一类带粘性项的抛物方程解的存在性和爆破性

该文研究具有多项式非线性项和粘性项的非线性抛物方程的初边值问题.在一定条件下,我们得到方程的弱解全局存在.在另一些条件下,我们得到该方程的解将在有限时刻爆破,并给出了爆破时间的上界,该上界受初始函数及其支集控制.该结论推广了Messaoudi在文献[15,16]中的工作.     

对一个非线性分数阶微分方程组爆破解的研究

The paper focuses on the blow-up solution of system of time-fractional differential equations
where ^cD₀₊^α, ^cD₀₊^β are Caputo fractional derivatives, n-1 < α < n, n-1 < β < n,A(t),B(t) are continuous functions. We obtain a system of the integral equations which is equivalent to the system of nonlinear partial differential equations with time-fractional derivative via the approach of Laplace transformation, and prove the local existence of solutions to the system of the integral equations. Secondly, this paper investigates the blow-up solutions to the a nonlinear system of fractional differential equations by making use of Hölder’s inequality and obtains a solution of system to blow up in a finite time, and gives an upper bound on the blow-up time.     

Blow-up of periodic solutions to linearly degenerate quasilinear hyperbolic systems

An example is given to show that for the Cauchy problem of n×n$n\times n$ homogeneous linearly degenerate quasilinear hyperbolic systems with periodic initial data, when n>2$n>2$, C¹$C^1$ solutions may blow up in a finite time, no matter how small and smooth the initial data are.     

A blow-up criterion for 3-D non-resistive compressible heat-conductive magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum. This blow-up criterion depends only on the gradient of velocity and the temperature, which is similar to the one for compressible Navier-Stokes equations.     

Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation

We present a pathfollowing strategy based on pseudo-arclength parametrization for the solution of parameter-dependent boundary value problems for ordinary differential equations. We formulate criteria which ensure the successful application of this method for the computation of solution branches with turning points for problems with an essential singularity. The advantages of our approach result from the possibility to use efficient mesh selection, and a favorable conditioning even for problems posed on a semi-infinite interval and subsequently transformed to an essentially singular problem. This is demonstrated by a Matlab implementation of the solution method based on an adaptive collocation scheme which is well suited to solve problems of practical relevance. As one example, we compute solution branches for the complex Ginzburg-Landau equation which start from non-monotone ‘multi-bump’ solutions of the nonlinear Schrödinger equation. Following the branches around turning points, real-valued solutions of the nonlinear Schrödinger equation can easily be computed.     

一类非线性波方程整体解的存在性和不存在性

This paper deals with the existence and uniqueness of the global solution of the initial boundary value problem of a class of wave equation. In the meantime, it gives the sufficient conditions of blow-up of the solution for the problem in finite time.     

临界半线性波动方程解的有限时间破裂

研究了带临界指标的半线性波动方程u_tt-Δu=（1+|x|²）^α|u|^p小初值Cauchy问题解的破裂,证明了该问题在p=p_c（n）时不存在整体解.     

A Quasilinear Parabolic System with Nonlocal Boundary Conditions and Localized Sources

This paper investigates the properties of solutions to a quasilinear parabolic system with nonlocal boundary conditions and localized sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and blow-up rate estimates are also derived.     

具有非线性边界条件半线性热方程组解的爆破性质

本文考虑一类半线性热方程组的解,给出了解爆破的充分必要条件,爆破速率和爆破点的位置。     

The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential

In this paper, we discuss a generalized Camassa–Holm equation whose solutions are velocity potentials of the classical Camassa–Holm equation. By exploiting the connection between these two equations, we first establish the local well-posedness of the new equation in the Besov spaces and deduce several blow-up criteria and blow-up results. Then, we investigate the existence of global strong solutions and present a class of cuspon weak solutions for the new equation.