t − s) Δ 2 u (x , s) d s + | u t | m (.) − 2 u t = | u | p (.) − 2 u. We discuss the blow-up in finite time with arbitrary positive initial energy and suitable large initial values if p (.) and the relaxation function g satisfies some conditions. Employing a different method for higher bounded positive initial energy, not only finite time blow-up for solutions proved but also the lower and upper bounds for blowing up time are gotten. [ABSTRACT FROM AUTHOR]

In this research, with help of the bilinear finite element, superconvergence analysis is proposed for nonlinear viscoelastic wave equation with strong damping. Firstly, a temporal discrete approximate scheme is established, and assisted by the method of first hypothesis and then proof, the temporal errors are given. Secondly, a new linearized second-order fully discrete scheme is presented, by using the technique of Interpolation and Ritz-prejection combination in the error estimations and analysis process, rigorous proofs are provided for the unconditional superclose estimates in H 1 -norm with order O (h 2 + τ 2) , here h and τ denote mesh size and time step, respectively. Finally, recurring to numerical examples, the correctness of the theoretical analysis is further demonstrated. [ABSTRACT FROM AUTHOR]

Initial-boundary value problems for semilinear wave equations with damping term and Neumann boundary conditions on the exterior domain are investigated. In sub-critical and critical cases, blow-up and upper bound lifespan estimates of solutions to the problems which contain power nonlinearity | u | q , derivative nonlinearity | u t | p , combined nonlinearity | u t | p + | u | q , respectively, are derived by applying test function technique and iterative method. The novelty is that lifespan estimates of solutions are associated with the well-known Strauss conjecture and Glassey conjecture. [ABSTRACT FROM AUTHOR]

The current paper develops and analyzes a numerical scheme for the space–time fractional stochastic nonlinear diffusion wave equations. The implicit scheme is based on the matrix transform technique for discretizing the Riesz-space fractional derivative, (3 − α) -order approximation to the Caputo-fractional derivative in time and Taylor's series method to linearize the nonlinear source term, and has been successfully applied to solve a class of nonlinear fractional diffusion wave equation. We prove that the implicit scheme is convergent with β-order in space and (3 − α) order in time, respectively. The optimum error estimates in the temporal-spatial direction and unconditional stability of the implicit scheme have been theoretically investigated. Moreover, several specific numerical experiments confirm the consistency and high efficacy of the provided algorithms, which minimizes the computational costs. [ABSTRACT FROM AUTHOR]

In terms of piezoelectric class (6 mm) crystal semiconductor, the propagation of shear surface waves on the half-space surface under initial stresses is studied. The initial stresses and material parameters are hypothesized to exponentially vary along x − direction only. Besides, the velocity equations of SH waves can be obtained based on the boundary conditions and the kinematic equations of the initially stressed graded piezoelectric semiconductor material, as well as the traction-free boundary conditions. Based on the numerical results, we explore how the semiconductivity, initial stresses as well as material gradient index impact the dispersive curves. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multi-dimensional wave equation with delay and dynamic boundary conditions, related to the Kelvin–Voigt damping. By using the Faedo–Galerkin approximations together with some priori estimates, we prove the local existence of solution. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. Exponential stability result of system is also established by constructing suitable Lyapunov functionals. [ABSTRACT FROM AUTHOR]

In this paper, a damped semilinear wave equation with logarithmic non-linearity is considered. Finite time blow-up criteria are established for solutions with both lower and higher initial energy, and an upper bound for the blow-up time is derived for each case. Moreover, by making full use of the strong damping term, a lower bound for the blow-up time is also obtained, for both subcritical and supercritical exponent. This partially extends some recent results obtained in [Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Analysis, Real World Applications, 51(2020), 102968] by Di et al. [ABSTRACT FROM AUTHOR]

The Hilbert transform H is a useful tool in the mathematical analysis of time-dependent partial differential equations in order to prove coercivity estimates in anisotropic Sobolev spaces in case of a bounded spatial domain Ω, but an infinite time interval (0 , ∞). Instead, a modified Hilbert transform H T can be used if we consider a finite time interval (0 , T). In this note we prove that the classical and the modified Hilbert transformations differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval (0 , T) onto R is used. This result is important when we deal with space–time variational formulations of time-dependent partial differential equations, and for the implementation of related space–time finite and boundary element methods for the numerical solution of parabolic and hyperbolic equations with the heat and wave equations as model problems, respectively. [ABSTRACT FROM AUTHOR]

We consider the boundary controllability of the structural acoustic model with variable coefficients. The structural acoustic model is a coupled partial differential equation, which comprises an acoustic wave equation in the interior domain, a Kirchoff plate equation on the boundary portion, with the coupling being accomplished across a boundary interface. In this model, the wave propagation medium and the plate material are all inhomogeneous. By the Riemannian geometry theory and the multiplier technique, our paper derives the exact controllability with two boundary controls under some checkable conditions and the exact–approximate boundary reachability with only one control for the boundary Kirchoff plate equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of periodic solutions for the asymptotically linear wave equations. We will establish an index theory for the linear wave equation and define a relative Morse index to measure the difference between the linearizations of the nonlinearity at the origin and at the infinity. Taking advantage of a generalized linking theorem for the strong indefinite functionals, we obtain existence and multiplicity of periodic solutions and the nonlinearity is not assumed to be C 1 . [ABSTRACT FROM AUTHOR]

In this paper, a new three-level implicit and fourth-order compact conservative difference scheme is developed for the generalized symmetric regularized long wave equations. The proposed scheme adopts a new time discretization method. The conservation of the new compact scheme is proved, and the discrete conservation laws are obtained. The priori estimation in the maximum norm is discussed by the discrete energy method and mathematical induction. Based on the priori estimation and Brouwer's fixed point theorem, the unique existence of the difference solution is proved. It is proved that the compact difference scheme is unconditionally convergent and stable in the maximum norm, and its convergence order is the fourth order in space and the second order in time. A decoupled and linearized iterative algorithm is proposed to calculate the nonlinear algebraic system generated by the compact scheme, and its convergence is proved. Several numerical experiments are performed to prove the efficiency and reliability of the proposed numerical scheme and to confirm the results obtained from the theoretical analysis. [ABSTRACT FROM AUTHOR]

The initial-boundary value problem for a wave equation with nonlinear averaged damping and nonlocal nonlinear source term is considered. We provide the sufficient conditions of finite time blow-up for weak solutions with suitable conditions on initial data by an ordinary differential inequality for an appropriately chosen functional. [ABSTRACT FROM AUTHOR]

In this paper, we study asymptotic properties of the maximum likelihood estimator (MLE) for the speed of a stochastic wave equation. We follow a well-known spectral approach to write the solution as a Fourier series, then we project the solution to a N-finite dimensional space and find the estimator as a function of the time and N. We then show consistency of the MLE using classical stochastic analysis. Afterward, we prove the asymptotic normality using the Malliavin–Stein method. We also study asymptotic properties of a discretized version of the MLE for the parameter. We provide this asymptotic analysis of the proposed estimator as the number of Fourier modes, N, used in the estimation and the observation time go to infinity. Finally, we illustrate the theoretical results with some numerical experiments. [ABSTRACT FROM AUTHOR]

We consider the integral type problem for linear and nonlinear abstract wave equations with operator coefficient in E-valued L p , 1 ≤ p ≤ ∞ spaces. The equation includes an abstract linear operator A that defined in a Banach space E. Here, the existence, uniqueness, uniform estimates of solution and quality properties of integral wave problem are derived. We also obtain the same properties for solutions of mixed integral problem for nonlocal and anisotropic classes, by choosing the space E and operator A. [ABSTRACT FROM AUTHOR]

We investigate the variable-coefficient wave equation with nonlinear acoustic boundary conditions and logarithmic nonlinearity source term. In addition, the damping is nonlinear, and plays a role only in part of the boundary. Using the semigroup theory, we state the existence and uniqueness of the local solutions. Accordingly, we prove that under some assumptions on the initial data, the local solution can be extended to be global. Particularly, it has been shown that the decay rates of the system are given implicitly as solutions to a first-order ODE by applying the Riemannian geometry method. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear viscoelastic equation. The aim is to expand the class of the function of relaxation h (t) that ensuring a general decay. We adopt the following commonly condition on relaxation function h (t) : h ′ (t) ≤ − ξ (t) χ (h (t)) , where ξ is a nonincreasing function and χ is an increasing and convex function on the whole [ 0 , ∞) instead of is convex only near the origin in the literature. [ABSTRACT FROM AUTHOR]

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree q = 0 , 1 , are considered for temporal semi-discretization for second-order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In the aftermath of the deadly 2018 Anak Krakatau tsunami (Indonesia) and associated confusions over its modeling and generation mechanism, there has been an urgent need for further studies to improve our understanding of landslide-generated tsunamis. Two important factors in accurate modeling of landslide tsunamis are the wave period and the initial wave amplitude. Here, we apply a physical modeling approach and develop an empirical equation to predict the dominant wave period generated by solid-block subaerial landslide tsunamis. Fifty-one laboratory experiments are conducted at different water depths and using four different concrete blocks for the sliding masses. The results are consequently employed to derive a predictive equation for the wave period of solid-block subaerial landslide tsunamis. An innovation of this study is that we apply data from different scales (laboratory and field scales) to produce our predictive equation. For field data, the data from the 2018 Anak Krakatau event is used. We compared our predictive equation with other previously-published equations. To confirm the validity of our predictive equation, it is applied for the prediction of the wave period of an independent landslide tsunami event whose data was not used for the derivation of the equation. [ABSTRACT FROM AUTHOR]

We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (M , g) of dimension N ≥ 3 , without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when M = R 3 , our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported. [ABSTRACT FROM AUTHOR]

Although Strichartz estimates have been widely applied for the nonlinear dispersive equations like Schr o ¨ dinger equation and wave equation in Besov spaces, it is rare in nonlinear kinetic type equations such as Vlasov-Poisson equation. In this paper, we are concerned about the dispersive estimates for the kinetic transport equation in Besov spaces, which could be applied to establish Strichartz estimates in Besov space and further may be exploited to investigate the solutions for Vlasov-Poisson equation and Boltzmann equation in Besov space. [ABSTRACT FROM AUTHOR]

The present paper deals with the propagation of Rayleigh surface waves in a nonlocal thermoelastic orthotropic layer which is lying over a nonlocal thermoelastic orthotropic half space. The problem is considered in the context of Green-Naghdi type III model of hyperbolic thermoelasticity and Eringen's nonlocal elasticity theory in the presence of voids. The problem is solved using eigenfunction expansion method. Frequency equation of Rayleigh waves is derived and different particular cases are also deduced. The effects of voids and nonlocality on different characteristics of Rayleigh waves are presented graphically. [ABSTRACT FROM AUTHOR]