Compactness of solutions to the Yamabe problem. III

For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates would imply the compactness of the set of solutions of the Yamabe equation on such manifolds.     

Global solutions and finite time blow up for damped Klein-Gordon equation

We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.     

Blow–Up Profiles in One–Dimensional. Semilinear Parabolic Problems

Partially supporte by CICYT Research Grant PB86–0112–00202 and EEC Contrast SCI–0019–c.

We consider the Cauchy Problem where p > 1 and U$sub:o$esub:(x) is continuous, nonnegative and bounded. Let u(x.t) be the solution of (1). (2). and assume that u blows up at t=T . and We then show that the blow–up set is discrete. Moreover, if x=0 is a blow–up point, one of the two following possibilities occurs. Either There exist c > 0 and an even number m such that     

有非局部源退化抛物方程组解的爆破

本文讨论一类具有非局部源退化抛物方程组．通过利用上下解方法得到解的全局存在和有限时刻爆破，给出爆破集是整个区域，而且得到了解的爆破率．     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

On the blow‐up phenomena for a 1‐dimensional equation of ion sound waves in a plasma: Analytical and numerical investigation

The initial‐boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately.     

Numerical Study of Blow‐Up Mechanisms for Davey–Stewartson II Systems

We present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and that the time‐dependent scaling behaves as in the Ozawa solution and not as in the stable blow‐up of standard L ² critical nonlinear Schrödinger equation. The blow‐up profile is given by a dynamically rescaled lump.     

Blow‐up profiles of solutions for the exponential reaction‐diffusion equation

We consider the blow‐up of solutions for a semilinear reaction‐diffusion equation with exponential reaction term. It is known that certain solutions that can be continued beyond the blow‐up time possess a non‐constant self‐similar blow‐up profile. Our aim is to find the final time blow‐up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.     

Global solutions and blow‐up problems to a porous medium system with nonlocal sources and nonlocal boundary conditions

This paper deals with a porous medium system with nonlocal sources and weighted nonlocal boundary conditions. The main aim of this paper is to study how the reaction terms, the diffusion terms, and the weight functions in the boundary conditions affect the global and blow‐up properties to a porous medium system. The conditions on the global existence and blow‐up in finite time for nonnegative solutions are given. Furthermore, the blow‐up rate estimates of the blow‐up solutions are also established. Copyright © 2011 John Wiley & Sons, Ltd.     

形变张量的特征值与Boussinesq方程组的正则性估计

讨论了二维及三维满足周期边界条件的Boussinesq方程初边值问题的局部正则解在有限时间内爆破的可能性.在二维情况下,用形变张量的特征值给出温度梯度的L2估计,从中看出若流体微团变形的速率大,则解爆破的可能性就大.在三维情况下,用形变张量的特征值和温度的偏导给出涡量的L2估计,从中发现若流体微团在大部分时间内一般是平面拉伸,且温度的偏导较小时,解爆破的可能性就大;若一般是线性拉伸,温度的偏导又不任意增大时,解爆破的可能性就小.     

Blow‐up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions

This paper deals with the blow‐up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow‐up occurs at some finite time. We also obtain upper and lower bounds for the blow‐up time when blow‐up occurs. Copyright © 2014 John Wiley & Sons, Ltd.     

非局部反应扩散方程的临界爆破指标

本文证明了一类来源于燃烧理论的非局部反应扩散方程的临界爆破指标的存在性，而且临界指标属于爆破情形     

Blow-up of optimal sets in the irrigation problem

We consider the minimization problem for an average distance functional in the plane, among all compact connected sets of prescribed length. For a minimizing set, the blow-up sequence in the neighborhood of any point is investigated. We show existence of the blow up limits and we characterize them, using the results to get some partial regularity statements.     

Numerical blow‐up for the porous medium equation with a source

We study numerical approximations of positive solutions of the porous medium equation with a nonlinear source, where m > 1, p > 0 and L > 0 are parameters. We describe in terms of p, m, and L when solutions of a semidiscretization in space exist globally in time and when they blow up in a finite time. We also find the blow‐up rates and the blow‐up sets, proving that there is no regional blow‐up for the numerical scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Global existence and bounds for blow‐up time in a class of nonlinear pseudo‐parabolic equations with a memory term

In this article, we consider the initial boundary value problem for a class of nonlinear pseudo‐parabolic equations with a memory term: Under suitable assumptions, we obtain the local and global existence of the solution by Galerkin method. We prove finite‐time blow‐up of the solution for initial data at arbitrary energy level and obtain upper bounds for blow‐up time by using the concavity method. In addition, by means of differential inequality technique, we obtain a lower bound for blow‐up time of the solution if blow‐up occurs.     

Blow‐up of smooth solutions to the Navier–Stokes–Poisson equations

The main purpose of this paper is concerned with blow‐up smooth solutions to Navier–Stokes–Poisson (N‐S‐P) equations. First, we present a sufficient condition on the blow up of smooth solutions to the N‐S‐P system. Then we construct a family of analytical solutions that blow up in finite time. Copyright © 2010 John Wiley & Sons, Ltd.     

Blow‐up phenomena in chemotaxis systems with a source term

This paper deals with a parabolic–parabolic Keller–Segel‐type system in a bounded domain of , {N = 2;3}, under different boundary conditions, with time‐dependent coefficients and a positive source term. The solutions may blow up in finite time t^?; and under appropriate assumptions on data, explicit lower bounds for blow‐up time are obtained when blow up occurs. Copyright © 2015 John Wiley & Sons, Ltd.     

Non-simultaneous blow up for coupled nonlinear diffusion equations with absorptions

This paper deals with coupled nonlinear diffusion equations with absorptions. We characterize the range of parameters for which non-simultaneous blow up occurs. We establish the necessary and sufficient conditions for the occurrence of non-simultaneous blow up with proper initial data. Moreover, we obtain the optimal condition under which any blow up is non-simultaneous.     

Blow‐up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis

The initial‐boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

非线性边界条件下具有变系数的热量方程解的存在性及爆破现象

考虑了定义在Ω上的有变系数的热量方程,其中Ω∝RN(N≥2)是一个有界的凸区域,并且方程具有非线性边界条件.利用微分不等式技术,首先推导了爆破一定发生的条件,并确定了爆破时间的上界.同时,通过对非线性项做一定的限制,得到了解的全局存在性.当爆破发生时,确定了爆破时间的下界.