A remark on the Beale-Kato-Majda criterion for the 3D MHD equations with zero kinematic viscosity

In this paper,we study the blow-up criterion of smooth solutions to the 3D magneto-hydrodynamic system in ˙ B 0 ∞,∞.We show that a smooth solution of the 3D MHD equations with zero kinematic viscosity in the whole space R 3 breaks down if and only if certain norm of the vorticity blows up at the same time.     

UNIFORM BLOW-UP PROFILES FOR HEAT EQUATIONS WITH COUPLING NONLOCAL SOURCES OF ASYMMETRIC MIXED TYPE NONLINEARITIES＊

This article deals with a nonlocal heat system subject to null Dirichlet boundary conditions,where the coupling nonlocal sources consist of mixed type asymmetric nonlinearities.We at first give the cri...     

Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source

This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source.Conditions for the existence of global or bl...     

Non-simultaneous blow-up and blow-up rates for reaction-diffusion equations

This paper considers blow-up solutions for reaction-diffusion equations, complemented by homogeneous Dirichlet boundary conditions. It is proved that there exist initial data such that one block or two (separated or contiguous) blocks of n components blow up simultaneously while the others remain bounded. As a corollary, a necessary and sufficient condition is obtained such that any blow-up must be the case for at least two components blowing up simultaneously. We also show some other exponent regions, where any blow-up of k(∈{1,2,…,n}) components must be simultaneous. Moreover, the corresponding blow-up rates and sets are discussed. The results extend those in Liu and Li [B.C. Liu, F.J. Li, Non-simultaneous blow-up of n components for nonlinear parabolic systems, J. Math. Anal. Appl. 356 (2009) 215-231].     

Pure Log Terminal Blow-Ups

The existence of a pure log terminal blow-up for a log terminal singularity and a criterion for a singularity to be weakly exceptional are proved.     

GLOBAL BLOWUP AND BLOWUP RATE ESTIMATES OF SOLUTIONS FOR A CLASS OF NONLINEAR NON-LOCAL REACTION-DIFFUSION PROBLEMS

In this paper, the global blowup properties of solutions for a class of non-linear non-local reaction-diffusion problems are investigated by the methods of the priorestimates. Moreover, the blowup rate estimate of the solution is given.     

Boundary blow-up solutions to elliptic systems of competitive type

We consider the elliptic system Δu=u^pv^q, Δv=u^rv^s in Ω, where p,s>1, q,r>0, and Ω⊂R^N is a smooth bounded domain, subject to different types of Dirichlet boundary conditions: (F) u=λ, v=μ, (I) u=v=+∞ and (SF) u=+∞, v=μ on ∂Ω, where λ,μ>0. Under several hypotheses on the parameters p,q,r,s, we show existence and nonexistence of positive solutions, uniqueness and nonuniqueness. We further provide the exact asymptotic behaviour of the solutions and their normal derivatives near ∂Ω. Some more general related problems are also studied.     

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

This paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball.     

On the Blow-up Criterion of Smooth Solutions to the MHD System in BMO Space

In this paper we study the blow-up criterion of smooth solutions to the incompressible magnetohydrodynamics system in BMO space. Let （u（x,t）,b（x,t）） be smooth solutions in （0, T）. It is shown that the solution （u（x, t）, b（x, t）） can be extended beyond t = T if （u（x,t）, b（x, t）） ∈ L^1 （0, T; BMO） or the vorticity （rot u（x, t）, rot b（x, t）） ∈ L^1 （0, T; BMO） or the deformation （Def u（x, t）, Def b（x, t）） ∈ L^1 （0, T; BMO）.