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.     

Well-posedness and blow-up phenomena for the generalized Degasperis-Procesi equation

In this paper, we mainly study the Cauchy problem of the generalized Degasperis-Procesi equation. We establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which blow up in finite time.     

具有梯度源和非局部源的反应扩散方程解的爆破时刻下界北大核心CSCD

对于反应扩散方程解的爆破时刻研究,不仅具有理论意义,而且与安全地控制生产,控制种群密度以及环境趋化治理等实际问题密切相关.该文考虑了一类具有梯度源和非局部源的反应扩散方程解的爆破时刻下界.首先,假设区域为高维空间中的具有光滑边界的有界凸区域;其次,通过构造合适的辅助函数,利用一阶微分不等式技术和Sobolev不等式,得出解在有限时刻发生爆破时的爆破时刻下界;最后,通过两个应用实例来解释说明文中所获得的抽象结论.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

Blow-up of smooth solutions to the compressible magnetohydrodynamic flows

In this paper, we prove the blow-up phenomena of smooth solutions to the Cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data. In particular, our results do not require that the initial data have compact support or contain vacuum in any finite region.     

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.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

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.     

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.     

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.     

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].     

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.     

Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier–Stokes equations and the classical Poisson–Nernst–Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi–Serrin type blow-up criteria are also established.     

Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition

In this paper we investigates the blow-up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition, we obtain the blow-up condition and its blow-up rate estimate.     

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.     

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.”     

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.     

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).     

Uniform blow-up rate for diffusion equations with localized nonlinear source

In this short paper, we investigate blow-up rate of solutions of reaction–diffusion equations with localized reactions. We prove that the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.