Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

Nonlinear functionals of the Boltzmann equation and uniform stability estimates

We present nonlinear functionals measuring physical space variation and L¹-distance between two classical solutions for the Boltzmann equation with a cut-off inverse power potential. In the case that initial datum is a small, smooth perturbation of vacuum and decays fast enough in the phase space, we show that these functionals satisfy stability estimates which lead to BV-type estimates and a uniform L¹-stability.     

Existence and multiplicity of semiclassical solutions for a Schrödinger equation

We establish the existence and multiplicity of solutions for the semiclassical nonlinear Schrödinger equation

     

Diffusive expansion for solutions of the Boltzmann equation in the whole space

This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space

(0.1)     

Low-regularity solutions of the periodic general Degasperis-Procesi equation

This paper studies low-regularity solutions of the periodic general Degasperis-Procesi equation with an initial value. The existence and the uniqueness of solutions are proved. The results are illustrated by considering the periodic peakons of the periodic general Degasperis-Procesi equation.     

On the Cauchy-Rassias stability of a generalized additive functional equation

Let X and Y be Banach spaces and f:X→Y an odd mapping. We solve the following generalized additive functional equation

     

Multiple solutions for some nonlinear Schrödinger equations with indefinite linear part

We consider a class of nonlinear Schrödinger equation with indefinite linear part in RN. We prove that the problem has at least three nontrivial solutions by means of Linking Theorem and (∇)-Theorem.     

Periodic solutions for a class of superquadratic Hamiltonian systems

We prove the existence of nontrivial critical points for a class of superquadratic nonautonomous second-order Hamiltonian systems by applying condition ^∗(C) to critical point theory, and some new solvability conditions of nontrivial periodic solutions are obtained.     

On the stability of the linear mapping in Banach modules

We prove the generalized Hyers-Ulam-Rassias stability of the linear mapping in Banach modules over a unital Banach algebra.     

Remark on uniqueness of mild solutions to the Navier-Stokes equations

We investigate a limiting uniqueness criterion to the Navier-Stokes equations. We prove that the mild solution is unique under the class , where bmo^-1 is the “critical” space including Lⁿ. As an application of uniqueness theorem, we also consider the local well-posedness of Navier-Stokes equations in bmo^-1.     

Boundary value problem for abstract first order differential equation with integral condition

The present paper is devoted to the study of a boundary value problem for abstract first order linear differential equation with integral boundary conditions. We obtain necessary and sufficient conditions for the unique solvability and well-posedness. We also study the Fredholm solvability. Finally, we obtain a result of the stability of solution with respect to small perturbation.     

Asymptotic stability for anisotropic Kirchhoff systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p(x)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.     

On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings

In this paper, several weak and strong convergence theorems are established for a modified Noor iterative scheme with errors for three asymptotically nonexpansive mappings in Banach spaces. Mann-type, Ishikawa-type, and Noor-type iterations are covered by the new iteration scheme. Our results extend and improve the recently announced ones [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453; Y.J. Cho, H. Zhou, G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004) 707-717] and many others.     

Global bifurcation for a class of degenerate elliptic equations with variable exponents

We are concerned with the following nonlinear problem

     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

An attractor for a nonlinear dissipative wave equation of Kirchhoff type

In this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial-boundary value problem of a quasilinear wave equation of Kirchhoff type with a standard dissipation ut.     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in R². Upper and lower bounds are derived for the L² decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument.     

The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation:

     

Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems

By computing the E-critical groups at θ and infinity of the corresponding functional of Hamiltonian systems, we proved the existence of nontrivial periodic solutions for the systems which may be resonant at θ and infinity under some new conditions. Some results in the literature are extended and some new type of theorems are proved. The main tool is the E-Morse theory developed by Kryszewski and Szulkin.