Exponential Nonlinearity Research Articles

Modified quasilinear equations with strongly singular and critical exponential nonlinearity

Modified quasilinear equations with strongly singular and critical exponential nonlinearity

Positive solutions to the planar logarithmic Choquard equation with exponential nonlinearity

In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11].

News Waves: Hard News, Soft News, Fake News, Rumors, News Wavetrains.

We discuss the spread of a piece of news in a population. This is modeled by SIR model of epidemic spread. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. The differential equation has an exponential nonlinearity and it can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. Exact solutions to these equations can be obtained by the Simple Equations Method (SEsM). Some of these exact solutions can be used to model a class of waves associated with the spread of the news in a population. The presence of exact solutions allow to study in detail the dependence of the amplitude and the time horizon of the news waves on the wave parameters, such as the size of the population, initial number of spreaders of the piece of the news, transmission rate, and recovery rate. This allows for recommendations about the change of wave parameters in order to achieve a large amplitude or appropriate time horizon of the news wave. We discuss five types of news waves on the basis of the values of the transmission rate and recovery rate-types A, B, C, D, and E of news waves. In addition, we discuss the possibility of building wavetrains by news waves. There are three possible kinds of wavetrains with respect of the amplitude of the wave: increasing wavetrain, decreasing wavetrain, and mixed wavetrain. The increasing wavetrain is especially interesting, as it is connected to an increasing amplitude of the news wave with respect to the amplitude of the previous wave of the wavetrain. It can find applications in advertising, propaganda, etc.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF TWO-PART DIFFERENTIAL EQUATIONS WITH EXPONENTIAL NONLINEARITY

For a two-term nonautonomous ordinary differential equation of the fourth order with an exponential nonlinearity of the form y(4) = α0p0(t)[1 + r(t)]eσy where α0 ∈ {−1,1}, σ ≠ 0, the function p0(t) is continuous or continuously differentiable and nonzero in some left neighbourhood of ω (ω ≤ +∞), r(t) is a continuous function such that limt↑ωr(t) = 0, the asymptotic behaviour at t↑ω of one class of Pω(Y0, λ0)-solutions is studied. For this equation, in [1], the necessary and sufficient conditions for the existence of such solutions were obtained in the case when λ0 ∈ R∖ {0, 1/2, 2/3, 1}. The proof of sufficient conditions for existence was carried out under some additional conditions that are quite strict. The aim of this paper is to improve the results obtained in [1] for sufficient conditions of existence. An attempt is made to extend the results of this paper to conditions that are less stringent. In contrast to [1], the proof of the main result in this paper assumes that there is a finite or infinite limit limt↑ω πω(t)q’(t). The equation under study is reduced to a system of equations for which it is necessary to determine the existence of solutions vanishing at infinity. This fact is established using the known results of [2]. The question of the number of solutions of the equation with the found asymptotic images is also solved.

Nonlocal hyperbolic Stokes system with variable exponent of nonlinearity

In this paper, we study the problem for a nonlinear hyperbolic Stokes system of the second order with an integral term.Sufficient conditions for the uniqueness of the weak solution of this problem are found in a bounded domain. The nonlinear term of the system contains a variable exponent of nonlinearity, which is a function of spatial variables.The problem is studied in ordinary Sobolev spaces and generalized Lebesgue spaces, which is quite natural in this case.

On nonlinear Sobolev equation with the Caputo fractional operator and exponential nonlinearity

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory.

Sign-changing solutions for a weighted Schrödinger–Kirchhoff equation with double exponential nonlinearities growth

This work is devoted to the existence of least energy nodal solutions for a nonlocal weighted Schrödinger–Kirchhoff problem, under boundary Dirichlet condition in the unit ball [Formula: see text] of [Formula: see text]. The nonlinearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. By using the constrained minimization in Nehari set, the quantitative deformation Lemma and degree theory results, we prove our existence result.

Effect of insulating dense pinning centers on pinning potential and transport critical current density of YBCO superconductors

Insulating phase included in the inter-granular region of a granular superconducting YBCO affects both the superfluid density and critical current density at the zero magnetic field. Even a highly dense population of about 40 % (weight%) of the BZO insulating particles is found to be unable to destroy the superconducting nature completely. We have studied current-voltage (IV) characteristics of several BZO of high percentage added YBCO bulk samples to extract superfluid density and critical current density. Change in the nonlinear exponent has been observed. Critical current density JcG is found to be very sensitive to inter-granular high density insulating BZO particles. The variation of the activation energy (U) with T is also investigated.

A globally convergent estimator of the parameters of the classical model of a continuous stirred tank reactor

In this paper we provide the first solution to the challenging problem of designing a globally convergent estimator for the parameters of the standard model of a continuous stirred tank reactor. Because of the presence of non-separable exponential nonlinearities in the system dynamics that appear in Arrhenius law, none of the existing parameter estimators is able to deal with them in an efficient way and, in spite of many attempts, the problem was open for many years. To establish our result we propose a novel procedure to obtain a suitable nonlinearly parameterized regression equation and introduce a radically new estimation algorithm—derived applying the Immersion and Invariance methodology—that is applicable to these regression equations. A further contribution of the paper is that parameter convergence is exponential and is guaranteed with weak excitation requirements. To achieve this remarkable property we rely on the utilization of a recently introduced parameter estimator that seamlessly combines a classical least-squares search with the dynamic regressor extension and mixing estimation procedure.

Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case

We study the nonlinear Schrödinger equation for the s-fractional p-Laplacian strongly coupled with the Poisson equation in dimension two and with p=2s, which is the limiting case for the embedding of the fractional Sobolev space Ws,p(R2). We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique.

Sign-changing solutions for Kirchhoff weighted equations under double exponential nonlinearities growth

Sign-changing solutions for Kirchhoff weighted equations under double exponential nonlinearities growth

Asymptotic Behavior and Classification of Solutions to Hartree Type Equations with Exponential Nonlinearity

Asymptotic Behavior and Classification of Solutions to Hartree Type Equations with Exponential Nonlinearity

Nonlinear transmission exponent for boundedness of solutions to a chemotaxis system with indirect signal production

This paper deals with the Neumann initial–boundary problem for a fully parabolic chemotaxis system with indirect signal production, and the size of phenotypic transmission exponent is identified to guarantee the global existence and boundedness of classical solutions to the problem with suitably regular initial data.

Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity

Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity

Investigation of Rheological and Geometric Properties Effect on Nonlinear Behaviour of Fluid Viscous Dampers

Global approval of the use of fluid viscous dampers to control the buildings response against dynamic loadings is growing. The idea behind incorporating additional dampers is that they will reduce most of the energy that is transmitted to the building during shaking event. The objective of this work is to identify and enhance the design parameters that control the nonlinear behaviour of fluid viscous damper subjected to sinusoidal excitation. For this, a numerical model of the flow inside the dissipater has been carried out based on finite volume method. A novel approach has been adopted to simulate elastic behaviour of the fluid, taking into account its compressibility by using the Murnaghan equation of state. A comparison between the calculations of the proposed model and the experimental tests was carried out. The model proved to be sufficiently accurate. A fluid flow analysis was then conducted to fully understand the internal mechanism of the damper. A parametric study was then performed by varying aspects such as dimensions, geometric relationships between components and fluid properties in order to better understand their effect on the non-linear behaviour of the device. The results highlight the relationship between the parameters governing the shear thinning behaviour of the fluid and the non-linearity exponent of the damper. This makes it possible to better control the non-linear behaviour of the device by selecting the appropriate silicone oil and the appropriate geometric dimensions of its components.

Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction

Abstract This paper is concerned with the following fractional (N/s)-Laplacian Choquard equation: ε N ⁢ ( - Δ ) N / s s ⁢ u + V ⁢ ( x ) ⁢ | u | N s - 2 ⁢ u = ε μ ⁢ ( 1 | x | N - μ ∗ F ⁢ ( u ) ) ⁢ f ⁢ ( u ) , x ∈ ℝ N , \varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)\lvert u\rvert^{\frac{N}{s}-2}u=% \varepsilon^{\mu}\Bigl{(}\frac{1}{\lvert x\rvert^{N-\mu}}\ast F(u)\Bigr{)}f(u)% ,\quad x\in{\mathbb{R}}^{N}, where ( - Δ ) N / s s {(-\Delta)_{N/s}^{s}} denotes the (N/s)-Laplacian operator, 0 < μ < N {0<\mu<N} , and V and f are continuous real functions satisfying some mild assumptions. Applying the weak growth conditions on the exponential critical nonlinearity f and without using the strictly monotone condition, we use some refined analysis and develop the arguments in the existing results to establish the existence of the ground state solution of the fractional (N/s)-Laplacian Choquard equation. Moreover, we also study the concentration phenomenon of the ground state solutions. As far as we know, our results seem to be new concerning the fractional (N/s)-Laplacian equation with the Choquard reaction.

Local/global existence analysis of fractional wave equations with exponential nonlinearity

In this paper, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension 1≤d≤3. Our proofs base on the analyticity of Mittag-Leffler functions, the framework of prior estimates and the type of exponential nonlinearity.

Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity

We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity f ( ρ ) = ρ σ f(\rho ) = \rho ^\sigma , where ρ = | ψ | 2 \rho =|\psi |^2 is the density with ψ \psi the wave function and σ > 0 \sigma >0 is the exponent of the semi-smooth nonlinearity. Under the assumption of H 2 H^2 -solution of the NLSE, we prove error bounds at O ( τ 1 2 + σ + h 1 + 2 σ ) O(\tau ^{\frac {1}{2}+\sigma } + h^{1+2\sigma }) and O ( τ + h 2 ) O(\tau + h^{2}) in L 2 L^2 -norm for 0 > σ ≤ 1 2 0>\sigma \leq \frac {1}{2} and σ ≥ 1 2 \sigma \geq \frac {1}{2} , respectively, and an error bound at O ( τ 1 2 + h ) O(\tau ^\frac {1}{2} + h) in H 1 H^1 -norm for σ ≥ 1 2 \sigma \geq \frac {1}{2} , where h h and τ \tau are the mesh size and time step size, respectively. In addition, when 1 2 > σ > 1 \frac {1}{2}>\sigma >1 and under the assumption of H 3 H^3 -solution of the NLSE, we show an error bound at O ( τ σ + h 2 σ ) O(\tau ^{\sigma } + h^{2\sigma }) in H 1 H^1 -norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional L 2 L^2 -stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of 0 > σ ≤ 1 2 0 > \sigma \leq \frac {1}{2} , and to establish an l ∞ l^\infty -conditional H 1 H^1 -stability to obtain the l ∞ l^\infty -bound of the numerical solution by using the mathematical induction and the error estimates for the case of σ ≥ 1 2 \sigma \ge \frac {1}{2} ; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.

Ground state solution for a weighted fourth-order Schrödinger equation with exponential growth nonlinearity

Ground state solution for a weighted fourth-order Schrödinger equation with exponential growth nonlinearity

A Decomposition for Borel Measures \(\mu \le \mathcal{H}^{s}\)

We prove that every finite Borel measure \( \mu \) in \( \mathbb R^N \) that is bounded from above by the Hausdorff measure \( \mathcal{H}^s \) can be split in countable many parts \( \mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \( \mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.