On the nonexistence of global solutions of the Hamilton-Jacobi equation

Although examples of nonexistence of global solutions of Hamilton-Jacobi equations have long been known, the general theory of nonexistence of global solutions (unlike the general theory of existence of global viscous solutions) does not yet exist.In the present paper, we use examples of model Hamilton-Jacobi equations to demonstrate how simple methods for the analysis of the problem of nonexistence of global solutions (in the corresponding function class) can be applied.     

On the solvability of a mathematical model for prion proliferation

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.     

一类半线性椭圆型方程正整体解的存在性

本文研究了一个半线性椭圆型方程正整体解的存在性问题.利用上下解方法克服了非齐次项出现带来的困难,获得了正整体解存在的必要条件,具体地给出了在此条件下正整体解所处的领域.     

Global solutions and blow-up solutions for a nonlinear string with boundary input and output

In this paper, we investigate the global solutions and blow-up solutions for a nonlinear string with boundary input and output, and prove that under some conditions, the system has global solutions and blow-up solutions, respectively.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Asymptotic convergence of solutions to the forest kinematic model

We study the asymptotic behavior of global solutions to forest kinetic model equations composed of young trees, old trees, and air-borne seeds. Under some parameter assumptions, we prove the asymptotic convergence of global solutions to a stationary solution. To this end, we show a non-smooth version of the Łojasiewicz–Simon gradient inequality on a suitable functional space and a certain norm estimate of the time derivative of global solutions.     

Strong solutions and trajectory attractors to the thin-film equation with absorption

We prove local and global in time existence of non-negative weak solutions to the thin-film equation with absorption and obtain sufficient conditions for extra regularity of these solutions. Moreover, for the class of global strong solutions, we show existence of a trajectory attractor.     

Global solutions of Yang–Mills–Higgs systems on four-dimensional Minkowski space: II. Some properties of solutions

The paper concerns the properties of global solutions to the Cauchy problem of Yang–Mills–Higgs systems in the temporal gauge. The following properties are considered: global analyticity of solutions, an approximation property, conservation of the topological class of solutions, scattering of a part of the energy of solutions through a light cone.     

一类Boussinesq型方程整体解的存在性和唯一性

证明一类6阶Boussinesq型方程Cauchy问题整体广义解和整体古典解的存在性和唯一性,给出解在有限时刻发生爆破的充分条件.     

Global weak solutions for a periodic two-component?μ-Hunter–Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component?μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component?μ-Hunter–Saxton system.     

广义Davey-Stewartson方程组的整体解及自相似解

本文研究Dalvey-Stewartson方程组的整体解与自相似解的存在性.首先,运用Ba- nach不动点定理得到一个关于解整体存在性的一般性定理,然后把一类特殊的初始值用到该存在性结果上去从而得到自相似解存在的结论.     

On the Global Solutions of the Higgs Boson Equation

In this article we study global in time (not necessarily small) solutions of the equation for the Higgs boson in the Minkowski and in the de Sitter spacetimes. We reveal some qualitative behavior of the global solutions. In particular, we formulate sufficient conditions for the existence of the zeros of global solutions in the interior of their supports, and, consequently, for the creation of the so-called bubbles, which have been studied in particle physics and inflationary cosmology. We also give some sufficient conditions for the global solution to be oscillatory in time.     

Global solvability and behavior of solutions of the cauchy problem for a system of two semilinear hyperbolic equations with dissipation

We obtain a sufficient condition for the existence of global solutions of Cauchy problems for systems of two semilinear hyperbolic equations with dissipation and find the decay rate of the solutions. We derive a sufficient condition for the absence of global solutions of systems of two semilinear hyperbolic inequalities. The proof of the absence of global solutions is performed on the basis of the test function method developed by S.I. Pokhozhaev, L. Veron, E. Mitidieri, et al.     

Self-similar solutions to a generalized Davey–Stewartson system

In this paper we study the global solvability and existence of self-similar solutions to a generalized Davey–Stewartson system. We first use the Banach fixed point theorem to establish a general global existence theorem. By using this general result to a class of special initial data we obtain the existence of global self-similar solutions.     

带状区域中渐近周期曲率流方程的整体解

该文研究了具有渐近周期系数的曲率流方程的Neumann边值问题.首先,考虑一列初值问题及其相应的全局解,通过一致的先验估计取一个收敛子列,得到其极限就是一个整体解的结论.其次,向负无穷时间方向进行重整化,使用强极值原理证明了整体解的唯一性.最后,为了研究整体解的ω-和α-极限,再次使用重整化方法,通过构造拉回函数、进行一致的先验估计以及Cantor对角化方法取收敛子列,得到整体解的ω-和α-极限都是极限问题的整体解,即它们都是周期行波的结论.     

Cauchy problem for a class of fourth-order semilinear pseudohyperbolic equations with structural damping

A sufficient condition for the existence of global solutions is established, and the rate of decay of solutions to the Cauchy problem for a semilinear pseudohyperbolic equation with structural damping is found. The nonexistence of global solutions is also investigated. An analogue of the critical Fujita exponent is obtained for the considered problem.     

Global Existence and Uniqueness of Solutions to Evolution p-Laplacian Systems with Nonlinear Sources

This paper presents the global existence and uniqueness of the initial and boundary value problem to a system of evolution p-Laplacian equations coupled with general nonlinear terms. The authors use skills of inequality estimation and themethod of regularization to construct a sequence of approximation solutions, hence obtain the global existence of solutions to a regularized system. Then the global existence of solutions to the system of evolution p-Laplacian equations is obtained with the application of a standard limiting process. The uniqueness of the solution is proven when the nonlinear terms are local Lipschitz continuous.     

Sufficient conditions for the existence of global solutions of delayed differential equations

The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The principle of monotone sequences is used as a main tool of investigation. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results.     

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotaztic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.     

The Cauchy-Darboux problem for the one-dimensional wave equation with power nonlinearity

The questions are studied of existence and uniqueness of a global solution to the Cauchy-Darboux problem for the one-dimensional wave equation with power nonlinearity. Under consideration are the existence of local solutions and the absence of global solutions.