Logarithmic Nonlinearity Research Articles

Multi‐bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

AbstractIn this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity where the magnetic potential , is a parameter and the nonnegative continuous function has the deepening potential well. Using the variational methods, we obtain that the equation has at least multi‐bump solutions when is large enough.

Fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity: Well-posedness, blow up and asymptotic stability

Fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity: Well-posedness, blow up and asymptotic stability

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Normalized solutions for critical Choquard equations involving logarithmic nonlinearity in the Heisenberg group

In this paper, we consider the existence and multiplicity of normalized solutions for critical Choquard equations involving logarithmic nonlinearity in the Heisenberg group. Under suitable assumptions, combined with the truncation technique, the concentration‐compactness principle, and the genus theory, we obtain the existence and multiplicity of the normalized solutions in the ‐subcritical case. As far as we know, the result of the paper is completely new in the Euclidean case.

Novel soliton solutions and phase plane analysis in nonlinear Schrödinger equations with logarithmic nonlinearities

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations.

On the Cauchy problem for logarithmic fractional Schrödinger equation

We consider the fractional Schrödinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space corresponding to the quadratic form domain of the fractional Laplacian, the energy space, and a space contained in the operator domain of the fractional Laplacian. For this last case, a finite momentum assumption is made, and the key step consists in estimating the Lie commutator between the fractional Laplacian and the multiplication by a monomial.

Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void

Abstract This article is devoted to the global existence and extinction behavior of the weak solution to a fast diffusion p p -Laplace equation with logarithmic nonlinearity and special medium void. By applying energy estimates approach, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, the global existence result is proved and the sufficient conditions on the occurrence of the extinction and nonextinction phenomena are given.

Existence of solutions for the $(p,N)$-Laplacian equation with logarithmic and critical exponential nonlinearities

This paper deals with the following $(p,N)$-Laplacian equation with logarithmic and critical exponential nonlinearities. Precisely, we study the problem \begin{equation*} \begin{cases} -\Delta_p u -\Delta_N u = |u|^{q-2}u \ln|u|^2 + \lambda f(u) & \text{in }\Omega,\\ u=0 & \text{on }\partial \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a bounded domain, $N \geq 2$, $1< p< N< q$, $\lambda > 0$ is a positive real parameter. By applying variational methods, we obtain the existence of solutions.

On the fractional pseudo-parabolic p(ξ)-Laplacian equation

In this paper, we investigate the existence of a global solution of the initial pseudo-parabolic p(ξ)-Laplacian equation with logarithmic nonlinearity, using the Aubin-Lions-Simon lemma and logarithmic inequality.

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Global Existence and Some Qualitative Properties of Weak Solutions for a Class of Heat Equations with a Logarithmic Nonlinearity in Whole $${\mathbb {R}}^{N}$$

Global Existence and Some Qualitative Properties of Weak Solutions for a Class of Heat Equations with a Logarithmic Nonlinearity in Whole $${\mathbb {R}}^{N}$$

On a class of capillarity phenomenon with logarithmic nonlinearity involving $$\theta (\cdot )$$-Laplacian operator

On a class of capillarity phenomenon with logarithmic nonlinearity involving $$\theta (\cdot )$$-Laplacian operator

Extinction and non-extinction of solutions for nonlocal fractional [formula omitted]-Kirchhoff problem with logarithmic nonlinearity

In this paper, we discuss the extinction and non-extinction properties of solutions for a class of Kirchhoff type parabolic problems involving fractional p-Laplacian and logarithmic nonlinearity. Based on energy estimates, embedding theorems, and certain ordinary differential inequalities, the global existence of solutions, and the extinction and non-extinction properties of global weak solutions are obtained. The results of this paper extend and complement previous research findings on this type of equation.

Crease instability in Gent-Gent hyperelastic materials

We study the crease instability of an incompressible Gent-Gent elastomer under general loading conditions. The Gent-Gent model has two distinct characteristics: the logarithmic nonlinearity triggered by I2 energy term and the strain hardening due to limiting chain extensibility. By performing finite element analysis combined with an evaluation of surface tension, we investigate the effects of the two characteristics on the critical creasing onset and the creasing onset delayed by surface tension. The results show that uniaxial compression induces logarithmic nonlinearity that accelerates the critical creasing onset, whereas equibiaxial compression hastens the initiation of strain hardening that delays the critical creasing onset. In addition, logarithmic nonlinearity mitigates the delay in creasing onset caused by surface tension under the uniaxial condition, and strain hardening suppresses creasing in cooperation with surface tension under the equibiaxial condition. Conversely, under the plane strain condition, two characteristics have little effect on the creasing onset delayed by surface tension.

On a class of Kirchhoff problems with nonlocal terms and logarithmic nonlinearity

On a class of Kirchhoff problems with nonlocal terms and logarithmic nonlinearity

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions

Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions

Large Deviations of Stochastic Heat Equations with Logarithmic Nonlinearity

Large Deviations of Stochastic Heat Equations with Logarithmic Nonlinearity

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

Abstract In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term and a logarithmic nonlinear term. This equation has been conceptualized within the context of interactions in liquid-gas systems, particularly in the context of island formation. The primary challenge lies in the departure from the original Cahn-Hilliard equation, as we no longer maintain conservation of the spatial average mean of the order parameter. This departure introduces the complexity in establishing uniform estimates for the solutions of the approximated problems concerning the regularization parameter, as it may potentially result in finite-time blow-up.