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Blow-up of Solution for a Nonlinear Hyperbolic Equation
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作者 WANG Yan-ping 《Chinese Quarterly Journal of Mathematics》 CSCD 2010年第4期578-581,共4页
This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local so... This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local solution of the problem. 展开更多
关键词 nonlinear hyperbolic equation blow-up of solution initial boundary valueproblem
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Blow-up of Solution for a Nonlinear Hyperbolic Equation
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作者 陈翔英 刘维先 《Chinese Quarterly Journal of Mathematics》 CSCD 2002年第2期106-110,共5页
In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficien... In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given. 展开更多
关键词 nonlinear hyperbolic equation initial boundary value problem local generalized solution blow_up of solution
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Blow-up of solution for a generalized Boussinesq equation 被引量:1
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作者 王艳萍 郭柏灵 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2007年第11期1437-1443,共7页
This paper studies the initial boundary value problem for a generalized Boussinese equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method. Moreov... This paper studies the initial boundary value problem for a generalized Boussinese equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method. Moreover, it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method. 展开更多
关键词 Boussinesq equation initial boundary value problem local solution blow-up of the solution
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Global Existence and Blow-up of Solutions to Multi-dimensional (n ≤ 5) Viscous Cahn-Hilliard Equation 被引量:5
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作者 黄锐 尹景学 +1 位作者 李颖花 王春朋 《Northeastern Mathematical Journal》 CSCD 2005年第3期371-378,共8页
In this paper we consider the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5, and established global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initia... In this paper we consider the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5, and established global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial data. 展开更多
关键词 viscous Cahn-Hilliard equation EXISTENCE blow-up
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On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation 被引量:1
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作者 Guoguang Lin Yunlong Gao Yuting Sun 《International Journal of Modern Nonlinear Theory and Application》 2017年第1期11-25,共15页
In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solutio... In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy. 展开更多
关键词 Nonlinear HIGHER-ORDER KIRCHHOFF TYPE Equation STRONG Damping Local solutions blow-up Initial Energy
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GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO A NONLOCAL EVOLUTION p-LAPLACE SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS 被引量:1
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作者 吴学凇 高文杰 曹建文 《Acta Mathematica Scientia》 SCIE CSCD 2011年第3期1001-1010,共10页
In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local exist... In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution. 展开更多
关键词 evolution p-Laplacian nonlinear boundary value problem nonlinear sources global existence blow-up
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GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS 被引量:1
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作者 徐润章 刘亚成 《Acta Mathematica Scientia》 SCIE CSCD 2010年第5期1793-1807,共15页
In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f(u)xx,x∈Ω,t〉0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t≥0, where Ω... In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f(u)xx,x∈Ω,t〉0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t≥0, where Ω = (0, 1). First, we obtain the existence of local Wkp solutions. Then, we prove that, if f(s) ∈ Ck+1(R) is nondecreasing, f(0) = 0 and |f(u)| ≤ C1|u|∫0uf(s)ds + C2, u0(x),u1(x) ∈ WkP(Ω) ∩ W01,P(Ω), k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u(x, t) ∈ W2,∞(0, T; WkP(Ω) ∩ W01,P(Ω)). Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed. 展开更多
关键词 Pochhammer-Chree equations initial boundary value problem Wkp solu-tion global existence blow-up
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Blow-Up of Solution to Cauchy Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation
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作者 Changming Song Li Chen 《Journal of Applied Mathematics and Physics》 2015年第7期834-838,共5页
We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond num... We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given. 展开更多
关键词 Singularly Perturbed Sixth-Order BOUSSINESQ EQUATION CAUCHY Problem blow-up of solution
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Global existence and blow-up of solutions to reaction-diffusion system with a weighted nonlocal source
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作者 蒋良军 王悦生 《Journal of Shanghai University(English Edition)》 CAS 2011年第6期501-505,共5页
In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists glob... In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists globally and blows up in finite time respectively,and then obtain the uniform blow-up rate in the interior. 展开更多
关键词 reaction-diffusion system nonlocal source uniform blow-up profile weight function simultaneous blow-up
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Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source
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作者 SUN PENG GAO WEN-JIE HAN YU-ZHU 《Communications in Mathematical Research》 CSCD 2012年第3期243-251,共9页
This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source. Conditions for the existence of global or blow-up solutions... This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, we prove that the unique solution has global blow-up property whenever blow-up occurs. 展开更多
关键词 blow-up moving localized source nonlocal boundary condition global blow-up
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Existence and Blow-up of Solutions for a Nonlinear Parabolic System with Variable Exponents
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作者 Gao Yun-zhu Gao Wen-jie 《Communications in Mathematical Research》 CSCD 2013年第1期61-67,共7页
In this paper, we study a nonlinear parabolic system with variable exponents. The existence of classical solutions to an initial and boundary value problem is obtained by a fixed point theorem of the contraction mappi... In this paper, we study a nonlinear parabolic system with variable exponents. The existence of classical solutions to an initial and boundary value problem is obtained by a fixed point theorem of the contraction mapping, and the blow-up property of solutions in finite time is obtained with the help of the eigenfunction of the Laplace equation and a delicated estimate. 展开更多
关键词 blow-up EXISTENCE nonlinear parabolic system variable exponent
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ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS
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作者 Hua CHEN Nian LIU 《Acta Mathematica Scientia》 SCIE CSCD 2021年第1期257-282,共26页
In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner t... In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family. 展开更多
关键词 coupled parabolic equations totally characteristic degeneracy singular potentials asymptotic stability blow-up
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Extinction and Non-simultaneous Blow-up of Solutions in Fast Diffusion Equations
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作者 LIU Bing-chen WANG Yu-xi WANG Lu 《Chinese Quarterly Journal of Mathematics》 2020年第2期199-213,共15页
In this paper,we deal with some fast di usion equations ut=um+au vp and vt=△v^n+bu^qv^βsubject to null Dirichlet boundary conditions.We prove that every solution vanishes in nite time for pq>(m-α)(n-β),m>αa... In this paper,we deal with some fast di usion equations ut=um+au vp and vt=△v^n+bu^qv^βsubject to null Dirichlet boundary conditions.We prove that every solution vanishes in nite time for pq>(m-α)(n-β),m>αand n>β,where the exact relation of initial data is determined with the aid of some Sobolev Embedding inequalities.If pd<(m-α)(n-β),m>αand n>β,we show the barriers of the initial data which lead to the non-extinction of solutions.For the case pq=(m-α)(n-β),the solutions vanish for small initial data.The results fill in a gap for the case pq<mn in Nonlinear Anal.Real World Appl.4(2013)1931-1937.The coecients a and b play a vital role in the existence of non-extinction weak solution provided that a and b are large enough.At last,we use the scaling methods to determine some exponent regions where one of the components would blow up alone for some suitable initial data. 展开更多
关键词 Fast di usion EXTINCTION Non-simultaneous blow-up
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Blow-up of Solution for the Nonlinear Sobolev-Galpern Equation
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作者 张恩彪 江成顺 《Chinese Quarterly Journal of Mathematics》 CSCD 2009年第3期432-438,共7页
In this paper,the initial boundary value problems of the nonlinear Sobolev-Galpern equation are studied.The existence,uniqueness of local solution for the problem are obtained by means of a special Green's functio... In this paper,the initial boundary value problems of the nonlinear Sobolev-Galpern equation are studied.The existence,uniqueness of local solution for the problem are obtained by means of a special Green's function and the contraction mapping principle.Finally,the blow-up of solution in finite time under some assumed conditions is proved with the aid of Jensen's inequality. 展开更多
关键词 nonlinear Sobolev-Galpern equations Green's function blow-up Jensen's inequality
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Blow-up of Solutions of Heat Flows for Harmonic Maps 被引量:4
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作者 丁伟岳 《数学进展》 CSCD 北大核心 1990年第1期80-92,共13页
Let (M,g) and (N,h)be two Riemannian manifolds. Consider the heat flow for harmonic maps from (M,g) into (N,h).We prove the following results Suppose dim M =3 and is a nontrivial homotopy class in C(M,N).Then there ex... Let (M,g) and (N,h)be two Riemannian manifolds. Consider the heat flow for harmonic maps from (M,g) into (N,h).We prove the following results Suppose dim M =3 and is a nontrivial homotopy class in C(M,N).Then there exists a constant ?o such that if and E(u0)<e, the solution of the heat flow with initial value u0 must blow up in finite time. We also present a sufficient condition which ensures that any global solutions subconverge to harmonic maps as t→∞. 展开更多
关键词 blow-up HARMONIC 映射 黎曼流形 热流动
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GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY 被引量:1
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作者 丁时进 李颖花 王喻 《Acta Mathematica Scientia》 SCIE CSCD 2024年第1期195-214,共20页
This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depen... This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth. 展开更多
关键词 Navier-Stokes/Allen-Cahn system density-dependent viscosity free boundary global solutions
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Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation
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作者 巩乾坤 王惠 王云虎 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第4期409-416,共8页
This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional ... This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs. 展开更多
关键词 lump solution rogue wave solution breather wave solution (2+1)-dimensional Hirota-Satsuma-Ito equation
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PDE Standardization Analysis and Solution of TypicalMechanics Problems
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作者 Ningjie Wang Yihao Wang +1 位作者 Yongle Pei Luxian Li 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第10期171-186,共16页
A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach throu... A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach through the standard forms of the PDE module in COMSOL.Two typical mechanics problems are exemplified:The deflection of a thin plate,which can be addressed with the dedicated finite element module,and the stress of a pure bending beamthat cannot be tackled.The procedure for the two problems regarding the three standard forms required by the PDE module is detailed.The results were in good agreement with the literature,indicating that the PDE module provides a promising means to solve complex PDEs,especially for those a dedicated finite element module has yet to be developed. 展开更多
关键词 Three standard forms expression input PDE module deflection solution stress solution
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THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE
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作者 金可 石影 谢华飞 《Acta Mathematica Scientia》 SCIE CSCD 2024年第2期583-608,共26页
In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are... In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems. 展开更多
关键词 Schrödinger system ground states solutions bound state solutions
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Solution approximations for a mathematical model of relativistic electrons with beta derivative
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作者 Ibrahim Yalcinkaya Orkun Tasbozan +1 位作者 Ali Kurt Hijaz Ahmad 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第3期469-485,共17页
The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with... The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative.For this purpose,both the modified extended tanh-function(mETF)method and the homotopy analysis method(HAM)are used.While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution.The advantage of this procedure is that discretization or normalization is not required.By applying the mETF,the exact solutions are obtained.Also,by applying the HAM semi-analytical results for the considered equation are acquired.In HAM?curve gives us a chance to find the suitable value of the for the convergence of the solution series.Also,comparative graphical representations are given to show the effectiveness,reliability of the methods.The results show that the m ETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative.This study can bring a new perspective for studies on fractional differential equations.On the other hand,it can be said that scientists can apply the considered methods for different mathematical models arising in physics,chemistry,engineering,social sciences and etc.which involves fractional differentiation.Briefly the results may cause a new insight who studies on relativistic electron modelling. 展开更多
关键词 Klein-Fock-Gordon(KFG)equation beta derivative semi-analytical solution exact solution
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