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blowup rate of positive solution of a semilinear parabolic system

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Chinese Science Bulletin 1998年第2期43卷 101-103页

作者： Zhigui Lin Chunhong Xie Nanjing Univ Dept Math Nanjing 210093 Peoples R China

It is dealt with positive solutions to the systems: u-t-u- XX =v+p, v-t-v XX =u+q in Ω×(0, T) subject to either initial conditions or boundary_value conditions. The exact blowup rate estimates of solution are establ... 详细信息

关键词： parabolic system Dirichlet problem blowup rate.

ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING:GLOBAL EXISTENCE,DECAY AND blowup OF THE SOLUTION

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Acta Mathematica Scientia 2013年第1期33卷 84-106页

作者： Abderrahmane ZARA Nasser-eddine TATAR Salem ABDELMALEK Depatment of Mathematics and Informatic Cheikh El Arbi Tébessi University12002 TébessaAlgeria Depatment of Mathematics and Statistics King Fahd University of Petroleum and MineralsDhahran 31261Saudi Arabia Depatment of Mathematics College of SciencesYanbuTaibah UniversitySaudi Arabia

In this paper we consider the Elastic membrane equation：with memory term and nonlinear boundary damping： Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions ：and a general decay for the energy are established using the multiplier technique. Also, ＇we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.

关键词： elastic membrane equation global existence boundary damping boundarysource general decay blowup

Analytical blowup Solutions to the Compressible Euler Equations with Time-depending Damping

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Acta Mathematicae Applicatae Sinica 2022年第3期38卷 568-578页

作者： Jian-wei DONG Guang-pu LOU Qiao ZHANG Zhengzhou University of Aeronautics Zhengzhou 450015China

In this paper,the analytical blowup solutions of the N-dimensional radial symmetric compressible Euler equations are *** previous results of the blowup solutions for the compressible Euler equations with constant damping are generalized to the time-depending damping *** generalization is untrivial because that the damp coefficient is a nonlinear function of time t.

关键词： compressible Euler equations time-depending damping blowup

Smooth Solution of Multi-dimensional Nonhomogeneous Conservation Law: Its Formula, and Necessary and Sufficient blowup Criterion

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Acta Mathematicae Applicatae Sinica 2023年第1期39卷 17-27页

作者： Gao-wei CAO Hui KAN Wei XIANG Xiao-zhou YANG Wuhan Institute of Physics and Mathematics Chinese Academy of SciencesInnovation Academy for Precision Measurement Science and TechnologyChinese Academy of SciencesWuhan 430071China City University of Hong Kong Hong KongChina

In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.

关键词： global smooth solution blowup multi-dimensional conservation law solution formula

Asymptotic behavior and blowup for two generalized Ginzburg-Landau type equations with several nonlinear source terms

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International Journal of Biomathematics 2017年第2期10卷 235-247页

作者： Shoubo Jin Zufeng Zhang Qun Zhang School of Mathematics and Statistics Suzhou University Suzhou 233000 Anhui P. R. China

In this paper, the long-time behavior and blowup of solutions for two generalized Ginzburg-Landau type equations with several nonlinear source terms are investigated. First, the conditions in which the solutions of the two equations are positive are ana- lyzed by using Green＇s function. According to the characteristics of nonlinear terms, four different functionals are constructed for solving the problems. Finally, we obtain the long-time behavior and finite-time blowup of solutions for the initial and boundary value problem by using these functionals.

关键词： Asymptotic behavior blowup Ginzburg-Landau type equations.

UNIFORM blowup PROFILES FOR DIFFUSION EQUATIONS WITH NONLOCAL SOURCE AND NONLOCAL BOUNDARY1

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Acta Mathematica Scientia 2004年第3期24卷 443-450页

作者： LinZhigui LiuYurong SchoolofMathematicalScience YangzhouUniversityYangzhou225002China

Long time behavior of solutions to semilinear parabolic equations with nonlo-cal nonlinear source ut-Δu=∫Ωg(u)dx in Ω×(0, T) and with nonlocal boundary con-dition u(x, t)=∫Ωf(x, y)u(y,t)dy on эΩ×(0, T) is studied. The authors establish local existence, global existence and nonexistence of solutions and discuss the blowup properties of solutions. Moveover, they derive the uniform blowup estimates for g(s)= s^p(p＞1) and g(s)=e^э under the assumption ∫Ωf(x,y)dy＜1 for x∈эΩ.

关键词： blowup nonlocal source nonlocal boundary condition 35K50 35B40

Nonexistence of Type Ⅱ blowup for Heat Equation with Exponential Nonlinearity

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Chinese Annals of Mathematics,Series B 2019年第2期40卷 309-320页

作者： Ruihong JI Shan LI Hui CHEN Geomathematics Key Laboratory of Sichuan Province Chengdu University of Technology Business School Sichuan University

This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = ?u + e^u in R^N. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9,and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen(2008) in a finite ball to the whole space.

关键词： Nonlinear heat equation Type II blowup Exponential nonlinearity

A Numerical Study of blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

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Numerical Mathematics(Theory,Methods and Applications) 2013年第2期6卷 364-383页

作者： Ronald D.Haynes Weizhang Huang Paul A.Zegeling Department of Mathematics and Statistics Memorial University of NewfoundlandSt.John’sNL A1C 5S7Canada Department of Mathematics University of KansasLawrenceKS 66045USA Department of Mathematics Utrecht UniversityUtrechtThe Netherlands

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method.A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic ***,the moving mesh method has finite time blowup when the underlying continuous problem *** situations where the continuous problem has infinite time blowup,the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points *** inadequacy of a uniform mesh solution is clearly demonstrated.

关键词： Heat flow harmonic map blowup moving mesh method finite difference

Moving collocation methods for time fractional differential equations and simulation of blowup

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Science China Mathematics 2011年第3期54卷 611-622页

作者： MA JingTang1 & JIANG YingJun2 1School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China 2Department of Mathematics and Scientific Computing, Changsha University of Science and Technology, Changsha 410076, China School of Economic Mathematics Southwestern University of Finance and Economics Chengdu China Department of Mathematics and Scientific Computing Changsha University of Science and Technology Changsha China

A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.

关键词： moving collocation methods time fractional differential equations blowup