Logarithmic Nonlinearity Research Articles

Dyonic matter equations, exact point-source solutions, and charged black holes in generalized Born-Infeld theory

We derive the equations of motion governing static dyonic matters, described in terms of two real scalar fields, in nonlinear electrodynamics of the Born--Infeld theory type. We then obtain exact finite-energy solutions of these equations in the quadratic and logarithmic nonlinearity cases subject to dyonic point-charge sources and construct dyonically charged black holes with relegated curvature singularities. In the case of quadratic nonlinearity, which is the core model of this work, we show that dyonic solutions enable us to restore electromagnetic symmetry, which is known to be broken in non-dyonic situations by exclusion of monopoles. We further demonstrate that in the context of k-essence cosmology the nonlinear electrodynamics models possess their own distinctive signatures in light of the underlying equations of state of the cosmic fluids they represent. In this context, the quadratic and logarithmic models are shown to resolve a density-pressure inconsistency issue exhibited by the original Born--Infeld model k-essence action function as well as by all of its fractional-powered extensions. Moreover, it is shown that the quadratic model is uniquely positioned to give rise to a radiation-dominated era in the early universe among all the polynomial models and other examples considered.

Improved growth estimate of infinite time blowup solution for a semilinear hyperbolic equation with logarithmic nonlinearity

In this short paper, we give an improved estimate of the infinite time blowup solution for the semilinear wave equation with logarithmic nonlinearity. Comparing the previous work, this improved growth estimate not only provides the information about the relationship between the power of the nonlinearity and the growth, but also gives a faster growth estimate. The proof of the main conclusions is carried out at the negative initial energy level, the sub-critical initial energy level, the critical initial energy level and the arbitrary positive initial energy level.

Variable exponent q(m)-Kirchhoff-type problems with nonlocal terms and logarithmic nonlinearity on compact Riemannian manifolds

Variable exponent q(m)-Kirchhoff-type problems with nonlocal terms and logarithmic nonlinearity on compact Riemannian manifolds

Global Existence and Extinction Behaviour for a Doubly Nonlinear Parabolic Equation with Logarithmic Nonlinearity

This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a series of ordinary differential inequalities, the global existence of the solution is obtained. Moreover, we give the sufficient conditions on the occurrence (or absence) of the extinction behaviour.

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Abstract We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) L^{\infty}_{t}(L^{\infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) L^{\infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

Logarithmic Harnack inequalities and gradient estimates for nonlinear $p$-Laplace equations on weighted Riemannian manifolds

In this paper, we prove the logarithmic Harnack inequalities for $L^p$-Log-Sobolev function on $n$-dimensional weighted Riemannian manifolds with $m$-Bakry-Émery Ricci curvature bounded below by $-K$ $(m\ge n, K\ge0)$. Under the assumption of nonnegative $m$-Bakry-Émery Ricci curvature, we obtain a global Li-Yau type gradient estimate and a Hamilton type estimate for the positive solutions to the weighted parabolic $p$-Laplace equation with logarithmic nonlinearity. As applications, the corresponding Harnack inequalities are derived.

General Decay Results for a Viscoelastic Euler–Bernoulli Equation with Logarithmic Nonlinearity Source and a Nonlinear Boundary Feedback

General Decay Results for a Viscoelastic Euler–Bernoulli Equation with Logarithmic Nonlinearity Source and a Nonlinear Boundary Feedback

Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic Nonlinearity of Variable Exponents

In this paper, a new pseudoparabolic equation with logarithmic nonlinearity of variable exponents is investigated. By using the energy functional and the classical potential well, we obtain the global existence and blow-up results of weak solutions with variable exponents. These results extend and improve some recent results in which the blow-up results were showed involving the logarithmic nonlinearity with variable exponents.

Hamiltonian formalism for nonlinear Schrödinger equations

We study the Hamiltonian formalism for second and fourth order nonlinear Schrödinger equations. In the case of the second order equation, we consider cubic and logarithmic nonlinearities. Since the Lagrangians generating these nonlinear equations are degenerate, we follow the Dirac–Bergmann formalism to construct their corresponding Hamiltonians. In order to obtain consistent equations of motion, the Dirac–Bergmann formalism imposes some set of constraints that contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of primary constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency condition. We show that for second order and fourth order nonlinear Schrödinger equations we only have primary constraints, and the form of nonlinearity or the order of derivatives does not change the constraint dynamics of the system. However, we observe that introducing new fields to treat higher derivatives in the Lagrangians of these equations changes the constraint dynamics, and secondary constraints are needed to construct a consistent set of Hamilton equations.

Uncertainty Relations in the Madelung Picture Including a Dissipative Environment.

In a recent paper, we have shown how in Madelung's hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment.

Initial boundary value problem for p-Laplacian type parabolic equation with singular potential and logarithmic nonlinearity

Initial boundary value problem for p-Laplacian type parabolic equation with singular potential and logarithmic nonlinearity

Fractional Schrödinger Equations with Logarithmic and Critical Nonlinearities

Fractional Schrödinger Equations with Logarithmic and Critical Nonlinearities

Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

Logarithmic Harnack inequalities and differential Harnack estimates for p-Laplacian on Riemannian manifolds

In this paper, we study logarithmic Harnack inequalities and differential Harnack estimates for p-Laplacian on Riemannian manifolds. We prove the logarithmic Harnack inequalities for Lp-log-Sobolev functions on Riemannian manifolds with Ricci curvature bounded below, which is related to the Lp-log-Sobolev constant. We obtain a new Li-Yau type differential Harnack estimate for the positive solution to p-Laplacian parabolic equation with logarithmic nonlinearity. These results generalize the works of Chung-Yau, F.-Y. Wang and X. Cao etc. for the classic L2 case.

On the p-Laplacian type equation with logarithmic nonlinearity: existence, decay and blow up

This work is deal with a problem of wave equation with p-Laplacian, strong damping and logarithmic source terms under initial-boundary conditions. The global existence of weak solution was proved for related to the equation. Global existence results of solutions are obtained using the potential well method, Galerkin method and compactness approach corresponding to the logarithmic source term. Besides, we established the energy functional decaying polynomially to zero as the time goes to infinity due to Nakao?s inequality and some precise priori estimates on logarithmic nonlinearity. For suitable conditions we proved the finite time blow up results of solutions. The proof is based on the concavity method, perturbation energy method and differential-integral inequality technique. Additionally, under suitable assumptions on initial data, the infinite time blow up result is investigated with negative initial energy.

Existence of global solutions and blow-up results for a class of p(x)−Laplacian heat equations with logarithmic nonlinearity

This paper?s main objective is to examine an initial boundary value problem of a quasilinear parabolic equation of non-standard growth and logarithmic nonlinearity by utilizing the logarithmic Sobolev inequality and potential well method. Results of global existence, estimates of polynomial decay, and blowing up of weak solutions have been obtained under certain conditions that will be stated later. Our results extend those of a recent paper that appeared in the literature.

Global existence and blow-up of solution to a class of fourth-order equation with singular potential and logarithmic nonlinearity

In this paper, we consider the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a fourth-order equation with strong damping and logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the method of Faedo–Galerkin approximation. By means of potential well method and Rellich inequality, we obtain the global existence and the decay estimate of global solutions under some appropriate conditions. Furthermore, we prove the finite time blow-up results of weak solutions, and establish the upper and lower bounds for blow-up time.

Numerical study of the logarithmic Schrödinger equation with repulsive harmonic potential

We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the nonlinear Schrödinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.

Delayed wave equation with logarithmic variable-exponent nonlinearity

<abstract><p>A delayed nonlinear wave equation with variable exponents of logarithmic type is discussed in this paper. In the presence of the logarithmic nonlinear source, we established a global existence result under sufficient conditions on the initial data only without imposing the Sobolev Logarithmic Inequality. After that, we established global results of exponential and polynomial types according to the range values of the exponents. At the end, we give a numerical study that supports our theoretical results.</p></abstract>

Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity

Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity