Logarithmic Wave Equation Research Articles

On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up

This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder fixed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.

Existence and exponential decay of a logarithmic wave equation with distributed delay

Existence and exponential decay of a logarithmic wave equation with distributed delay

Logarithmic Wave Equation Involving Variable-exponent Nonlinearities:well-posedness and Blow-up

Logarithmic Wave Equation Involving Variable-exponent Nonlinearities:well-posedness and Blow-up

Local existence, global existence and decay results of a logarithmic wave equation with delay term

In this article, we deal with a logarithmic wave equation with strong damping and delay. Firstly, we prove the local existence by utilizing the semigroup theory. Later, we obtain the global existence of solutions by using the well‐depth method. Moreover, under appropriate assumptions on the weight of the delay and that of strong damping, we get the exponential decay.

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.

Global nonexistence for logarithmic wave equations with nonlinear damping and distributed delay terms

This paper is concerned with global nonexistence of solutions for a logarithmic wave equation with nonlinear damping and distributed delay terms. Due to the simultaneous presence of nonlinear damping and logarithmic source terms, we have difficulty in use of the concavity method. Applying the energy estimates, we show the global nonexistence of solutions with not only non-positive initial energy but also positive initial energy.

Logarithmic wave-mechanical effects in polycrystalline metals: theory and experiment

Schrodinger-type wave equations with logarithmic nonlinearity occur in hydrodynamic models of Korteweg-type materials with capillarity and surface tension, which can undergo liquid-solid or liquid-gas phase transitions. One of the predictions of the theory is a periodic pattern of density inhomogeneities occurring in the form of either bubbles (topological phase), or cells (non-topological phase). Such inhomogeneities are described by solitonic solutions of a logarithmic wave equation, gaussons and kinks, in the vicinity of the liquid-solid phase transition. During the solidification process, these inhomogeneities become centers of nucleation, thus shaping the polycrystalline structure of the metal grains. The theory predicts a Gaussian profile of material density inside such a cell, which should manifest in a Gaussian-like profile of microhardness inside a grain. We report experimental evidence of large-scale periodicity in the structure of grains in the ferrite steel S235/A570, copper C-Cu/C14200, austenite in steel X10CrNiTi18-10/AISI 321, and aluminium-magnesium alloy 5083/5056; and also Gaussian-like profiles of microhardness inside an averaged grain in these materials.

Acoustic oscillations in cigar-shaped logarithmic Bose–Einstein condensate in the Thomas–Fermi approximation

We consider the dynamical properties of density fluctuations in the cigar-shaped Bose–Einstein condensate described by the logarithmic wave equation with a constant nonlinear coupling by using the Thomas–Fermi and linear approximations. It is shown that the propagation of small density fluctuations along the long axis of a condensed lump in a strongly anisotropic trap is essentially one-dimensional, while the trapping potential can be disregarded in the linear regime. Depending on the sign of nonlinear coupling, the fluctuations either take the form of translationally symmetric pulses and standing waves or become oscillations with varying amplitudes. We also study the condensate in an axial harmonic trap, by using elasticity theory’s notions. Linear particle density and energy also behave differently depending on the nonlinear coupling’s value. If it is negative, the density monotonously grows along with lump’s radius, while energy is a monotonous function of density. For the positive coupling, the density is bound from above, whereas energy grows monotonously as a function of density until it reaches its global maximum.

Soliton solutions of logarithmic wave equation and their application for polycrystalline metals

Wave equations with logarithmic nonlinearity are applied to Korteweg-type materials which can undergo liquid-solid or liquid-gas phase transitions. One of predictions of the theory is a periodical pattern for inhomogeneities of density, which can occur in the form of bubbles or cells. Such inhomogeneities are described by soliton and solitary wave solutions of the logarithmic wave equation in the vicinity of a liquid-solid phase transition. During the solidification process, these inhomogeneities become centers of nucleation of grains. Previous works were dealing with generic natural silicate materials in geophysics, such as magmas in volcanic conduits, where the (approximately) periodical flows and structures were observed. Here we report an experimental evidence of a large-scale periodicity in structure of grains in the structural steel S235/A570 Grade 36, copper C-Cu/C14200, stainless steel X10CrNiTi18-10/AISI 321, and aluminium-magnesium alloy 5083/5056.

Logarithmic viscoelastic wave equation in three-dimensional space

ABSTRACT The initial-boundary value problem of logarithmic viscoelastic wave equation in three space dimensions is studied. The existence of local and global solutions for this problem are proved by means of the contraction mapping criterion and potential well method. Meanwhile, the blow-up of solutions in the unstable set is also obtained.

Matrix logarithmic wave equation and multi-channel systems in fluid mechanics

We formulate the mapping between a large class of nonlinear wave equations and flow equations for barotropic fluid with internal surface tension and capillary effects. Motivated by statistical mechanics and multi-channel physics arguments, we focus on wave equations with logarithmic nonlinearity, and further generalize them to matrix equations. We map the resulting equation to flow equations of multi-channel or multi-component Korteweg-type materials. For some special cases, we analytically derive Gaussian-type matrix solutions and study them in the context of fluid mechanics.

Stability and Metastability of Trapless Bose-Einstein Condensates and Quantum Liquids

AbstractVarious kinds of Bose-Einstein condensates are considered, which evolve without any geometric constraints or external trap potentials including gravitational. For studies of their collective oscillations and stability, including the metastability and macroscopic tunneling phenomena, both the variational approach and the Vakhitov-Kolokolov (VK) criterion are employed; calculations are done for condensates of an arbitrary spatial dimension. It is determined that that the trapless condensate described by the logarithmic wave equation is essentially stable, regardless of its dimensionality, while the trapless condensates described by wave equations of a polynomial type with respect to the wavefunction, such as the Gross-Pitaevskii (cubic), cubic-quintic, and so on, are at best metastable. This means that trapless “polynomial” condensates are unstable against spontaneous delocalization caused by fluctuations of their width, density and energy, leading to a finite lifetime.

Abstract Cauchy problem for weakly continuous operators

We discuss the solvability of the abstract Cauchy problem u′(t)=Au(t) for t>0 and u(0)=u0∈D, where D is a weakly closed subset of a Banach space Y continuously embedded in the underlying Banach space X and A is a nonlinear operator from D into X satisfying a local weak continuity condition described by a family of functionals. We introduce a new type of weak compactness on the level sets of functionals and give a necessary and sufficient condition for the existence of a weakly continuously differentiable solution satisfying a growth condition. We apply the abstract result obtained to the Cauchy problem for a nonlinear Schrödinger equation and a mixed problem for a logarithmic wave equation.

Exponential Decay of Energy for a Logarithmic Wave Equation

In this paper we consider the initial boundary value problem for a class of logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the logarithmic wave equation with linear damping and some suitable initial data. The results extend the early results.

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR A LOGARITHMIC WAVE EQUATION ARISING FROM Q-BALL DYNAMICS

In this paper we investigate an initial boundary value prob- lem of a logarithmic wave equation. We establish the global existence of weak solutions to the problem by using Galerkin method, logarithmic Sobolev inequality and compactness theorem. In this paper we study the global existence of weak solutions for the initial boundary value problem 8 utt − �u + u − ulogjuj 2 + ut + juj 2 u = 0, (x,t) 2 � (0,T), u = 0, (x,t) 2 @ � (0,T), u(x,0) = u 0 (x), ut(x,0) = u 1 (x), x 2 ,

Volume element structure and roton-maxon-phonon excitations in superfluid helium beyond the Gross-Pitaevskii approximation

We propose a theory which deals with the structure and interactions of volume elements in liquid helium II. The approach consists of two nested models linked via parametric space. The short-wavelength part describes the interior structure of the fluid element using a non-perturbative approach based on the logarithmic wave equation; it suggests the Gaussian-like behaviour of the element's interior density and interparticle interaction potential. The long-wavelength part is the quantum many-body theory of such elements which deals with their dynamics and interactions. Our approach leads to a unified description of the phonon, maxon and roton excitations, and has noteworthy agreement with experiment: with one essential parameter to fit we reproduce at high accuracy not only the roton minimum but also the neighboring local maximum as well as the sound velocity and structure factor.

Quantum Bose liquids with logarithmic nonlinearity: self-sustainability and emergence of spatial extent

The Gross–Pitaevskii (GP) equation is a long-wavelength approach widely used to describe the dilute Bose–Einstein condensates (BEC). However, in many physical situations, such as higher densities, it is unlikely that this approximation suffices; hence, one might need models which would account for long-range correlations and multi-body interactions. We show that the Bose liquid described by the logarithmic wave equation has a number of drastic differences from the GP one. It possesses the self-sustainability property: while the free GP condensate tends to spill all over the available volume, the logarithmic one tends to form a Gaussian-type droplet—even in the absence of an external trapping potential. The quasi-particle modes of the logarithmic BEC are shown to acquire a finite size despite the bare particles being assumed to be point-like, i.e. the spatial extent emerges here as a result of quantum many-body correlations. Finally, we study the elementary excitations and demonstrate that the background density changes the topological structure of their momentum space which, in turn, affects their dispersion relations. Depending on the density, the latter can be of the massive relativistic, massless relativistic, tachyonic and quaternionic type.