Global existence and nonexistence of solutions for the nonlinear pseudo‐parabolic equation with a memory term

In this paper, we study the initial boundary value problem of nonlinear pseudo‐parabolic equation with a memory term with initial conditions and Dirichlet boundary conditions. By the combination of the Galerkin method and Potential well theory, the existence of global solutions is derived. Moreover, not only the finite time blow up of solutions with the negative initial energy (E(0) < 0) but also the finite time blow up results with the nonnegative initial energy (0≤E(0) < d_k) are obtained by using Concavity method and Potential well theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion‐type source term. Based on a low‐frequency and high‐frequency decomposition, Green's function method and the classical energy method, we not only obtain L² time‐decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u₀(x) satisfies the smallness condition on , but not on . Furthermore, by taking a time‐frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation

ABSTRACT

In this paper, a nonlinear viscoelastic kirchhoff equation in a bounded domain with a time varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galarkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.     

The existence and uniqueness of the local solution for a Camassa-Holm type equation

A shallow water equation of Camassa-Holm type, containing nonlinear dissipative effect, is investigated. Using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself, we establish the existence and uniqueness of its local solution in Sobolev space H^s(R) with . Meanwhile, a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order Sobolev space H^s with are presented.     

Global existence of solutions to a magnetohydrodynamic‐omega model

In this paper, some sufficient conditions in terms of the magnetic field are established to guarantee global existence of solutions to a magnetohydrodynamic‐omega model.     

Novel existence and uniqueness criteria for periodic solutions of a Duffing type p-Laplacian equation

In this study, we investigate a Duffing type p-Laplacian equation. Some new criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation are given by using the Manásevich–Mawhin continuation theorem and some analysis techniques. Our results improve and extend some known results from the literature.     

Global existence and blow‐up of the solutions for the multidimensional generalized Boussinesq equation

In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the multidimensional generalized Boussinesq equation are obtained. Furthermore, the blow‐up of the solution for the Cauchy problem of the generalized Boussinesq equation is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

The uniqueness and existence of solutions for the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation

In this paper, we study the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are too few. On the basis of the Green's function for the 3D homogeneous Helmholtz equation in a step‐index waveguide without perturbation, we introduce a generalized (out‐going) Sommerfeld–Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3D Helmholtz equation satisfying our radiation condition. Copyright © 2012 John Wiley & Sons, Ltd.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.     

Global non‐existence of solutions of a class of wave equations with non‐linear damping and source terms

In this paper we consider the non‐linear wave equation a,b>0, associated with initial and Dirichlet boundary conditions. We prove, under suitable conditions on α,β,m,p and for negative initial energy, a global non‐existence theorem. This improves a result by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833), who requires that the initial energy be sufficiently negative and relates the global non‐existence of solutions to the size of Ω. Copyright © 2004 John Wiley & Sons, Ltd.     

一类非局部Cahn-Hilliard方程弱解的存在唯一性

研究一类对流非局部Cahn-Hilliard方程的Neumann问题.通过一致Schauder估计和Leray-Schauder不动点定理,得到了该问题经典解的存在唯一性.进而,利用弱收敛方法得到了该问题弱解的存在唯一性.     

Global existence and uniqueness to the Cauchy problem of the BGK equation with infinite energy

The Bhatnagar–Gross–Krook model of the Boltzmann equation is of great importance in the kinetic theory of rarefied gases. Various existence and uniqueness results have been built under the boundedness of energy. In this paper, we will establish several global existence results to the Bhatnagar–Gross–Krook equation with infinite energy. It heavily relies on a new moments lemma and a new existence and uniqueness theorem of weighted velocity‐spatial L^∞ solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

New results on the existence of periodic solutions for a higher‐order Li énard type p‐Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher‐order Li énard type p‐Laplacian differential equation as follows Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of a particle method for the one‐dimensional Vlasov‐Maxwell system

In this article we present a particle method for solving numerically the one‐dimensional Vlasov‐Maxwell equations. This method is based on the formulation by characteristics. We perform the error analysis and we investigate the properties of this scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Low regularity solutions,blowup, and global existence for a generalization of Camassa–Holm‐type equation

We consider a generalization of Camassa–Holm‐type equation including the Camassa–Holm equation and the Novikov equation. We mainly establish the existence of solutions in lower order Sobolev space with . Then, we present a precise blowup scenario and give a global existence result of strong solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

On local existence and blowup of solutions for a time‐space fractional diffusion equation with exponential nonlinearity

In this paper, we are concerned with local existence and blowup of a unique solution to a time‐space fractional evolution equation with a time nonlocal nonlinearity of exponential growth. At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, the blowup result of the solution in finite time is established by the test function method with a judicious choice of the test function.     

The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations

Let H,V and K be separable Hilbert spaces. In this paper we consider the existence and uniqueness of energy solutions to the following stochastic evolution equation:

where is a linear bounded operator with coercivity, monotone condition and hemicontinuity, and are measurable functions and satisfy the local non-Lipschitz condition proposed by the author [T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Equations 96 (1992) 152–169].