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In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0 < | v | < 1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases. [ABSTRACT FROM AUTHOR]

We present some embedding results of fractional D s , 2 (R N) space with radial potential. To overcome the difficulty of non-compactness, we obtain the Lions-type theorems by means of the refined Sobolev inequality with the Morrey norm. Then we apply the obtained results to study the existence of ground state solution for fractional equations with radial potential. [ABSTRACT FROM AUTHOR]

We are concerned with the well-posed problem of a strongly degenerate parabolic equation. A kind of entropy solutions in the BV sense is introduced and a reasonable partial boundary value condition is provided. By means of Kruzkov's bi-variables method, the stability of entropy solutions is proved under this partial boundary value condition. Through the Crocco transformation, the Prandtl boundary layer system is transformed to an anisotropic degenerate parabolic equation with a nonlinear boundary value condition. We propose a new analytical method by restricting some geometric properties on the spatial domain and establish the well-posedness of the BV solution to this anisotropic degenerate parabolic equation with the initial value condition. [ABSTRACT FROM AUTHOR]

For the stability of the non-Newtonian fluid equation ∂ u ∂ t − div (a (x) | ∇ u | p − 2 ∇ u) − ∑ i = 1 N b i (x) D i u + c (x , t) u = f (x , t) , where a (x) | x ∈ Ω > 0 , a (x) | x ∈ ∂ Ω = 0 and b i (x) ∈ C 1 (Ω ‾) , we know that the degeneracy of a (x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of semilinear elliptic equations with a potential function − Δ u = λ | x | − α u − | x | σ u p in Ω ∖ { 0 } , where λ , σ ∈ R , α > 0 , p > 1 , and Ω ⊂ R N (N ≥ 3) is a bounded smooth domain with 0 ∈ Ω. We establish the existence, nonexistence and asymptotic behavior of positive solutions when the potential function | x | − α has strong singularity at the origin. When the potential function | x | − α has weak singularity at the origin, we obtain some qualitative properties of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂Ω and the others are positive on Ω ‾ , we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain Ω, is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear degenerate parabolic equation is considered. By proposing a new partial boundary value condition and constructing special test functions, we prove the stability of the solutions by means of the Kruzkov bi-variables method. Moreover, we show that if there is an interaction between the diffusion term and the convection term, the stability can be established without any boundary value condition. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the hyperbolic–parabolic mixed type equations with the non-homogeneous boundary condition. If it is degenerate on the boundary, the part of the boundary whose boundary value should be imposed, is determined by the entropy condition from the convection term. If there is no convection term in the equation, we show that the stability of solutions can be proved without any boundary condition. If the equation is completely degenerate, we show that the stability of solutions can be established just based on the partial boundary condition. [ABSTRACT FROM AUTHOR]

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations − div A (x , u , D u) = μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p -Laplacian with respect to Du for the singular case p ∈ (1 , 2 − 1 / n ] , whose discontinuity in the x -variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u -variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good- λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces. [ABSTRACT FROM AUTHOR]

For a class of subelliptic p -harmonic systems with the subcritical growth defined in Carnot groups, we prove that the weak solutions belong to Γ loc 1 , α -regularity with respect to the Carnot–Caratheodory distance provided that p is close to 2. As a consequence, we make it clear that the critical growth of nonlinearity for p -harmonic type systems in Carnot groups is just a sharp borderline of partial regularity of the small “excess” energy due to the counterexamples of p -harmonic maps with p close to 2. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with a second order non-autonomous Hamiltonian system on time scales Under certain conditions, the existence and multiplicity of periodic solutions are obtained for this Hamiltonian system on time scales by using the saddle point theory, the least action principle as well as the three-critical-point theorem. In addition, the existence of homoclinic orbit is obtained as a limit of -periodic solutions of a given sequence of Hamiltonian system on time scales by means of the mountain pass theorem and the standard minimizing argument. [Copyright &y& Elsevier]

In this paper, we study an epidemic system in a periodic patchy environment with bilinear incidence. It is assumed that all the parameters share the same period N ∈ N. When the basic reproduction number R 0 > 1 , we show that for each c ≥ c ⁎ the system admits a nontrivial traveling wave solution, where c ⁎ is the minimum speed, and there is no nontrivial traveling wave solution when R 0 < 1 , or R 0 > 1 and c ∈ (0 , c ⁎). The uniqueness of the endemic equilibrium and the convergence of traveling wave solutions at +∞ are also discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, first we survey some recent advances in the study of traveling wave solutions to the Burgers–Korteweg–de Vries equation and some comments are given. Then, we study a Burgers–Korteweg–de Vries-type equation with higher-order nonlinearities. A qualitative analysis to a two-dimensional autonomous system which is equivalent to the Burgers–KdV-type equation is presented, and indicates that under certain conditions, the Burgers–Korteweg–de Vries-type equation has neither nontrivial bell-profile solitary waves, nor periodic waves. Finally, a solitary wave solution is obtained by means of the first-integral method which is based on the ring theory of commutative algebra. [Copyright &y& Elsevier]

In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with an initial–boundary value problem for a class of non-Newtonian fluids. The viscosity term of momentum equation possesses singularity and nonlinearity, and the initial vacuum is allowed. The main feature which distinguishes this paper from other related works lies in the fact that we study the global strong solution of the compressible non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals, and show that there exists a unique global-in-time strong solution to the problem while the initial energy is small. [ABSTRACT FROM AUTHOR]

Cholera is a virulent intestinal infection caused by Vibrio cholerae strains. To explore multiple effects of spatial structure, seasonality and human behavior change on cholera transmission, we establish a reaction-advection-diffusion model with a general boundary condition at the downstream end of the river. We derive the basic reproduction number R 0 and discuss its asymptotic behaviors. We further show that R 0 is a strictly decreasing function of the bacterial loss c , which reveals that the bacterial loss at the downstream end of the river due to water flux reduces the risk of cholera transmission. Then we derive a threshold-type result in terms of R 0. More precisely, (i) if R 0 < 1 , the disease-free periodic solution is globally attractive; (ii) if R 0 > 1 , the disease is uniformly persistent; (iii) if R 0 = 1 , the disease-free steady state is globally asymptotically stable in the absence of the advection term, in which the proof is quite challenging due to the introduction of human behavior change, involving some subtle inequalities and estimates. Numerically, an application is demonstrated by investigating the cholera outbreak in Somalia from 1995 to 2016. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n . It proves that for n = 3 , 4 and 5, there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively, and in each case this upper bound is sharp. Moreover, for any n > 5 , there are at most [ n − 1 2 ] critical periods induced by periodic orbits of the unperturbed cubic system. An example is given to show that the upper bound in the case of n = 11 can be reached. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal diffusive single species model with Allee effect. We prove that the model admits positive traveling wave solutions connecting the equilibrium 0 to some unknown positive steady state if and only if the wave speed c ≥ 2 r , where r > 0 is the intrinsic rate of increase of the species. For the sufficient large wave speed c , we show that the unknown steady state is the unique positive equilibrium. For two types of convolution kernel functions, we investigate the change of the wave profile as the non-locality increases, and illustrate that the unknown steady state may be a positive periodic solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system: { − Δ u + λ u + K ( x ) ϕ u = Q ( x ) | u | p − 2 u   in R 3 , − Δ ϕ = K ( x ) u 2   in R 3 , where λ > 0 , 2 < p < 6 , and both K ( x ) and Q ( x ) are nonnegative and uniformly continuous functions on R 3 . We show that the number of positive solutions is dependent on the profile of Q ( x ) . Some novel results are presented which improve and generalize the existing ones in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we are concerned with the regularity in Morrey spaces for weak solutions of a class of degenerate elliptic equations when the coefficient matrices satisfy certain VMO conditions in x uniformly with respect to u and the lower order terms satisfy a natural growth condition. Interior Hölder continuity of weak solutions is also derived with the improvement of the given data regularities. [Copyright &y& Elsevier]

Abstract: The stability and bifurcation of a van der Pol-Duffing oscillator with the delay feedback are investigated, in which the strength of feedback control is a nonlinear function of delay. A geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and Hopf bifurcations. The Hopf bifurcation curves and multi-stable regions are obtained as two parameters vary. Some weak resonant and non-resonant double Hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the “death island” regions simultaneously. By applying the center manifold theory, the normal forms near the double Hopf bifurcation points, as well as classifications of local dynamics are analyzed. Furthermore, some quasi-periodic and chaotic motions are verified in both theoretical and numerical ways. [Copyright &y& Elsevier]

In this paper, we are concerned with a sixth-order nonlinear Schrödinger equation. By applying the factorization formula and an analytical method, we prove the existence and uniqueness of global solution for the Cauchy problem of the sixth-order nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of infinitely many sign-changing solutions of nonlinear Schrödinger systems in R N by applying the perturbation method and the method of invariant sets of descending flow. The main feature which distinguishes this paper from other related works (such as [13]) lies in the fact that coefficients of nonlinearities are dependent on x rather than constants. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study dynamics of a prey–predator system under the impulsive control. Sufficient conditions of the existence and the stability of semi-trivial periodic solutions are obtained by using the analogue of the Poincaré criterion. It is shown that the positive periodic solution bifurcates from the semi-trivial periodic solution through a transcritical bifurcation. A strategy of impulsive state feedback control is suggested to ensure the persistence of two species. Furthermore, a steady positive period-2 solution bifurcates from the positive periodic solution by the flip bifurcation, and the chaotic solution is generated via a cascade of flip bifurcations. Numerical simulations are also illustrated which agree well with our theoretical analysis. [Copyright &y& Elsevier]