NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS

In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈R×H^(1)(R^(N))to the general Kirchhoff problem-M■,satisfying the normalization constraint f_(R)^N u^2dx=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical.

From Dirac’s Aether to the Dirac Equation

In 1951, Dirac proposed a formalism for a Lorentz invariant Aether with a vacuum state that contains all possible velocity states at each space-time point. Dirac showed no explicit path from the Aether towards the Quantum Mechanics. In this paper, we demonstrate that Dirac’s proposed Aether can be described by a lattice of possible events in space-time built in the local Lorentz frame. The idealised case of single velocity state leads to the famous Dirac equation for a plane wave state and is compatible with quantum statistics. On the lattice, possible space-time events are connected by the Dirac spinors which provide the probability of observing an event. The inertial mass of a particle is shown to be equivalent to the density of possible events on the lattice. Variation of the lattice density of events modifies the metric and provides a space-time curvature leading to the Hilbert action associated with general relativity. In classical limit, the perturbation in the density of possible events of the Aether is proportional to the Newtonian gravitational potential.

Generally Unitary Solution to a System of Matrix Equations over the Quaternion Field

Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence of and the expression for the generally unitary solution of the system are derived.

N-Soliton Solutions of General Nonlinear Schrdinger Equation with Derivative

The bilinear equation of the genera/nonlinear Schrodinger equation with derivative （GDNLSE） and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrodinger equation with derivative （DNLSE） and its multisoliton solutions are given by reduction.

EXACT SOLUTIONS FOR GENERAL VARIABLE-COEFFICIENT KdV EQUATION

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.

Solution of general dynamic equation for nanoparticles in turbulent flow considering fluctuating coagulation

A new averaged general dynamic equation （GDE） for nanoparticles in the turbulent flow is derived by considering the combined effect of convection, Brownian diffusion, turbulent diffusion, turbulent coagulation, and fluctuating coagulation. The equation is solved with the Taylor-series expansion moment method in a turbulent pipe flow. The experiments are performed. The numerical results of particle size distribu- tion correlate well with the experimental data. The results show that, for a turbulent nanoparticulate flow, a fluctuating coagulation term should be included in the averaged particle GDE. The larger the Schmidt number is and the lower the Reynolds number is, the smaller the value of ratio of particle diameter at the outlet to that at the inlet is. At the outlet, the particle number concentration increases from the near-wall region to the near-center region. The larger the Schmidt number is and the higher the Reynolds num- ber is, the larger the difference in particle number concentration between the near-wall region and near-center region is. Particle polydispersity increases from the near-center region to the near-wall region. The particles with a smaller Schmidt number and the flow with a higher Reynolds number show a higher polydispersity. The degree of particle polydispersity is higher considering fluctuating coagulation than that without considering fluctuating coagulation.

GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS

A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.

GENERAL DECAY FOR A QUASILINEAR SYSTEM OF VISCOELASTIC EQUATIONS WITH NONLINEAR DAMPING

In this paper, we consider a system of coupled quasilinear viscoelastic equa- tions with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.

GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND NONLINEAR BOUNDARY DAMPING-SOURCE INTERACTIONS

A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions ira- posed on the source and the damping, we establish uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.

GLOBAL CLASSICAL SOLUTIONS TO THE 3-D ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL INITIAL ENERGY

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.

A New General Equation of State

A new general equation of state is presented, which can be used to express not only common cubic equations of state, but also quartic equations of state and so on. Main advantage of the new equation over the previous general equations is that it is in simple form, and is easy to manipulate mathematically.

Exact Solutions and Invariant Sets to General Reaction-Diffusion Equation

In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.

EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation−(a+b∫_(R^(3))|∇u|^(2))Δu+V(x)u=f(u)in R^(3),with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions.Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity,especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

Numerical method of the Riemann problem for two-dimensional multi-fluid flows with general equation of state

Based on the classical Roe method, we develop an interface capture method according to the general equation of state, and extend the single-fluid Roe method to the two-dimensional （2D） multi-fluid flows, as well as construct the continuous Roe matrix for the whole flow field. The interface capture equations and fluid dynamic conservative equations are coupled together and solved by using any high-resolution schemes that usually suit for the single-fluid flows. Some numerical examples are given to illustrate the solution of 1D and 2D multi-fluid Riemann problems.

Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.

CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

Dark and multi-dark solitons in the three-component nonlinear Schrodinger equations on the general nonzero background

We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrodinger systems, and require more in-depth studies in the future.

Solutions to the generalized Sylvester matrixequations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.

Blowup of Solutions to the Non-Isentropic Compressible Euler Equations with Time-Dependent Damping and Vacuum

This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data is assumed to be radially symmetric and the initial density contains vacuum, we obtain that classical solution, especially the density, will blow up on finite time. The results also reveal that damping can really delay the singularity formation.