关于一类拟线性拟抛物方程解的Blow─up问题

本文利用Ｆｏｕｒｉｅｒ空间的比较原理研究一类拟线性拟抛物方程解的Ｂｌｏｗ－ｕｐ问题，并给出了其解在有限时刻Ｂｌｏｗ－ｕｐ的条件。     

理想磁流体动力学方程组经典解的破裂

本文给出了理想磁流体动力学方程组的经典解在初始扰动适当大的情况下破裂的结果.文[1]证明了描述多方理想可压缩气体运动的欧拉系统的经典解在初始扰动适当大的情况下破裂的结果.本文将利用和文[1]相似的方法证明所得定理.     

Semidiscretization in Space of Nonlinear Degenerate Parabolic Equations with Blow-up of the Solutions

Semidiscretization in space of nonlinear degenerate parabolic equations of nondivergent form is presented, under zero Dirichlet boundary condition. It is shown that semidiscrete solutions blow up in finite time. In particular, the asymptotic behavior of blowing-up solutions, is discussed precisely.     

带梯度项的双重退缩抛物方程正解的Blow-up

研究了RN中一般区域上的一族带非线性梯度项的双重退缩抛物方程解的Blow-up性质.通过构造适当的辅助函数,利用特征函数法和不等式技巧,给出了其齐次Dirichlet边值问题的正解产生Blow-up的充分条件:利用能量方法,证明了其Cauchy问题非平凡整体解的不存在性.方法也适用于研究其它带非线性源的退缩非线性抛物方程解的Blow-up问题.     

一类非线性发展方程初边值问题解的Blow—up

本文利用Fourier变换方法,研究了一类非线性拟双曲方程的初边值问题的解的bolw-up问题,并给出了其解在有限时间内bolw-up的条件。     

Blow-up for the Timoshenko-type equation with variable exponents

The finite time blow-up of solutions to a nonlinear Timoshenko-type equation with variable exponents is studied. More concretely, we prove that the solutions blow up in finite time with positive initial energy. Also, the existence of finite time blow-up solutions with arbitrarily high initial energy is established. Meanwhile, the upper and lower bounds of the blow-up time are derived. These results deepen and generalize the ones obtained in [Nonlinear Anal. Real World Appl., 61: Paper No. 103341, 2021].     

Blow-up phenomena for a family of Burgers-like equations

By introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis-Procesi equation. On the line R, we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L^∞-norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis-Procesi equation, see Remark 4.1.     

On the Cauchy problem for the periodic generalized Degasperis-Procesi equation

We mainly study the Cauchy problem of the periodic generalized Degasperis-Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.     

一类弱耗散双组份Hunter-Saxton系统的爆破与爆破率

研究了一类周期弱耗散双组份Hunnter-Saxton系统的爆破现象.首先,给出了此类Hunnter-Saxton系统解的局部适定性及其精确的爆破机制;其次,证明了在一定的初始值下Hunnter-Saxton系统强解的几个爆破结果;最后,给出了HunnterSaxton系统强解的爆破率.     

三维不可压磁流体方程组的显式爆破解

该文构造了三维磁流体方程组的若干分离变量型和自相似型显式爆破解.     

Threshold and generic type I behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|^p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N?3 and . Our first goal is to analyze a threshold behavior for solutions with initial data u₀=λv, where v∈C∩H¹ and v?0, v?0. It is known that there exists λ^?>0 such that the solution converges to 0 as t→∞ if 0<λ<λ^?, while it blows up in finite time if λ?λ^?. We show that there exist at most finitely many exceptional values λ₁=λ^?<λ₂λk such that, for all λ>λ^? with λ≠λj (j=1,2,…,k), the blow-up is complete and of type I with a flat local profile. Our method is based on a combination of the zero-number principle and energy estimates. In the second part of the paper, we employ the very same idea to show that the constant solution κ attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. Using this result, we derive a sharp criterion for no blow-up.     

Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources

This paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions.     

Global weak solutions and blow-up structure for the Degasperis-Procesi equation

In this paper we study several qualitative properties of the Degasperis-Procesi equation. We first established the precise blow-up rate and then determine the blow-up set of blow-up strong solutions to this equation for a large class of initial data. We finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions.     

Blow-up for hyperbolic systems in diagonal form

We prove a blow-up result for classical solutions of the Cauchy problem for a nonlinear hyperbolic system in one space dimension. The initial conditions are periodic and the system is supposed to be in diagonal form. We give an estimate of the lifespan of the classical solutions. Received May 2000     

Blow-up vs. spurious steady solutions

In this paper, we study the blow-up problem for positive solutions of a semidiscretization in space of the heat equation in one space dimension with a nonlinear flux boundary condition and a nonlinear absorption term in the equation. We obtain that, for a certain range of parameters, the continuous problem has blow-up solutions but the semidiscretization does not and the reason for this is that a spurious attractive steady solution appears.

     

A class of degenerate diffusion equations with mixed boundary conditions

This paper is concerned with a class of degenerate diffusion equations subject to mixed boundary conditions. Under some structure conditions, we discuss the blow-up property of local solutions and estimate the bounds of “blow-up time.”     

On blow-up solutions for systems of semilinear heat equations with quadratic nonlinearities

This article deals with blow-up solutions of the Cauchy–Dirichlet problem for system of semilinear heat equations with quadratic non-linearities. Sufficient conditions for the existence of blow-up solutions are established. Sets of initial values for these solutions as well as upper bounds for corresponding blow-up time are determined. Furthermore, an application to the Lotka-Volterra system with diffusion is also discussed. The result of this article may be considered as a continuation and a generalization of the results obtained in (Baris, J., Baris, P. and Ruchlewicz, B., 2006, On blow-up solutions of nonautonomous quadratic differential systems. Differential Equations, 42, 320–326; Baris, J., Baris, P. and Wawiórko, E., 2006, Asymptotic behaviour of solutions of Lotka-Volterra systems. Nonlinear Analysis: Real World Applications, 7, 610–618; Baris, J., Baris, P. and Ruchlewicz, B., 2006, On blow-up solutions of quadratic systems of differential equations. Sovremennaya Matematika. Fundamentalnye Napravleniya, 15, 29–35 (in Russian); Baris, J. and Wawiórko, E., On blow-up solutions of polynomial Kolmogorov systems. Nonlinear Analysis: Real World Applications, to appear).     

Blow-up analysis for a cosmic strings equation

In this paper we develop a blow-up analysis for solutions of an elliptic PDE of Liouville type over the plane. Such solutions describe the behavior of cosmic strings (parallel in a given direction) for a W-boson model coupled with Einstein's equation. We show how the blow-up behavior of the solutions is characterized, according to the physical parameters involved, by new and surprising phenomena. For example in some cases, after a suitable scaling, the blow-up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the “uniformization” of the Riemann sphere with conical singularities.