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In Fritz John's famous paper (1979), he discovered that for the wave equation $\Box u=|u|^p$, where $1, Comment: 11 pages, 3 figures

In this short paper, we extend the result of Shinbrot (1974) to an incompressible fluid with shear dependent viscosity. It is shown that a weak solution to the equations of non-Newtonian fluids lying in a space L q ( 0 , T ; L p ) satisfies an energy equality, where 2 r r − 1 ≤ p ≤ 2 r r − 2 and 1 p + 1 q ≤ r − 1 r , if r > 2 ; p ≥ 2 r r − 1 and r − 1 p + 1 q = r − 1 2 , if 2 ( n + 1 ) n + 2 r ≤ 2 . In particular, our result implies that the weak solution must satisfy the energy equality when r ≥ 3 n + 2 n + 2 , which is consistent with the known fact.

In this short paper, we consider the formation of singularities for the solutions of the Euler–Poisson equations with damping with attractive forces in R n ( n ≥ 1 ) and the repulsive forces in R . Under the appropriate assumptions on initial data, we can obtain the divergence of the velocity will blowup in the finite time.

In this short paper, the Cauchy problem for the Le Roux system is studied and the global weak solutions are obtained by using a new technique from the div–curl lemma in the compensated compactness theorem, where the regular B V estimate is assumed for one characteristic field. This work extends the previous work, (LeVeque and Temple, 1985), which provided the global solutions when both the characteristic fields are B V bounded by using the Godunov’s method.

In this short paper, we focus on the dependence on initial data of stochastic Camassa–Holm equation. The results show that the nonlinear noise will have an impact on the property of solutions with respect to initial data, but the linear noise will have no effect.

The stability problem on perturbations near the hydrostatic equilibrium of the Boussinesq equations has recently attracted considerable attention and there are substantial developments. This paper establishes the global H 1 -stability for a 2D Boussinesq system with partial dissipation. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. This paper assesses the stability of the temperature gradient when there is a horizontal thermal diffusion.

In this paper, a complex cubic–quintic Ginzburg–Landau equation (CCQGLE) is investigated. Using the asymmetric method, the analytic one-soliton solution of the CCQGLE is obtained for the first time. Through analyzing the solutions obtained, the transmission of the soliton is controlled by changing the values of related parameters. Results of this paper contribute to obtain the analytic soliton solution of the higher-order CCQGLE.

This paper formulates a new switched two species chemostat model and discusses the coexistence behavior in the chemostat. A complete classification on the single-species chemostat is carried out firstly, where the stationary distribution with ergodicity is derived to exist and be unique. Then, based on the obtained stationary distribution and the comparison theorem, we put forward some sufficient conditions for the coexistence of microorganisms in the two species chemostat with Markov switchings. Moreover, when the species coexist in the deterministic chemostat for each state and have the same break-even concentrations for all states, they are proved to coexist still in the switched chemostat, which randomized the results of the classical deterministic chemostat. Results in this paper show that Markov switchings can contribute to coexistence of the two species.

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.

In this paper, we remove the restriction A ′ ( x ) ≥ 0 in the paper ‘Lu (2011)’, the restriction z 0 ( x ) ≤ 0 or w 0 ( x ) ≤ 0 in the paper ‘Klingenberg and Lu (1997)’, and obtain the global existence of entropy solutions to the isothermal gas dynamics system in a divergent nozzle with friction.

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana–Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana–Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial–boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.

The primary work of this paper is to investigate twin iterative positive solutions of a class of fractional q -difference Schrodinger equations appeared in Li et al. (2015). By employing a completely different approach—monotone iterative approach, this paper not only establishes excellent conditions to ensure the existence of the twin iterative positive solutions, but also constructs two convergent monotone explicit iterative schemes to seek them. The obtained theorems are well illustrated with the aid of an example.

In this paper, we consider the wave equation of variable coefficients with acoustic boundary conditions in domains with nonlocally reacting boundary. This work is devoted to prove the energy decay for the wave equation of variable coefficients having some boundary reacting conditions. This paper is to improve our previous result in Ha (2016) by applying the variable coefficients case.

Considering the complexity of random variations in ecosystem, a n-species Gilpin-Ayala competition system with nonlinear noises is studied in this paper. Using our developed ϵ -stochastic criterion method in eliminating nonlinear perturbations, we construct several suitable Lyapunov functions to obtain the threshold for the existence and uniqueness of an ergodic stationary distribution, which reflects population coexistence in a long term. Moreover, we establish the sufficient conditions for population extinction. Finally, several numerical simulations are performed and our analytical results are discussed by comparison with the existing papers.

In Jung and Roh (2017), we investigated some properties of approximate solutions of the second-order inhomogeneous linear differential equations. In this paper as an application of above paper we will study the stability of the time independent Schrodinger equation with a potential box of finite walls.

This paper deals with the study of a nonlinear eigenvalue problem driven by a new class of non-homogeneous differential operators with variable exponent and involving a nonlinear term with variable growth. The framework in the present paper corresponds to the case of small perturbations of the nonlinear term. Combining variational arguments with energy estimates, we establish the existence of eigenvalues in a neighborhood of the origin. Our abstract setting includes several models described by nonhomogeneous differential operators, including the case of the capillarity equation.

This paper gives the upper bounds of blow-up time and blow-up rate for a semi-linear edge-degenerate parabolic equation, and the results extend the results of a recent paper Chen and Liu (2016).

In this paper, we consider a stochastic Kawasaki disease model, which is perturbed by both white noise and color noise. The sufficient condition for the existence of a unique ergodic stationary distribution is established by using Khasminskii’s theorem and constructing stochastic Lyapunov functions with regime switching. A brief summary is given at the end of this paper.

This paper deals with the study of combined effects of logarithmic and critical nonlinearities for the following class of fractional p -Kirchhoff equations: M ( [ u ] s , p p ) ( − Δ ) p s u = λ | u | q − 2 u ln | u | 2 + | u | p s ∗ − 2 u in Ω , u = 0 in R N ∖ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, N > s p with s ∈ ( 0 , 1 ) , p ≥ 2 , p s ∗ = N p ∕ ( N − p s ) is the fractional critical Sobolev exponent, and λ is a positive parameter. The main result establishes the existence of nontrivial solutions in the case of high perturbations of the logarithmic nonlinearity (large values of λ ). The features of this paper are the following: (i) the presence of a logarithmic nonlinearity; (ii) the lack of compactness due to the critical term; (iii) our analysis includes the degenerate case, which corresponds to the Kirchhoff term M vanishing at zero.

A Kirchhoff-type parabolic equation with logarithmic nonlinearity is studied in this paper. By employing the potential well theory and some differential inequality techniques, a new blow-up condition, the upper bound of the blow-up time, and the lower bound of the growth rate of blow-up solutions are obtained. The effect of the parameters a , b , θ in the assumption of Kirchhoff function M is simultaneously considered, while Ding and Zhou only considered the effect of the parameters b , θ . Thus, the main result in the present paper improves a recent blow-up result.

In this paper, a novel D2Q9 lattice Boltzmann model with BGK operator (LBGK) is proposed for the incompressible time-fractional Navier–Stokes equations with Caputo-type fractional derivative. First the fractional derivative is divided into the history part and the local part, in which the former is approximated using the efficient algorithm for the evaluation of the Caputo fractional derivative, while the latter is simply approximated by rectangle formula to keep the time-dependent characteristics of Navier–Stokes equations as the evolution equations. Then, a LBGK model is constructed for Navier–Stokes equations after approximation of the Caputo derivative. Through Chapman–Enskog analysis the macroscopic equations can be recovered from this model in the small Mach number limit. At the end of this paper, a numerical example with analytic solutions is carried out to show that the LBGK model is efficient.

In this paper, we consider the Cauchy problem of the evolution p -Laplacian equation, and reveal the fact that there exists an equivalence relation between the ω -limit set of solutions and the ω -limit set of initial values. This relation can be used to prove different asymptotic behaviors of solutions, and two examples are given at the end of this paper.

In this paper we investigate oscillations of conjoined bases of linear Hamiltonian differential systems related via symplectic transformations. Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for their Hamiltonians. The main result of the paper presents new explicit relations connecting the multiplicities of proper focal points of Y ( t ) and the transformed conjoined basis Y ( t ) = R − 1 ( t ) Y ( t ) , where the symplectic transformation matrix R ( t ) obeys some additional assumptions on the rank of its components. As consequences of the main result we formulate the generalized reciprocity principle for the Hamiltonian systems without normality. The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.

We investigate some properties of approximate solutions for the second-order inhomogeneous linear differential equations, y ′ ′ ( x ) + α y ′ ( x ) + β y ( x ) = r ( x ) , with complex constant coefficients. And, as an application of our results, we will see the time independent Schrodinger equations. This paper was motivated by the paper, Li and Shen (2010).

The aim of this paper is to obtain different criteria which ensure that the one parameter family of n th -order linear differential equations, given by the following expression (1) T n [ M ] u ( t ) ≡ u ( n ) ( t ) + a 1 ( t ) u ( n − 1 ) ( t ) + ⋯ + a n − 1 ( t ) u ′ ( t ) + ( a n ( t ) + M ) u ( t ) = 0 , t ∈ I ≡ [ a , b ] , is not disconjugate for every M ∈ R . Three different sufficient criteria, which warrant that such a property holds, are presented. Moreover, a characterization of this property is given. To finish the paper, there are shown three examples where the different criteria are applied.

This paper considers both quenching and blowup phenomena to the coupled parabolic equations with zero Dirichlet boundary. Here, one component of the solution represents the density of some chemical and the other denotes its temperature in some ignition process. All the singular phenomena have been obtained including simultaneous and nonsimultaneous blow-up or quenching, which are classified completely by the exponents. The results extend the ones in the previous paper (Liu and Chan, 2011).

In this paper, we consider the Kirchhoff type problem with radial potentials and locally L i p s c h i t z functional and there is no subcritical growth and no Ambrosetti–Rabinowitz type condition. This is in contrast with some recent papers concerning similar problems.

In this paper we present an intuitive explanation for the non-uniqueness of the traveling wave speed in the Fisher equation, showing a similar non-uniqueness property in the case of inviscid traveling waves. More precisely, we prove that traveling waves of the Fisher equation with wave speed c > 0 converge to the inviscid traveling wave with speed c > 0 as the diffusion vanishes. A complete diagram that shows the relation between the diffusive and inviscid traveling waves is given in this paper.

In this paper, we investigate the initial value problem for the three dimensional nematic liquid crystal flows with two directional viscosity. The main purpose of this paper is to establish global well-posedness of classical small solutions. The proof is mainly based on the energy estimate, anisotropic type Sobolev inequality and bootstrapping argument.

This paper is a continuous work of Liu et al. (2017, first order) and Xu et al. (2019, second order) for affine-periodic solutions to ordinary differential equations. It is a hard problem to obtain satisfied extremum principles. In this paper, we give several extremum principles (Theorem 2.1 and Lemma 2.2) for affine-periodic problems, especially for the case of higher order systems. By these extremum principles, we partly establish the existence of affine-periodic solutions for higher order ordinary differential equations.

In this paper, we construct some localized waves for the Bogoyavlenskii–Kadomtsev–Petviashili (BKP) and (3+1)-dimensional KP equations with the aid of the Hirota bilinear method and long wave limit. The BKP equation, as a modification of the KP equation, describes the shallow long waves and the (3+1)-dimensional KP equation can be applied to describe the propagation of nonlinear waves in fluids. Breather and breather-type kink soliton for the BKP equation are obtained from the two-soliton solutions; interactions between breather and kink soliton or between two breathers are obtained from the three-soliton solutions; interactions among one breather and two kink solitons, among two breathers and one kink soliton, or among three breathers are obtained from the four-soliton solutions. Furthermore, based on the explicit solutions with appropriate parameters, breathers of the (3+1)-dimensional KP equation have different behaviors in the different planes, and lump solutions are constructed via the long wave limit method. Results obtained in this paper may be helpful for understanding the propagation of localized waves.

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.

The main object of the present paper is to derive several sufficient conditions for close-to-convexity, starlikeness, and convexity of certain (normalized) analytic functions. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided.

It was shown in [1,2] that surface water waves in a water tunnel can be described by systems of the form η t +u x +(uη) x +au xxx −bη xxt =0, u t +η x +uu x +cη xxx −du xxt =0 , where a, b, c, and d are real constants. In this paper, we show that to find an exact traveling-wave solution of the system, it is suffice to find a solution of an ordinary differential equation, and the solution of the ordinary differential equation in a prescribed form can be found by solving a system of nonlinear algebraic equation. The exact solutions for some of the systems are presented at the end of the paper.

In this paper, we study the Cauchy problem of the Cahn–Hilliard equation, and first reveal that the complicated asymptotic behavior of solutions can happen in high-order parabolic equation.

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term − Δ u + b → ( x ) ⋅ ∇ u + V ( x ) u = H v ( x , u , v ) i n R N , − Δ v − b → ( x ) ⋅ ∇ v + V ( x ) v = H u ( x , u , v ) i n R N . Under a local super-quadratic condition on the nonlinearity, we obtain a new existence result of nontrivial solutions by using variational method. Here we do not need the global super-quadratic condition, and in this case our result allows the nonlinearity to be super-quadratic at some domains and asymptotically quadratic at other domains. In the proofs we apply some special techniques to demonstrate the link geometry of the energy functional.

In this paper, we consider a nonlinear hyperbolic equation. We establish a blow up result for the initial and boundary value problem. Furthermore, the upper bound of the blow up time is derived.

This paper studies the decay estimate of solutions to the 3D generalized magnetohydrodynamic system in the critical space χ 1 − 2 α ( R 3 ) with 1 2 ≤ α ≤ 1 . We obtain the temporal decay rate ( 1 + t ) − ( 5 4 α − 1 ) for a small initial data global solution which was established by Ye (2016) and improve the temporal decay rate.

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form $\nabla\cdot\vec{C}(x,\nabla u(x))=0$, in a bounded smooth domain $\Omega$. We assume that $\overrightarrow{C}(x,\vec{p})=\gamma(x)\vec{p}+\vec{b}(x)|\vec{p}|^2+\mathcal{O}(|\vec{p}|^3)$, by expanding $\overrightarrow{C}(x,\vec{p})$ around $\vec{p}=0$. We give a reconstruction method for $\gamma$ and $\vec{b}$ from the Dirichlet to Neumann map defined on $\partial\Omega$., Comment: NA

In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.

The analysis of the existence of solutions for the systems of polynomial equations is much more difficult to handle in general. Nonetheless, we can still confirm the existence of some particular solutions, especially for some special structured tensor equations. In this paper, we theoretically prove that a system of H + -tensor equations with the positive right-hand side has a unique positive solution by proposing a positive increasing map. We also prove that this system also has a negative solution. Numerical examples are presented to verify our results are true.

Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

Based on the basic conclusions of the Riemann–Hilbert method for solving the initial value problem of the complex short-pulse equations, the general form of the two-soliton solutions of the complex short pulse equation is given in this paper. Under the two different assumptions of the scattering coefficient, the expression of the two-soliton solutions of the equation is given specifically.

In this paper, we investigate a fourth-order coupled nonlinear Schrodinger equation, which describes the propagation of ultrashort optical pulses in the birefringent optical fiber. Lax pair is constructed through the symbolic computation, along with the corresponding Darboux transformation (DT) which is different from that given in the existing literatures. Rogue wave solutions are obtained by virtue of the DT. Three different types of the rogue waves based on the solutions are exhibited, i.e., the anti-eye-shaped, eye-shaped four-petaled ones. Moreover, we find that the wave structure is affected by the relative background amplitudes and certain parameter in the solutions, and the range of the first-order rogue wave along the x axis increases with the increase of the fourth order term γ , while the range of the rogue wave along the t axis decreases.

In this paper, we present an efficient nonmonotone adaptive cubic regularization method with line search for solving large-scale unconstrained optimization problem. Its global convergence is analyzed. Numerical experiments are performed to show the efficiency and effectiveness of the proposed method and its superiority to the other existing methods.

In the paper, a new sufficient condition for the oscillation of all solutions of linear differential equations with general deviating argument is established. An example numerically solved in MATLAB is also given to illustrate the applicability and the strength of a newly obtained condition over the known ones.

In this paper, we study the approximation system to the steady compressible Navier–Stokes equations. It is proved that the density has a positive lower bound. And we use the lower bound to study the distribution of the density for a family of elliptic equations. A further investigation gives a typical upper bound of the density, and its limit functions only fit the solutions with the velocity divergence free.

This paper is concerned with the long-term behavior for the strong solution of weakly dissipative wave equation with sub-quintic nonlinearity. With the help of Strichartz estimate, we can establish an energy estimate for the strong solution. Then the global attractor is obtained in ( H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) × H 0 1 ( Ω ) . Furthermore, translational regularity of the attractor is established.

This paper is concerned with a chemotactic system modeling the coral broadcast spawning given by u t + U ⋅ ∇ u = Δ u − χ ∇ ⋅ ( u ∇ v ) − μ u 2 , v t + U ⋅ ∇ v = Δ v − v + u in a bounded domain Ω ⊂ R n ( n ≥ 1 ) under Neumann boundary conditions. We provide a rather simpler proof of the non-trivial bounded classical solution on the decay profile. In addition, we also obtain the optimal decay rate of ∇ v ( ⋅ , t ) in L p ( Ω ) as t → ∞ .

In this paper, we develop a new square phase-field crystal model using the L 2 -gradient flow approach, where the total mass of atoms is conserved through a nonlocal Lagrange multiplier. We construct a fast, provably unconditionally energy stable, second-order scheme by using the recently developed SAV approach with the stabilization technique, where an extra stabilization term is added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with the classical Cahn–Hilliard type square phase-field crystal model and the non-stabilized SAV scheme for simulating some benchmark numerical examples, we demonstrate the robustness of the new model, as well as the stability and the accuracy of the developed scheme, numerically.