Blow-up of solutions to the DGH equation

In this paper, firstly we find best constants for two convolution problems on the unit circle via a variational method. Then we apply the best constants on a nonlinear integrable shallow water equation (the DGH equation) to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Finally, we discuss the blow-up phenomena for the nonperiodic case.     

一个局部波动方程组的解的全局存在性与爆破

我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p, (e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x), (e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率.     

Blow-up and global existence for a nonlocal degenerate parabolic system

This paper investigates the blow-up and global existence of nonnegative solutions of the system

     

Blow-up and global existence to a degenerate reaction-diffusion equation with nonlinear memory

In this paper, we consider a degenerate reaction-diffusion equation

     

Blow-up of solutions to a nonlinear dispersive rod equation

In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Mathematics Subject Classification (2000) 30C70, 37L05, 35Q58, 58E35     

BLOW-UP AND GLOBAL EXISTENCE FOR A COUPLED SYSTEM OF DEGENERATE PARABOLIC EQUATIONS IN A BOUNDED DOMAIN

This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut=△ul up1vq1 and vt=△vm up2vq2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1-(l-P1)(m-q2), the initial data, and the domainΩ.     

Blow-up and decay criteria for a model of chemotaxis

For a system of equations introduced by Jäger and Luckhaus (1992) [6] as a model of chemotaxis, the questions of blow-up and global existence criteria are investigated. Specifically, for a convex region, a lower bound for the blow-up time is derived if the solution blows up, and explicit criteria to ensure non-blow-up are determined.     

Darboux transformation and explicit solutions for two integrable equations

A new N-fold Darboux transformation for two integrable equations is constructed with the help of a gauge transformation for the spectral problem proposed by Qiao [Z.J. Qiao, Phys. Lett. A 192 (1994) 316-322]. By the Darboux transformation, explicit soliton and multi-soliton solutions for the two equations are obtained. In particular, soliton and complexiton solutions are shown through some figures.     

Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities

This paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. As the results of the interaction among the multi-coupled nonlinearities in the system, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.     

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.     

On the global existence of solutions to an aggregation model

In this paper we consider a reaction-diffusion-chemotaxis aggregation model of Keller-Segel type with a nonlinear, degenerate diffusion. Assuming that the diffusion function f(n) takes values sufficiently large, i.e. takes values greater than the values of a power function with sufficiently high power (f(n)?δnp for all n>0, where δ>0 is a constant), we prove global-in-time existence of weak solutions. Since one of the main features of Keller-Segel type models is the possibility of blow-up of solutions in finite time, we will derive the uniform-in-time boundedness, which prevents the explosion of solutions. The uniqueness of solutions is proved provided that some higher regularity condition on solutions is known a priori. Finally, computational simulation results showing the effect of three different types of diffusion function are presented.     

一类带有非局部源的退化反应扩散方程组解的整体存在与爆破

研究了一类具有齐次Dirichlet边界条件和带有非局部反应项的退化抛物方程组解的性质.用正则化的方法证明了局部解的存在唯一性,用上下解方法,得到了解的全局存在与爆破的充分条件.     

On the global existence and wave-breaking criteria for the two-component Camassa-Holm system

Considered herein is a two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. A wave-breaking criterion for strong solutions is determined in the lowest Sobolev space Hs, by using the localization analysis in the transport equation theory. Moreover, an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this Sobolev space Hs.     

Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources

This paper deals with the global existence and blow-up of nonnegative solution of the degenerate reaction-diffusion system with nonlinear localized sources involved a product with local terms. We investigate the influence of localized sources and local terms on global existence and blow up for this system. Moreover, we establish the precise blow-up estimates. Finally, for the special case p₁=p₂=0, we show the blow-up set is whole region and the uniform blow-up profiles are obtained. These extend a resent work of Chen and Xie in [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. Comput. 159 (2004) 79-93], which considered the single equation with localized sources.     

Chemotaxis with logistic source: Very weak global solutions and their boundedness properties

We consider the chemotaxis system

     

非局部退化拟线性抛物型方程组解的爆破和整体存在性

该文采用弱上下解方法和正则化技巧,研究了一类非局部退化抛物型方程组解的爆破和整体存在性,给出了爆破指标,并对非退化情形m=n=1,p_1=q_1=0,p_2q_21给出了一致爆破速率.     

Boundedness of global solutions for a porous medium system with moving localized sources

This paper deals with a class of porous medium systems with moving localized sources u_t=u^r₁(Δu+af(v(x₀(t),t))),v_t=v^r₂(Δv+bg(u(x₀(t),t))) with homogeneous Dirichlet boundary conditions. It is shown that under certain conditions, solutions of the above system blow up in finite time for large a and b or large initial data while there exist global positive solutions to the above system for small a and b or small initial data. Moreover, in the one dimensional space case, it is also shown that all global positive solutions of the above problem are uniformly bounded.