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

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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.     

带Robin边界条件的拟线性抛物方程解的整体存在性与爆破

主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界.     

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.     

一类具有对数非线性项的四阶薄膜方程新的爆破结果

该文侧重研究一类具有对数非线性项的四阶薄膜方程解的爆破现象.目前,此类问题解的爆破结果来看都依赖于井的深度d.本文中,我们建立与井的深度d无关的新爆破结论且给出爆破时间的上界.     

一类具非局部源抛物型方程组解的一致爆破性质

考虑带有齐次Dirichlet边界条件,具非局部源项的半线性抛物型方程组正解的爆破性质,首先给出了该问题的解在有限时刻爆破的充分条件,以及解的两个分量同时爆破的必要条件,并建立了解的一致爆破模式.     

On blow-up solutions of polynomial Kolmogorov systems

We investigate the existence problem for blow-up solutions of the polynomial Kolmogorov systems. We find sets of initial values of the blow-up solutions. We also consider a method of finding upper bounds for the blow-up time of these solutions.     

Critical non-linear dispersive equations: global existence, scattering, blow-up and universal profiles

We discuss recent progress in the understanding of the global behavior of solutions to critical non-linear dispersive equations. The emphasis is on global existence, scattering and finite time blow-up. For solutions that are bounded in the critical norm, but which blow-up in finite time, we also discuss the issue of universal profiles at the blow-up time.     

On the finite time blow-up for filtration problems with nonlinear reaction

We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow-up of solutions subject to sufficiently large initial data provided that the reaction term “overpowers” the nonlinear diffusion in a certain sense. Secondly, under related assumptions on the nonlinearities, we show that initial data above positive stationary state solutions will always lead to finite time blow-up.     

一类带记忆项的非经典热方程的爆破问题

本文考虑了一类带记忆项的非经典热方程,证明解会在有限时间爆破,而且爆破只会发生在边界.主要结论是:首先利用Green函数与Banach压缩映射定理,建立了问题的经典解;其次,利用经典解,证明了解是有限时间爆破的;最后,证明了一个关于非经典热方程解的性质,利用这个性质,证明了解是在边界上爆破的.     

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.     

Blow-up time and boundary layer for solutions in parabolic equations with different diffusion

This paper deals with parabolic equations with different diffusion coefficients and coupled nonlinear sources, subject to homogeneous Dirichlet boundary conditions. We give many results about blow-up solutions, including blow-up time estimates for all of the spatial dimensions, the critical non-simultaneous blow-up exponents, uniform blow-up profiles, blow-up sets, and boundary layer with or without standard conditions on nonlocal sources. The conditions are much weaker than the ones for the corresponding results in the previous papers.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

Uniform blow-up profiles and boundary layer for a parabolic system with localized sources

This paper deals with the Dirichlet problem for a parabolic system with localized sources. We first obtain some sufficient conditions for blow-up in finite time, and then deal with the possibilities of simultaneous blow-up under suitable assumptions. Moreover, when simultaneous blow-up occurs, we also establish the uniform blow-up profiles in the interior and estimate the boundary layer.     

Blow-up solutions of cubic differential systems

We investigate the existence problem for blow-up solutions of cubic differential systems. We find sets of initial values of the blow-up solutions. We also discuss a method of finding upper estimates for the blow-up time of these solutions. Our approach can be applied to systems of partial differential equations. We apply this approach to the Cauchy-Dirichlet problem for systems of semilinear heat equations with cubic nonlinearities.     

On the blow-up of solutions of equations of hydrodynamic type under special boundary conditions

We use the nonlinear capacity method to prove the blow-up of solutions of initial-boundary value problems of hydrodynamic type in bounded domains. We present sufficient boundary conditions ensuring the blow-up of the solution of an equation that is globally solvable under the classical boundary conditions. We estimate the blow-up time of solutions under given initial conditions. Note that it is the first result concerning blow-up for one of the problems considered.     

Local solvability and a priori estimates for classical solutions to an equation of Benjamin-Bona-Mahony-Bürgers type

We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary-value problems on the half-line for a nonlinear equation similar to Benjamin-Bona-Mahony-Bürgers-type equation. We also derive an a priori estimate that implies sufficient blow-up conditions for the second boundary-value problem. We obtain analytically an upper bound of the blow-up time and refine it numerically using Richardson effective accuracy order technique.     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.