Decay Of Solutions Research Articles

Testing by wave packets and modified scattering in nonlinear dispersive pde’s

Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely. An interesting example is that of problems with cubic nonlinearities in one space dimension. The method of testing by wave packets was introduced by the authors as a tool to efficiently capture the asymptotic equations associated to such flows, and thus establish the modified scattering mechanism in a simpler, more efficient fashion, and at lower regularity. In these expository notes we describe how this method can be applied to problems with general dispersion relations.

Dispersive decay of small data solutions for the KdV equation

Dispersive decay of small data solutions for the KdV equation

Energy decay analysis for Porous elastic system with thermoelasticity of type III: A second spectrum approach

Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green–Naghdi law of heat conduction into our study. To begin with, we use the Faedo–Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.

Mechanistic insights into the AIE properties of BF2 complexes of N-acetyl 2-aminobenzothiazoles as rotorless polyheteroaromatics

A series of four novel BF2 complexes of N-acetyl 2-aminobenzothiazoles featuring rigid rotorless polyheteroaromatic structures have been designed and synthesized. All of these compounds exhibited AIE properties with excellent solid-state fluorescence quantum yields up to 81.7%. Theoretical calculations revealed the characteristics of their ground-state structures and excitation processes. Their seemingly abnormal nonradiative decay in solutions was investigated from both direct vibrational relaxation and S0/S1 surface crossing perspectives by the two-channel model. The minimum energy conical intersection (MECI) located by SF-TDDFT method indicated that the CI is mainly induced by the deformation of the BF2 complex ring, leads to elevated energy level, and thus is not responsible for the fluorescence quenching. The potential energy curve (PEC) constructed along the distortion angle θ of BF2 complex ring showed that when θ reaches 50° (the dark-state region approaching MECI), the energy barrier is only 0.11 eV, suggesting that the deformed dark state could serve as an effective deactivation channel via ultrafast internal conversion (IC). Crystallographic analysis illustrated that the F atoms perpendicular to the polyheterocycle could effectively prohibit detrimental π-π stacking and enhance intermolecular interactions, thereby resulting in efficient solid-state emission.

Quasilinear wave equations on Schwarzschild–de Sitter

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.

Stability and rate of decay for solutions to stochastic differential equations with Markov switching

In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results. For more information see https://ejde.math.txstate.edu/Volumes/2024/01/abstr.html

Luminescence mechanism analysis of a TADF molecule showing peculiar thermal behavior.

In recent years, much attention has been paid to the development of thermally activated delayed fluorescence (TADF) materials with short delayed-fluorescence lifetimes to improve the device performances of OLEDs. In principle, by reducing the highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) overlap, while the energy difference between S1-T1 (ΔEST) and activation energy (Ea) can be reduced, and the reverse intersystem crossing rate constant (kRISC) can be accelerated, a decrease in the radiative rate constant happens, necessitating an advanced molecular design. Furthermore, a molecule based on heptazine as a parent skeleton has recently been found to have a peculiar temperature dependence of luminescence decay, suggesting a negative gap (NG) material. In this report, we show that 9-[1,4]benzoxaborino[2,3,4-kl]phenoxaborine-7-yl-1,3,6,8-tetramethyl-9H-carbazole (TMCz-BO), a donor-acceptor linked TADF molecule with a very short delay lifetime of 750 ns, exhibits a peculiar thermal behavior similar to that of NG materials based on the temperature dependence of its luminescence decay in solution.

Изменение структуры, механических и коррозионных свойств сплава системы Mg–Zn–Zr, подвергнутого равноканальному угловому прессованию

Magnesium alloys are considered promising materials for the production of bioresorbable implants. Their main disadvantages are low strength and corrosion resistance in biological environment. In the work, the authors studied the effect of severe plastic deformation using the equal channel angular pressing (ECAP) method on the structure, mechanical properties, and corrosion resistance of the Mg–8.6Zn–1.2Zr magnesium alloy. It was identified that one ECAP cycle at 400 °C leads to a substantial hardening of the Mg–8.6Zn–1.2Zr alloy by ~10 %, up to 330 MPa. Structural studies showed that dynamic recrystallisation plays a significant role in the structure transformation. ECAP leads to the formation of a bimodal structure with large deformed grains with an average transverse size of 20±4 µm and recrystallised grains with an average transverse size of 6±2 µm. It was found that with a decrease in the strain temperature up to 250 °С, the process of deformation-induced decay of the supersaturated solid solution takes place. Electrical conductivity of a sample after ECAP at 400 °C amounted 29±2 % according to the International Annealed Copper Standard (IACS), while second ECAP cycles lead to an increase in the electrical conductivity up to 32±2 % IACS. Using the electrochemical corrosion method, the authors found that one ECAP cycle at 400 °C leads to a slight decrease in the corrosion resistance of the alloy under study compared to the initial state. The study showed that the corrosion current increases from 24 to 32 µA/cm2, while the subsequent ECAP cycle at 250 °С increases the corrosion current more than twice (up to 57 µA/cm2).

Energy decay and blow-up of solutions to a coupled non-degenerate Kirchhoff-Heat system with source term

Energy decay and blow-up of solutions to a coupled non-degenerate Kirchhoff-Heat system with source term

An approximation method for exact controls of vibrating systems with numerical viscosity

We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell’s “stabilizability implies controllability” principle and a finite elements method of order θ with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order θ for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.

Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity

Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity

Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source

Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source

Decay of solutions for second gradient viscoelasticity with type Ⅱ heat conduction

Decay of solutions for second gradient viscoelasticity with type Ⅱ heat conduction

Exponential Decay of Solutions of Damped Wave Equations in One Dimensional Space in the Lp Framework for Various Boundary Conditions

We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.

Local Exponential Stabilization of Rogers–McCulloch and FitzHugh–Nagumo Equations by the Method of Backstepping

In this article, we study the exponential stabilization of some one-dimensional nonlinear coupled parabolic-ODE systems, namely Rogers–McCulloch and FitzHugh–Nagumo systems, in the interval (0, 1) by boundary feedback. Our goal is to construct an explicit linear feedback control law acting only at the right end of the Dirichlet boundary to establish the local exponential stabilizability of these two different nonlinear systems with a decay e−ωt, where ω ∈ (0, δ] for the FitzHugh–Nagumo system and ω ∈ (0, δ) for the Rogers–McCulloch system and δ is the system parameter that presents in the ODE of both coupled systems. The feedback control law, derived by the backstepping method forces the exponential decay of solution of the closed-loop nonlinear system in both L2(0, 1) and H1(0, 1) norms, respectively, if the initial data is small enough. We also show that the linearized FitzHugh–Nagumo system is not stabilizable with exponential decay e−ωt, where ω > δ.

Analysis of Functional and Neutral Differential Equations via Lyapunov Functionals

We employ Lyapunov types functions and functionals and obtain sufficient conditions that guarantee the boundedness and the exponential decay of solutions, stability and exponential stability of the zero solution in nonlinear delay and neutral differential systems. The theory is illustrated with several examples. 2000 Mathematics Subject Classification: Primary 39A13, 39A23; Secondary 34K42

Propagation of regularity and decay of solutions to nonlinear dispersive equations

Propagation of regularity and decay of solutions to nonlinear dispersive equations

Asymptotic analysis of two thermoelastic plates with dissipative histories

In this paper we study the time decay of the solutions for the problems determined by two plates where the dissipation mechanisms are given by the history of the material. To be precise we consider the thermo-viscolestic plate with heat conduction of the Green-Naghdi type II and the thermoelastic plate when the heat equation is described by the history-dependent Moore-Gibson-Thompson equation. In both cases we prove the well-posedness of the problems by means of semigroup theory. In the first case we also prove that the solutions decay in an exponential way by means of Prüss characterizations of exponential stable semigroups. In the second case we prove that the solutions decay in a polynomial way with optimal rates of decay, which is proved by Tomilov-Borichev characterizations of polynomial stable semigroups.

Long-time behavior of stochastic Hamilton-Jacobi equations

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion of noise speeds up the decay of solutions, and (ii) the long-time convergence of solutions to spatially inhomogeneous stochastic Hamilton-Jacobi equations. A number of motivating examples about nonlinear stochastic partial differential equations are presented in the appendix.

Axion inflation in the strong-backreaction regime: decay of the Anber-Sorbo solution

Axion inflation coupled to Abelian gauge fields via a Chern-Simons-like term of the form ϕFF~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\phi F\\overset{\\sim }{F} $$\\end{document} represents an attractive inflationary model with a rich phenomenology, including the production of magnetic fields, black holes, gravitational waves, and the matter-antimatter asymmetry. In this work, we focus on a particular regime of axion inflation, the so-called Anber-Sorbo (AS) solution, in which the energy loss in the gauge-field production provides the dominant source of friction for the inflaton motion. We revisit the AS solution and confirm that it is unstable. Contrary to earlier numerical works that attempted to reach the AS solution starting from a regime of weak backreaction, we perform, for the first time, a numerical evolution starting directly from the regime of strong backreaction. Our analysis strongly suggests that, at least as long as one neglects spatial inhomogeneities in the inflaton field, the AS solution has no basin of attraction, not even a very small one that might have been missed in previous numerical studies. Our analysis employs an arsenal of analytical and numerical techniques, some established and some newly introduced, including (1) linear perturbation theory along the lines of ref. [1], (2) the gradient expansion formalism (GEF) developed in ref. [2], (3) a new linearized version of the GEF, and (4) the standard mode-by-mode approach in momentum space in combination with input from the GEF. All these methods yield consistent results confirming the instability of the AS solution, which renders the dynamics of axion inflation in the strong-backreaction regime even more interesting than previously believed.