基于交叉变分的非线性Klein—Gordon方程解的整体存在和爆破

研究非线性Klein-Gordon方程的初边值问题,运用位势井方法,在E(0)d的情况得到了方程解的整体存在和爆破.在临界能量状态得到了整体解的存在性与不存在性.最后使用凸性方法,得到某些具有高初始能量解的爆破.     

耗散Boussinesq方程弱解的爆破

本文主要研究一类耗散Boussinesq方程的初边值问题的弱解的有限时间爆破.我们主要研究了当初值落在位势井内时,弱解在有限时间爆破的充分必要条件,并给出爆破时间的下界估计.本文是对WANG和SU(2016)的文章的一个补充.     

位势井及其对具临界初始能量的非牛顿渗流方程的应用

该文研究具临界初始能量的非牛顿渗流方程的初边值问题.通过构建新的位势井及其井外集合并证明它们的不变性,得到了解的整体存在性与有限时间爆破.     

一类非线性Klein-Gordon方程解的整体存在和爆破的条件

本文研究了一类非线性Klein-Gordon方程的初边值问题.通过引进能量泛函和与之对应的势井,采用Galerkin方法得到了解整体存在和爆破的充分条件,并给出了解爆破时生命跨度的上界估计.     

一类非线性四阶波动方程初边值问题解的有限时间爆破

研究四阶带有阻尼项的非线性波动方程的解的初边值问题,利用位势井方法,证明了当初值满足一定条件时解发生爆破.将有关该系统爆破性质的研究结果一般化,通过证明得到了该系统较好的性质.     

具有阻尼项的非线性KG方程的柯西问题

首先,介绍位势井及位势井族,并由定义证明其性质;其次,证明整体解的不变性;最后得到了整体解的存在定理,并得到了有限时间内解的爆破.     

一类强阻尼波方程解的存在性和爆破性

讨论一类强阻尼波方程解的局部存在性,并利用势井理论研究解的整体存在性和爆破性.     

非线性Schr\"{o}dinger 方程解的整体存在和爆破的门槛结果

研究了非线性 Schr\"{o}dinger 方程的柯西问题. 通过引进位势井及其外部集合, 得到了解的整体存在性和爆破的门槛结果.     

一类具一般非线性源项的波动方程解的Blow-up

利用位势井理论和凸性方法研究了具有一般非线性源项的波动方程的解,证明了初边值问题的解在有限时间爆破.     

带双调和记忆项的四阶非线性伪抛物方程解的整体存在性和不存在性（英文）

本文致力于带双调和记忆项的四阶非线性伪抛物方程初边值问题的研究.通过应用伽辽金方法、势井理论和相关估计,推导出了整体弱解的存在性.此外,通过应用凹性方法、势井理论和不稳定集的定义,不仅得到了具有非正初始能量(E(0)≤0)的弱解在有限时间爆破的结果,而且得到了具有正初始能量(0E(0)d_θ)的弱解在有限时间爆破的结果.     

Dispersive blow-up for solutions of the Zakharov-Kuznetsov equation

The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that solutions of the linear problem present this kind of singularities. Then we show that the corresponding solutions of the nonlinear problem present dispersive blow-up inherited from the linear component part of the equation. Similar results are obtained for the generalized Zakharov-Kuznetsov equation.     

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.     

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.     

Resolving singularities with Cartan’s prolongation

A standard method for resolving a plane curve singularity is the method of blow-up. We describe a less-known alternative method which we call prolongation, in honor of Cartan’s work in this direction. This method is known to algebraic geometers as Nash blow-up. With each application of prolongation the dimension of the ambient space containing the new “prolonged” singularity increases by one. The new singularity is tangent to a canonical plane field on the ambient space. Our main result asserts that the two methods, blow-up and prolongation, yield the same resolution for unibranched singularities. The primary difficulties encountered are around understanding the prolongation analogues of the exceptional divisors from blow-up. These analogues are called critical curves. Most of the critical curves are abnormal extremals in the sense of optimal control theory as it applies to rank 2 distributions (2 controls). Dedicated to V. I. Arnol’d and his creative force     

Multiple blow-up rates in a coupled heat system with mixed-type nonlinearities

This paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed-type nonlinearities. At first, the critical exponent is obtained, and simply described via a characteristic algebraic system introduced by us. Then, as the main results of the paper, three blow-up rates are established under different dominations of nonlinearities for the one-dimensional case, and represented in another characteristic algebraic system. In particular, it is observed that unlike those in previous literature on parabolic models with absorptions, two of the multiple blow-up rates obtained here do depend on the absorption exponents. In the known works, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates. To our knowledge, this is the first example of absorption-dependent blow-up rates, exploiting the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system. It is also proved that the blow-up of solutions in the model occurs on the boundary only.     

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.     

Blow-up phenomena for a class of fourth order parabolic equation

A class of fourth order parabolic equation is studied in this paper. Some related blow-up results are obtained by applying the potential well theory, the concavity method and a series of differential-integral inequality techniques. More precisely, under some proper assumptions, the upper and lower bounds of the blow-up time and the growth rate for blow-up solutions are estimated. Moreover, a new blow-up condition independent of the depth of the potential well is found. These results complement the recent results obtained in Han (2018).     

Properties of non-simultaneous blow-up solutions in nonlocal parabolic equations

This paper deals with blow-up solutions in parabolic equations coupled via nonlocal nonlinearities, subject to homogeneous Dirichlet conditions. Firstly, some criteria on non-simultaneous and simultaneous blow-up are given, including four kinds of phenomena: (i) the existence of non-simultaneous blow-up; (ii) the coexistence of non-simultaneous and simultaneous blow-up; (iii) any blow-up must be simultaneous; (iv) any blow-up must be non-simultaneous. Next, total versus single point blow-up are classified completely. Moreover, blow-up rates are obtained for both non-simultaneous and simultaneous blow-up solutions.     

On the Well-Posedness for Stochastic Schr(o)dinger Equations with Quadratic Potential

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schr(o)dinger equations.The local and global well-posedness are proved with values in the space ∑(Rn) =｛f ∈ H1(Rn),｜ · ｜f ∈ L2(Rn)}.When the nonlinearity is focusing and L2-supercritical,the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential.Especially for the repulsive case,the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time.Thus,compared with the deterministic equation for the repulsive case,the blow-up condition is stronger on average,and depends on the regularity of the noise.If φ =0,our results coincide with the ones for the deterministic equation.     

Asymptotic analysis of a coupled nonlinear parabolic system

This paper deals with asymptotic analysis of a parabolic system with inner absorptions and coupled nonlinear boundary fluxes. Three simultaneous blow-up rates are established under different dominations of nonlinearities, and simply represented in a characteristic algebraic system introduced for the problem. In particular, it is observed that two of the multiple blow-up rates are absorption-related. This is substantially different from those for nonlinear parabolic problems with absorptions in all the previous literature, where the blow-up rates were known as absorptionindependent. The results of the paper rely on the scaling method with a complete classification for the nonlinear parameters of the model. The first example of absorption-related blow-up rates was recently proposed by the authors for a coupled parabolic system with mixed type nonlinearities. The present paper shows that the newly observed phenomena of absorptionrelated blow-up rates should be due to the coupling mechanism, rather than the mixed type nonlinearities.