Exponential Nonlinearity Research Articles

(p,N) equations with critical exponential nonlinearities in [formula omitted

The paper deals with the existence of nontrivial solutions for (p,N) equations in RN with critical exponential growth. The main features and novelty of the paper are the (p,N) growth of the elliptic operator, combined with the double lack of compactness. In order to state a multiplicity result, we introduce a tricky step analysis based on the application of a completely new Brézis and Lieb type lemma for exponential nonlinearities.

Nonlinear Electrical Properties of ZnO-V2O5 Based Rare Earth (Er2O3) Added Varistors

The densification, microstructure and nonlinear electrical properties of ZnO-V2O5-MnO2-Nb2O5 varistor ceramics have been investigated with different amounts (0–2 mol.%) of Er2O3 addition. The ball milled powder mixtures have been sintered at 900°C, 1100°C and 1300°C for 1 h. The microstructure shows the presence of ZnO grains as the primary phase with secondary phases like Zn3(VO4)2, ErVO4, Zn4V2O9, Er- and Mn-rich in the intergranular layers. Generation of ErVO4 and Er-rich secondary phases at triple points and grain boundaries is found to inhibit grain growth resulting in the significant reduction of ZnO grain size, although this slightly diminishes the densification process. At a particular sintering temperature, the size of ZnO grains gets reduced over ten times when Er2O3 addition is increased from 0 mol.% to 2 mol.%. The nonlinear electrical property of the varistors improves with Er2O3 addition up to 0.5 mol.% and further addition (1–2 mol.%) does not appear to be beneficial in the chosen sintering temperatures. For varistors sintered at 1100°C, average ZnO grain size reduces from 10.8 μm in Er2O3 free sample to 7.2 μm in 0.5 mol.% Er2O3 added sample; this, in turn, significantly improves nonlinear electrical properties. The breakdown field increases from 1588 V cm−1 to 2585 V cm−1 with concurrent enhancement of nonlinear exponent value from 16 to 26 when Er2O3 concentration is raised from 0 mol.% to 0.5 mol.%.

Laminar Power-Law Fluid Flow in a T-Shaped Channel at Given Pressure Differences

The steady laminar flow of an incompressible power-law fluid in a plane T-shaped channel is studied at given pressure differences between the inflow and outflow sections. A non-Newtonian fluid, whose rheological behavior is governed by the Ostwald—de Waele law is called the power-law fluid. The mathematical formulation of the problem includes the equations of motion and continuity. The no-slip condition is imposed on solid walls. The problem solution is obtained using a finite-difference method invoking the SIMPLE procedure. A parametric investigation of the kinematic and dynamic flow parameters is carried out for different values of the relevant parameters of the problem. The flow regime map is constructed for different pressure differences between the inflow and outflow boundaries and different nonlinearity exponents of the rheological model.

Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

AbstractLetΩ⊂ ℝ2be a bounded domain with smooth boundary andb(x) > 0 a smooth function defined on∂Ω. We study the following Robin boundary value problem:$$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\ &u>0 &\quad& \text{in }{\it\Omega},\\ &\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &\quad& \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$whereνdenotes the exterior unit vector normal to∂Ω, 0 <λ< +∞ andp> 1 is a large exponent. We construct solutions of this problem which exhibit concentration asp→ +∞ and simultaneously asλ→ +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.

Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

Existence of Solutions to Elliptic Equation with Exponential Nonlinearities and Singular Term

Existence of Solutions to Elliptic Equation with Exponential Nonlinearities and Singular Term

Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity

We study the long-time behavior of the solutions of the partly dissipative reaction diffusion systems of the FitzHugh–Nagumo type with exponential growth nonlinearity. More precisely, we prove the existence of weak solutions, the regularity of the global attractor and the exponential stability of stationary solutions of the systems.

Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearityht=∇⋅(1|∇h|∇eδEδh)=∇⋅(1|∇h|∇e−∇⋅(∇h|∇h|)) where total energy E=∫|∇h| is the total variation of h. Using a logarithmic correction for total energy E=∫|∇h|ln⁡|∇h|dx and gradient flow structure with a suitable defined functional, we prove the one dimensional evolution variational inequality solution preserves a positive gradient hx which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity hxx+ to happen.

Positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term

We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension N involving the N-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any N>2.

Fragility and Cooperative dynamics correlations in GexSe1-x chalcogenide glasses

In our thermal studies correlation between Cooperativity and Angell's fragility index neatly delineates homogeneous floppy, heterogeneous isostatic, and semi-homogeneous stressed rigidity-phases of the GexSe1-x system. These phases are manifestation of network evolutions due to doping of Ge into selenium polymer chains. In present experiments/data analysis distinct power law exponents (positive and negative) in fragility and cooperative dynamics correlation are observed. Their explanations are provided for three phases across GexSe1-x glass series. Temperature scanning study by ramp method was performed and data was analyzed with Vogel-Fulcher type. Isothermal measurements were performed and data was analyzed with Kohlarausch stretched exponential type. However, the Kohlrausch-Williams-Watts (KWW) nonlinearity exponent β parametrizing stretched-exponential relaxations [~ exp.{−(t/τ)β}] does not discern between the three phases, in its covariance with mAngell. In present study theoretical models for confirmation the result for correlations among properties of glasses are being suggested.

Local existence and nonexistence for reaction-diffusion systems with coupled exponential nonlinearities

We study the reaction-diffusion system with coupled exponential nonlinearities{∂tu=Δu+ep1u+p2vin Ω×(0,T),∂tv=Δv+eq1u+q2vin Ω×(0,T),u(x,t)=v(x,t)=0on ∂Ω×(0,T),u(x,0)=u0(x),v(x,0)=v0(x)in Ω, where T>0, pi≥0 and qi≥0(i=1,2) with (p1,p2)≠(0,0) and (q1,q2)≠(0,0). The domain Ω is RN(N≥1) or a bounded domain in RN with C2 boundary and the initial functions u0 and v0 are nonnegative and measurable. For each (p1,p2,q1,q2), we obtain integrability conditions of (u0,v0) which explicitly determine the existence/nonexistence of a local in time nonnegative classical solution. For the existence result, we can take a wider class of initial functions than previously suggested. Our analysis can be applied to other nonlinearities including superexponential ones.

An Exponential Jerk System: Circuit Realization, Fractional Order and Time Delayed Form with Dynamical Analysis and Its Engineering Application

Simple dynamical systems are of interest always. In this paper, we propose a simple jerk system with one exponential nonlinearity. Dynamical properties of the proposed system are investigated. To show the practical realizability of the proposed system we implement the exponential jerk system using off the shelf components. Fractional order and time delays are considered as complex analysis patterns of nonlinear systems. We investigate the fractional order time delayed exponential jerk system. For numerical analysis we use the modified Adomian Decomposition Method. To show the engineering importance of the proposed system, we derive a pseudo random number generator based on it.Various test results are presented to show the randomness of the system.

Numerical simulation for homogeneous–heterogeneous reactions and Newtonian heating in the silver-water nanofluid flow past a nonlinear stretched cylinder

The present exploration aims to deliberate silver-water nanofluid flow with homogeneous–heterogeneous reactions and magnetic field impacts past a nonlinear stretched cylinder. The novelty of the presented work is enhanced with the addition of Newtonian heating, heat generation/absorption, viscous dissipation, nonlinear thermal radiation and joule heating effects. The numerical solution is established via Shooting technique for the system of ordinary differential equations with high nonlinearity. The influences of miscellaneous parameters including nanoparticles volume fraction magnetic parameter nonlinearity exponent curvature parameter conjugate parameter heat generation/absorption parameter radiation parameter Eckert number strength of homogeneous reaction strength of heterogeneous reaction and Schmidt number on axial velocity, temperature profile, local Nusselt number, and skin friction coefficient are discussed via graphical illustrations and numerically erected tabulated values. It is examined that the velocity field diminishes while the temperature profile enhances for mounting values of the magnetic parameter. An excellent concurrence is achieved when our obtained numerical calculations are compared with an already published paper in limiting case; hence dependable results are being presented.

Characterization of Pore Throat Size Distribution in Tight Sandstones with Nuclear Magnetic Resonance and High-Pressure Mercury Intrusion

Characterization of pore throat size distribution (PTSD) in tight sandstones is of substantial significance for tight sandstone reservoirs evaluation. High-pressure mercury intrusion (HPMI) and nuclear magnetic resonance (NMR) are the effective methods for characterizing PTSD of reservoirs. NMR T2 spectra is usually converted to mercury intrusion capillary pressure for PTSD characterization. However, the conversion is challenging in tight sandstones due to tiny pore throat sizes. In this paper, the linear conversion method and the nonlinear conversion method are investigated, and the error minimization method and the least square method are proposed to calculate the conversion coefficients of the linear conversion method and the nonlinear conversion method, respectively. Finally, the advantages and disadvantages of these two different conversion methods are discussed and compared with field case study. The research results show that the average linear conversion coefficients of the 20 tight sandstone core plugs collected from Yanchang Formation, Ordos Basin of China is 0.0133 μm/ms; the average nonlinear conversion coefficient is 0.0093 μm/ms and the average nonlinear conversion exponent is 0.725. Although PTSD converted from NMR spectra by the nonlinear conversion method is wider than that obtained from linear conversion method, the nonlinear conversion method can retain the characteristic of bi-modal distribution in PTSD.

Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states. Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system, the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.

Bäcklund Transformations for Nonlinear Differential Equations and Systems

In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases were considered. The BTs construction is based on the method proposed by Clairin. The solutions of the considered equations have been found using the BTs, with a unified algorithm. In addition, the work develops the Clairin’s method for the system of two third-order equations related to the integrable perturbation and complexification of the Korteweg-de Vries (KdV) equation. Among the constructed BTs an analog of the Miura transformations was found. The Miura transformations transfer the initial system to that of perturbed modified KdV (mKdV) equations. It could be shown on this way that, considering the system as a link between the real and imaginary parts of a complex function, it is possible to go to the complexified KdV (cKdV) and here the analog of the Miura transformations transforms it into the complexification of the mKdV.

Estimates of Decadal Climate Predictability From an Interactive Ensemble Model

AbstractDecadal climate predictability has received considerable scientific interest in recent years, yet the limits and mechanisms for decadal predictability are currently not well known. It is widely accepted that noise due to internal atmospheric dynamics at the air‐sea interface influences predictability. The purpose of this paper is to use the interactive ensemble (IE) coupling strategy to quantify how internal atmospheric noise at the air‐sea interface impacts decadal predictability. The IE technique can significantly reduce internal atmospheric noise and has proven useful in assessing seasonal‐to‐interannual variability and predictability. Here we focus on decadal timescales and apply the nonlinear local Lyapunov exponent method to the Community Climate System Model comparing control simulations with IE simulations. This is the first time the nonlinear local Lyapunov exponent has been applied to the state‐of‐the‐art coupled models. The global patterns of decadal predictability are discussed from the perspective of internal atmospheric noise and ocean dynamics.

Impact of exponential saturable nonlinearity on modulation instability in silicon-on-Insulator nano-waveguides

We present a theoretical study of modulation instability (MI) in silicon-on-insulator (SOI) nano-waveguides in the presence of exponential saturable nonlinearity (SNL). The study incorporates two-photon absorption and free-carrier effects in the normal dispersion regimes at telecommunication wavelength. Also, we have shown the impact of SNL on the power dependence of MI gain spectra, and carrier density. Further, the effects of carrier lifetime and fourth order dispersion on MI gain spectra have been discussed in the presence of SNL.

Nonclassical Symmetries of a System of Nonlinear Reaction-Diffusion Equations

We study the conditional symmetry of a system of nonlinear reaction-diffusion equations and establish the existence of operators of conditional symmetry for systems of nonlinear reaction-diffusion equations with any number of independent variables and find these operators in the explicit form. The exact solutions of nonlinear reaction-diffusion equations with exponential nonlinearity are constructed.

Entropy and renormalized solutions of anisotropic elliptic equations with variable nonlinearity exponents

The Dirichlet problem is considered in arbitrary domains for a class of second-order anisotropic elliptic equations with variable nonlinearity exponents and right-hand sides in . It is proved that an entropy solution exists in anisotropic Sobolev spaces with variable exponent. It is proved that the entropy solution obtained is a renormalized solution of the problem under consideration. Bibliography: 37 titles.