非线性Schroeding方程Cauchy问题解的Blow—Up的质量集中

We prove in this paper the blow-up theorem of the generalized solution.and its L^2 mass concentration theorem on the set of blow-up points near the blow-up time of the Cauchy problem for nonlinear Schroedinger equations and give the structure of the set of the blow-up points and the estimates on the blow-up time.Furthermore we extend and improve some results in [2] and [5].     

Exact Blow-up Solutions forMultidimensional Landau--Lifshitz Equations

We present exact blow-up solutions for multidimensional Landau-Lifshitz equations. It isshown that for any prescribed blow-up time there are exact C-solutions which are blow-up atthe blow-up time and that the solutions are smooth except at the blow-up time.In 1986, Zhou and Guo in [3] proved the global existence of weak solution for generalizedLandau-Lifshitz equations without Gilbert term in multidimensions. They consider the homogeneous boundary problemwith the initial value conditionfor …     

Blow up behavior for a semilinear parabolic equation with localized source

In this paper, we investigate the blow-up behavior of solutions of a parabolic equation with localized reactions. We completely classify blow-up solutions into the total blow-up case and the single point blow-up case, and give the blow-up rates of solutions near the blow-up time which improve or extend previous results of several authors. Our proofs rely on the maximum principle, a variant of the eigenfunction method and an initial data construction method.     

具有一般非线性耦合项的抛物型方程组的爆破时间估计

This paper deals with some parabolic equations,where the reaction and the boundary flux are taken of some general forms.We study the explicit blow-up time estimates according to the different coupled relationship,including the lower and upper bounds of blow-up time for every dimension of space domains.As examples,the results could be used to so many completely coupled models.     

Profiles of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Cauchy problem for Gross-Pitaevskii equation.In terms of Merle and Raphёel＇s arguments as well as Carles＇ transformation,the limiting profiles of blow-up solutions are obtained.In addition,the nonexistence of a strong limit at the blow-up time and the existence of L2 profile outside the blow-up point for the blow-up solutions are obtained.     

BLOW-UP RATE OF POSITIVE SOLUTION OF UNIFORMLY PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

This paper deals with the blow-up of positive solutions of the uniformly pa-rabolic equations ut = Lu + a(x)f(u) subject to nonlinear Neumann boundary conditions . Under suitable assumptions on nonlinear functi-ons f, g and initial data U0(x), the blow-up of the solutions in a finite time is proved by the maximum principles. Moreover, the bounds of "blow-up time" and blow-up rate are obtained.     

BLOW-UP AND BLOW-UP RATE FOR A COUPLED SEMILINEAR PARABOLIC SYSTEM WITH NONLOCAL BOUNDARIES

This paper deals with the blow-up properties of positive solutions to a coupled semilinear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. Under appropriate hypotheses, the global existence and finite time blow-up of solutions are proved. Moveover, the upper bound of blow-up rate is obtained.     

GLOBAL BLOW-UP FOR A LOCALIZED DEGENERATE AND SINGULAR PARABOLIC EQUATION WITH WEIGHTED NONLOCAL BOUNDARY CONDITIONS

This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, the global blow-up behavior and the uniform blow-up profile of blow-up solutions are also described. We find that the blow-up set is the whole domain {0, a}, including the boundaries, and this differs from parabolic equations with local sources case or with homogeneous Dirichlet boundary conditions case.     

Uniform Blow-up Behavior for Degenerate and Singular Parab olic Equations

This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blow-up exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases.     

Moving finite element methods for time fractional partial differential equations

With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuniform meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations (FPDEs). The unconditional stability and convergence rates of 2-α for time and r for space are proved when the method is used for the linear time FPDEs with α-th order time derivatives. Numerical examples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.     

Evolutions of the momentum density,deformation tensor and the nonlocal term of the Camassa–Holm equation

Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term.     

一类非线性抛物系统解的爆破现象

研究了具有依赖于时间的系数的非线性抛物方程解的爆破现象.对已知数据项进行一定的假设并设置一些辅助函数,应用微分不等式技术,得到了方程的解发生爆破的条件.当爆破发生时,分别推导了方程在二维区域和三维区域上解的爆破时间的下界.     

Blow－Up and Mass Concentration of Solutions to the Cauchy Problem for Nonlinear Schrodinger Equations

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Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II

This paper deals with the blow-up of the solution to a semilinear second-order parabolic equation with nonlinear boundary conditions. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

On the Behaviors of Solutions Near Possible Blow-Up Time in the Incompressible Euler and Related Equations

We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier–Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of the scaling invariant norms, we derive the possible blow-up behaviors of the above quantities, from which we obtain new type of blow-up criteria and some necessary conditions for the finite time blow-up.     

Blow-up solution near the traveling waves for the second-order Camassa–Holm equation

This paper studies the blow-up solution and its blow-up rate near the traveling waves of the second-order Camassa–Holm equation. The sufficient condition for the existence of blow-up solution is obtained by a rather ingenious method. Applying the extended pseudo-conformal transformation, an equivalent proposition of the solution breaking in finite time near the traveling waves is constructed. The relation is established between the blow-up time and rate of the solution and the residual’s.     

Global existence and blow-up of solutions of nonlocal diffusion problems with free boundaries

In this paper, we investigate some nonlocal diffusion problems with free boundaries. We first give the existence and uniqueness of local solution by the ODE basic theory and the contraction mapping principle. Then we provide a complete classification for the global existence and finite time blow-up of solutions. Moreover, estimates of blow-up rate and blow-up time are also obtained for the blow-up solution.