Logarithmic Nonlinearity Research Articles

A perturbation of the Cahn–Hilliard equation with logarithmic nonlinearity

Our aim in this paper is to study a perturbation of the Cahn–Hilliard equation with nonlinear terms of logarithmic type. This new model is based on an unconstrained theory recently proposed in [5]. We prove the existence, regularity and uniqueness of solutions, as well as (strong) separation properties of the solutions from the pure states, also in three space dimensions. We finally prove the convergence to the Cahn–Hilliard equation, on finite time intervals.

Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.

Blow-Up of Solution of Lamé Wave Equation with Fractional Damping and Logarithmic Nonlinearity Source Terms

In this work, by the use of a semigroup theory approach, we provide a global solution for an initial boundary value problem of the wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. In addition to this, we establish a blow-up result for the solution under the condition of non-positive initial energy.

Stability of ground states of nonlinear Schrodinger systems

In this article, we study existence and stability of ground states for a system of two coupled nonlinear Schrodinger equations with logarithmic nonlinearity. Moreover, global well-posedness is verified for the Cauchy problem in \(H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})\) and in an appropriate Orlicz space.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/76/abstr.html

Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity

Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity

Infinite-time blowup and global solutions for a semilinear Klein–Gordan equation with logarithmic nonlinearity

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein–Gordon equation characterized by logarithmic nonlinearity, specifically in the form . Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions and in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein–Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations.

Existence of a positive bound state solution for logarithmic Schrödinger equation

We investigate the existence of bound state solutions for the logarithmic Schrödinger equation−Δu+V(x)u=ulog⁡u2inRN,N≥1, where V(x)∈C(RN) has a limit at infinity. In the case when the ground state is not attained, we construct a higher critical value for the variational functional. The classical variational methods cannot be applied directly to deal with the above problem due to nonsmoothness. To recover the smoothness, we use the superlinear power-law perturbation of the logarithmic nonlinearity.

Initial boundary value problem of pseudo‐parabolic Kirchhoff equations with logarithmic nonlinearity

In this paper, we consider the initial boundary value problem for a pseudo‐parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite‐time blow‐up for the weak solutions with initial energy . When the initial energy , we find another criterion for the vanishing solution and blow‐up solution. We also establish the decay rate of the global solution and estimate the life span of the blow‐up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem.

Existence and Multiplicity of Solutions for a Mixed Local–Nonlocal System with Logarithmic Nonlinearities

Existence and Multiplicity of Solutions for a Mixed Local–Nonlocal System with Logarithmic Nonlinearities

Existence of solutions of a viscoelastic p(x)-Laplacian equation with logarithmic nonlinearity

Existence of solutions of a viscoelastic p(x)-Laplacian equation with logarithmic nonlinearity

Well‐posedness and asymptotic behavior for a p‐biharmonic pseudo‐parabolic equation with logarithmic nonlinearity of the gradient type

AbstractThis paper is concerned with the well‐posedness and asymptotic behavior for an initial boundary value problem of a pseudo‐parabolic equation with p‐biharmonic operator and logarithmic nonlinearity of the gradient type. The existence of the global weak solution is established by combining the technique of potential‐well and the method of Faedo–Galerkin approximation. Meantime, by virtue of the improved logarithmic Sobolev inequality and modified differential inequality, we obtain the results on infinite and finite time blow‐up and derive the lifespan of blow‐up solutions in various energy levels. Furthermore, the extinction phenomenon with extinction time is presented.

Analytical simulations of the Q-ball dynamics model in theoretical physics

We encounter scalar fields everywhere in nature, although some may seem fainter than in the early universe. The models of this field having the logarithmic non-linearity are considered in inflation cosmology and supersymmetric field theories especially Q-ball dynamics model, quantum mechanics and nuclear physics. Till now, numerical studies and theoretical analysis of the considered problems have been studied. Purpose of the study is to obtain the analytical solutions of the Q-ball dynamics model via the ansatz-based method. The results have been seen in the literature for the first time to the best of our knowledge. The resulting solutions are useful in interpreting the study of wave propagation and have many applications in physics and the multidisciplinary audience.

Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source term

We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity utt−Δut−Δpu+ut=|u|q−2uln|u|,x∈Ω,t>0 in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.

Global solution and blow-up for a class of Δγ-biharmonic parabolic equations with logarithmic nonlinearity

In this paper, we study a class of [Formula: see text]-biharmonic parabolic equations with logarithmic nonlinearity. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique and the Sobolev inequality, we obtain the existence of the unique global weak solutions. In addition, we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow-up of weak solutions.

WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR FOR THE DISSIPATIVE P-BIHARMONIC WAVE EQUATION WITH LOGARITHMIC NONLINEARITY AND DAMPING TERMS

This paper concerns with the initial and boundary value problem for a p-biharmonic wave equation with logarithmic nonlinearity and damping terms. We establish the well-posedness of the global solution by combining Faedo–Galerkin approximation and the potential well method, and derive both the polynomial and exponential energy decay by introducing an appropriate Lyapunov functional. Moreover, we use the technique of differential inequality to obtain the blow-up conditions and deduce the life-span of the blow-up solution.

Well-Posedness and Asymptotic Behavior for the Dissipative p-Biharmonic Wave Equation with Logarithmic Nonlinearity and Damping Terms

Well-Posedness and Asymptotic Behavior for the Dissipative p-Biharmonic Wave Equation with Logarithmic Nonlinearity and Damping Terms

Nonlocal Kirchhoff-type problem involving variable exponent and logarithmic nonlinearity on compact Riemannian manifolds

Nonlocal Kirchhoff-type problem involving variable exponent and logarithmic nonlinearity on compact Riemannian manifolds

Analysis of large time asymptotics of the fourth‐order parabolic system involving variable coefficients and mixed nonlinearities

In a general domain, we discuss an initial value problem for the symmetry generalized ‐coupled system of fourth‐order parabolic equations with space–time‐dependent diffusion coefficients and mixed nonlinearities. For the power‐type nonlinearities containing the sub‐critical Sobolev exponents, we achieve the local existence, a unique continuation, global existence, or finite‐time blow‐up of solutions to the system in time‐weighted Sobolev space. For the logarithmic nonlinearities containing the critical Sobolev exponents, we investigate the local existence in time‐weighted Sobolev space.

Well-posedness and Asymptotic Behavior for a Pseudo-parabolic Equation Involving $p$-biharmonic Operator and Logarithmic Nonlinearity

This paper deals with the well-posedness and asymptotic behavior for a pseudo-parabolic equation involving $p$-biharmonic operator and logarithmic nonlinearity under Navior boundary condition. By combining Galerkin approximation, the method of potential well, the technique of differential inequality and improved logarithmic Sobolev inequality, we establish the local and global solvability, infinite and finite time blow-up phenomena of weak solutions in different energy levels. Moreover, we obtain the growth rate of weak solutions, life span in different energy cases and also give a result of extinction phenomenon.

Well-posedness results for a class of nonlinear reaction-diffusion equations with memory

In this paper, we are interested to study the initial value problem for nonlinear heat equation with memory. For our problem, we show that the local existence theory related to the finite time blow-up is also obtained for the problem with logarithmic nonlinearity. We also obtain the blowup continuation property of the mild solution. The principal techniques based on some Sobolev embeddings with \(L^p\) spaces.