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

In this paper,the authors consider the stabilization and blow up of the wave equation with infinite memory,logarithmic nonlinearity and acoustic boundary conditions.The authors discuss the existence of global solutions for the initial energy less than the depth of the potential well and investigate the energy decay estimates by introducing a Lyapunov function.Moreover,the authors establish the finite time blow up results of solutions and give the blow up time with upper bounded initial energy.

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.

Two model equations with a second degree logarithmic nonlinearity and their Gaussian solutions

In the paper,we try to study the mechanism of the existence of Gaussian waves in high degree logarithmic nonlinear wave motions.We first construct two model equations which include the high order dispersion and a second degree logarithmic nonlinearity.And then we prove that the Gaussian waves can exist for high degree logarithmic nonlinear wave equations if the balance between the dispersion and logarithmic nonlinearity is kept.Our mathematical tool is the logarithmic trial equation method.

Dark incoherent spatial solitons in logarithmically saturable nonlinear media

This paper studies numerically the dark incoherent spatial solitons propagating in logarithmically saturable nonlinear media by using a coherent density approach and a split-step Fourier approach for the first time. Under odd and even initial conditions, a soliton triplet and a doublet are obtained respectively for given parameters. Simultaneously, coherence properties associated with the soliton triplet and doublet are discussed. In addition, if the values of the parameters are properly chosen, five and four splittings from the input dark incoherent spatial solitons can also form. Lastly, the grayness of the soliton triplet and that of the doublet are studied, in detail.

具有对数非线性项的伪抛物型p-Laplacian型方程的初边值问题

In this paper,we study the initial boundary value problem of pseudo-parabolic p-Laplacian type equation,which be use to model some important physical and biological phenomena.By using the potential well method,we obtain the global existence,asymptotic behavior and blow up results of weak solution with subcritical initial energy.Then we also extend these results to the critical initial energy.

Fractional Schrodinger Equations with Logarithmic and Critical Nonlinearities

In this paper,we study a class of the fractional Schrodinger equations involving logarithmic and critical nonlinearities.By using the Nehari manifold method and the concentration compactness principle,we show that the above problem admits at least one ground state solution and one ground state sign-changing solution.Moreover,by using variational methods,we prove that how the coefficient function of the critical nonlinearity affects the number of positive solutions.The main feature which distinguishes this paper from other related works lies in the fact that it is the first attempt to study the existence and multiplicity for the above problem involving both logarithmic and critical nonlinearities.